THERMODYNAMICS AND BIOLOGICAL SYSTEMS

11 THERMODYNAMICS AND BIOLOGICAL SYSTEMS 11.1

INTRODUCTION

Systems may exhibit two different types of behavior: (i) the tendency towards maximum disorder or (ii) the spontaneous appearance of a high degree of organization in space, time, and/or function. The best examples of the latter are dissipative systems at nonequilibrium conditions, such as the B6nard cell, the tricarboxylic acid (TCA) cycle, ecosystems, and living systems. As living systems grow and develop, a constant supply of energy is needed for reproduction and survival in changing conditions. Organized structures require a number of coupled metabolic reactions and transport processes that control the rate and timing of life processes. Schrodinger proposed that these processes appear to be at variance with the second law of thermodynamics, which states that a finite amount of organization may be obtained at the expense of a greater amount of disorganization in a series of interrelated (coupled) spontaneous changes. Biochemical reaction cyclic processes maintain the biological cell in nonequilibrium state by controlling the influx of reactants and efflux of products. Biological systems do not decay towards an equilibrium state, but instead increase in size, developing organized structures and complexity. An evolved and adapted biological system converts energy in an efficient manner for the transport of substances across a cell membrane, the synthesis and assembly of proteins, muscular contraction, reproduction, and survival. The source of energy is adenosine triphosphate (ATP), which is produced by oxidative phosphorylation in the inner membrane of the mitochondria. Kinetic equations and statistical models can describe such processes satisfactorily. However, these procedures often require detailed information, which may be unavailable. The nonequilibrium thermodynamics theory may be a useful approach to describe energy pathways and coupling in a quantitative manner, evaluate the stoichiometry in partially coupled systems, and formulate the efficiency of energy conversion in bioenergetics. Nonequilibrium thermodynamics formulations may provide a new approach of analyzing the results of experimental studies and guide the design of new experimental methods relating to biological energy conversions. The linear nonequilibrium thermodynamics theory is valid for systems close to equilibrium, and does not require detailed information about the mechanisms of biological process, although a complete analysis requires a quantitative description of the mechanisms of energy conversion. This chapter starts with a simplified analysis of biological processes using the basic tools of physics, chemistry, and thermodynamics. It provides a brief description of mitochondria and energy transduction in the mitochondrion. The study of proper pathways and multi-inflection points in bioenergetics are summarized. We also summarize the concept of thermodynamic buffering caused by soluble enzymes and some important processes of bioenergetics using the linear nonequilibrium thermodynamics formulation.

11.2

SIMPLIFIED ANALYSIS IN LIVING SYSTEMS

Living systems consist of many subsystems with characteristic functions and outputs. Communication among these various functions and outputs leads to an organized system that can be maintained by a constant supply of energy and matter from the outside. Therefore, living systems represent nonequilibrium open systems with various thermodynamic forces and flows. The second law of thermodynamics suggests that living systems are capable of creating order through various coupled (interrelated) chemical cycles and transport processes. The relations between the flows and forces can provide quantitative information on the characteristics and level of organization without detailed knowledge about the mechanism of coupling. Many chemical cycles and transport processes are the result of controlled energy conversions, and the principles of thermodynamics can describe the efficiency of energy conversions and the exchange of energy and matter between living systems and the environment.

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Thermodynamicsand biological systems

The analysis of real biological systems may be introduced in idealized simplifications using the principles of physics, chemistry, biology, thermodynamics, and kinetics. The following examples are the simple application of these principles in describing some biological processes.

Example 11.1 Cell electric potentials In living systems, ions in the intracellular phase and the extracellular phase produce a potential difference of about 80 mV between the two phases. The intracellular phase potential is negative (Garby and Larsen, 1995). Determine the difference in electrical potential energy per mole positive monovalent ion, e.g., Na+, between the two phases. Electric potential differences can exist in living systems. Changes in electrical potential energy are produced by ions in solutions whose electric potentials change across cell membranes. The potential energy per mole species i, Epi, at the potential 6 is obtained from Epi =

ziF~

With the valence of the ion z = + 1, and F Faraday constant (F = 96,485 C/mol), we have (EpNa)ext -- (EpNa)int "- z i F ( ~ e x t

- ~tint) --

1(96485)(80 X 10 -3) = 7718.8 J / m o l

This result illustrates one of the nonequilibrium conditions maintained by ion transport.

Example 11.2 Excess pressure in the lungs The lungs have a large surface area (approximately 100 m 2 in adults). During inspiration, the incoming air fills the canals in the bronchial tree that ends with the near-spherical alveoli, across whose walls transport of oxygen and carbon dioxide takes place (Garby and Larsen, 1995). The single alveolus can be considered an elastic membrane covered by a thin film. At equilibrium, between the surface tension crs and the excess pressure AP of the air inside a spherical surface with radius R, we find A P - 2°'s R

(a)

If we assume that O's is constant, then an increasing radius would reduce excess pressure, and the alveoli would collapse when the excess pressure fell below a certain value. This kind of instability is not common, since o-s does not remain constant in a normal lung but increases with increasing radius such that AP always increases with increasing radius. This behavior of O's is caused by a surface-active agent (phospholipids) in the liquid film. The concentration of the surface-active agent decreases in the interface between liquid and air when the surface of the film distends, and therefore both surface tension and excess pressure increase. The hysteresis of this process is controlled by the diffusion of matter between the free interface and interior of the liquid. The surface tension is about 0.06 N/m for water and about 0.05 N/m for blood plasma, and it can vary between 0.002 and 0.04 N/m for the liquid films of alveoli, and the total area changes by a factor of 5. With these data on surface tension and for a maximal area of 100 m 2, the total surface energy (o'sA) varies between the values (o-sA)l = (0.04)(100) = 4 J (o-sA)2 : (0.002)(20): 0.04 J Assuming that the hysteresis over a cycle is 25% of the maximal energy, that the power due to surface tension at a respiration frequency of 12 min -1 becomes

The radii of the --~300 million alveoli vary in the interval R = 0.06--0.15 mm. The excess pressures (Eq. (a)) to distend an alveolus and hence surface tensions can vary between 2(0.04) (~)1

-- O. 15 / 1 0 0 0

= 533Pa

11.2 Simplified analysis in living systems

(AP)2 =

2(0.002) 0.06 / 1000

543

= 66 Pa

These relations show the equality between the work of the excess pressure and the increase in surface energy.

Example 11.3 Enthalpy and work changes of blood due to the pumping work of the heart Calculate the change in enthalpy of blood when subjected to an isothermal increase in pressure of 16 kPa (120 mmHg) (Garby and Larsen, 1995). Assuming that the blood is an incompressible fluid, the density becomes constant, and the second term in the total differential dU= (OU/OS)vdS + (OU/OV)sdVvanishes. From the definition ofenthalpy H= U+ PV, we have dH = CvdT + VdP

(a)

This relation shows that the first term dominates when a liquid is heated, and the second dominates when isothermal pumping takes place. From Eq. (a), assuming the density of water (1000 kg/m 3) is similar to the density of blood, and V= 1/p, we have H 2 - H 1 = V(P2 - P~) = 10- 3 (16000) = 16 J/kg Assuming a pulse rate of 70 beats per minute at rest and a stroke volume of 70 mL, the power is obtained from

/

rn(H 2 - H1) - 0.07 ~

(16) = 1.3 W

This is about 1000 times the power associated with the kinetic energy of the flow. The increase in pressure originates in the heart, and here we calculate the work added to the blood as useful energy. The pump is understood as a system that supplies energy to the fluid in the form of an increase in pressure. So, the blood can circulate through the closed circuit of blood vessels despite the associated pressure drops. The pump power Wp supplied by a muscle can be estimated by th(H 2 -

HI) + AEki n + AEpo t = 0 "+"~"p

Assuming that the process is steady, and kinetic z~kEkin and potential ~ p o t energies are negligible (the same area and velocity in the inflow and outflow), the equation above reduces to rn(H 2 - H, ) = c) + Wp For an isothermal process, we have (b) Therefore, the pump power can be calculated only when the heat flow rate is known. From the general entropy balance equation dS= deS+ diS, we conclude that for an incompressible and isothermal process, we have des = -diS. This relation shows the equality between the dissipated heat flow and internal entropy production and hence the loss of power is c) - ~/'loss.Therefore, Eq. (b) becomes

/

r h ( p (P2-P1)+Eloss =Wp where Elos~is the power loss due to friction and associated internal backflow. If the pump is reversible, ~+loss = 0, and we have

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Thermodynamicsand biological systems

The isothermal mechanical efficiency for this pump becomes

Tip--

/~rp,rev I//Zp

In turn, by using the metabolic energy expenditures EM needed by the muscle to affect the mechanical motion, we can define the metabolic or chemo-mechanical efficiency of the muscle by

T~ch -- /~M

Combining the two efficiencies, and representing the conversion of chemical energy into useful (reversible) energy received by the fluid, we may define the net chemo-mechanical pump efficiency:

T] : 7~p~ch --

l/~rp,rev /~M

For external muscle work, the chemo-mechanical efficiency is about 0.2-0.25.

11.2.1

Biological Fuels

Biological fuels can be categorized in three groups: carbohydrates (CH), representing a mixture of mono-, di-, and polysaccharides, fats (F), and proteins (Pr) (Garby and Larsen, 1995). The fuel value is equal to the negative reaction enthalpy. Carbohydrates and fats can be completely oxidized, while proteins can only be partially oxidized. Therefore, protein has a lower fuel value. The oxidation enthalpies per unit mass differ only slightly within each group. Table 11.1 shows the reaction enthalpies and stoichiometries of biological fuels. The respiratory quotient shows the ratio of carbon dioxide elimination and oxygen uptake associated with oxidation. The last column in Table 11.1 is the thermal energy equivalency of oxygen, which is a measure of the chemical energy that is liberated per mole of oxygen consumed. It is possible to measure only the uptake of oxygen and the rates of elimination of carbon dioxide and nitrogen. The energy expenditure with the reaction enthalpies is

]E : Z(nAHr)i : (hAHr)cH -F (nAHr) F + (nAHr)Pr : (-0) + ( - W )

(11.1)

i

Example 11.4 Energy expenditure in small organisms A small organism has a heat loss o f - q = 1.52 W and performs external work W = 0.02 N m/s. Calculate that part of the total energy expenditure that originates from its internal circulation, which involves the pumping of 120 mL/min of fluid against a pressure drop of 3.34 kPa with a net chemo-mechanical efficiency of 10%. The energy expenditure of the pump is /~ = - q + (-W) = 1.52 + 0.02 = 1.54 W

Table 11.1 Biological fuel parameters Fuel

CH Fat Pr

Specific reaction enthalpy (kJ/g) - 17 -39 -17

CH: carbohydrate; Pr: protein.

Source: Garby and Larsen, (1995)

Specific turnover 02 (mmol/g)

CO2 (g/g)

N (mmol/g)

33.3 90.6 43.3

33.3 63.8 34.43

0.16

Respiratory quotient = n(CO2)/n(02)

Energy equivalency (kJ/mol 02)

1.00 0.70 0.79

511 431 393

11.2 Simplified analysis in living systems

545

The reversible pump power becomes mp,re v

- ( 0"12X10-3 ] (3340) = 0.0067W 60

Therefore, the net chemo-mechanical pump efficiency becomes

Ep l/Vp,rev/0.1 0 . 0 0 6 7 / 0 . 1 ~ / - EM

EM

1.54

= 0.043

Consider an amphipod with a body weight of l0 p~g consuming 4.0 × 10 - 9 mol oxygen every hour at steady state and eliminating 3.6 x 10-9 mol carbon dioxide, 0.4 x 10 -9 mol N (as ammonia), and 0.1 x 10 -9 mol lactic acid. The external work power is 50 × 10 -9 W. The heat loss of the animal may be calculated when the following four net reactions contribute to the energy expenditure" C6H12O 6 + 6 0 2 ~ 6 C O 2 + 6 H 2 0 - 2 8 7 0 k J / m o l C6H1206

--~ 2 C 3 H 6 0 3

- 100 kJ/mol

C55H1040 6 + 780 2 ~ 55CO 2 + 52H20 - 34300 kJ/mol

C32H48010N 8 + 330 2 ~ 32CO 2 + 8NH 3 + 12H20 - 14744 kJ/mol Therefore, the energy expenditure (Garby and Larsen, 1995) from Eq. (11.1) with the given reaction enthalpies is L- = 2870(h)G + 100(rt)G__.La + 34300(n)F + 14744(n)p r where G refers to the combustion of glucose, G ~ La to the metabolism of glucose to lactic acid (anaerobic process), F to the combustion of fat, and Pr to the combustion of protein. The mass conservations are ho~ = 6n G + 78n F + 33fipr = 4 X10 -9 mol/h _

hL~ -- 2n~_~La -- 0.1X 10 - 9 mol/h

rico;

= 6h~ + 55n F + 32hpr = 3.6X10 -9 mol/h

fin -- 8/)/Pr -- 0 . 4

x 10 - 9 m o l / h

From Eq. (11.1) and solving the above equations, we have ~" = [2870(0.194)~ + 100(0.05)~ ~La + 34300(0.0152)F + 14744(0.05)p r ] X 10 - 9 = 1820 X 10 -9 kJ/h = 0.506 I~W Therefore, the heat loss is ( - q ) = L ' - ( - / ~ ) - 0 . 5 0 6 - 0 . 0 5 0 : 0.461~W

Example 11.5 Energy expenditure in an adult organism An adult organism has an oxygen uptake of about 21.16mol over 24h, and the associated elimination of carbon dioxide and nitrogen is 16.95mol and 5.76g, respectively (Garby and Larsen, 1995). If the adult has performed 0.9 MJ of external work over the same period and his energy expenditure at rest is L'0 = 65 W, estimate his energy expenditure, heat loss, and net efficiency for the external work.

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Thermodynamicsand biological systems

The energy expenditure may be calculated from the energy balance. Assume that (i) carbohydrate (CH), fat (F), and protein (Pr) are the only compounds involved in the oxidation process; (ii) the other compounds are stationary; and (iii) the uptake and elimination of oxygen, carbon dioxide, and nitrogen is instantaneous. From the first law of thermodynamics, we have /~ = E ( / ; / ~ r ) i i

= ( / ; / ~ r ) C H + (/:/~-/r)F + ( / ; / ~ r ) P r = ( _ q ) + (__~f/r)

From Table 11.1, we get 17hca + 39h F + 17npr = q + W = / ~

(a)

The conservation of mass requires that ho2 = 33.3hcH + 90.6h F + 43.3hpr mmol/s hco 2 = 33.3hcn +63.8h F + 34.4hpr mmol/s

th N -- 0.16/;/pr g/s

From these equations, we obtain

npr

5.76 0.16

- 36 g/day, hcn = 194 g/day, and tiv = 145 g/day

From Eq. (a), we have 17(194)+ 39(145)+ 17(36) = - q + ( - W ) =/~ = 9566 kJ/day = 110.7W Since the work is - 0 . 9 MJ/day, the heat loss is ( - q ) = 9566 - 900 = 8666 kJ/day = 100 W A net efficiency is defined by

rtnet -- E _ E0

900 = 0.22 9 5 6 6 - 5500

where E0 is the energy expenditure during resting and/~ is the energy expenditure during the performing of external work (-W).

Example 11.6 Oxidation of glucose (a) We wish to estimate the reaction enthalpy for the isothermal and isobaric oxidation of glucose at 310 K and 1 atm. (b) Estimate the energy expenditure for oxidation of 390 g/day glucose at steady conditions. C6H120 6 (aq) + 60 2 (g) ~ 6CO 2 (g) + 6H 2O(1) If the control volume is a piece of tissue, the reaction above may take place in an aqueous solution (aq). Consider an aqueous solution of 0.01 mol/L glucose; the partial pressure of carbon dioxide and oxygen are 0.07 and 0.21 atm, respectively. From Table BS, we obtain the enthalpy of formations for the components of the reaction above at the standard state (298 K and 1 atm) AH ° = 6 ( - 393) + 6(-286) - (1264) - 6(0) = - 2 8 1 0 kJ/mol

547

11.2 Simplified analysis in living systems

From Table B8, we see that the difference in enthalpy between solid and dissolved glucose is 11 kJ/mol, while the differences in enthalpy of formation for gas and dissolved matter is 20 kJ/mol for carbon dioxide and 10 kJ/mol for oxygen. For all the components in aqueous solution, we find from Table B9. AHr° = - 2870 kJ/mol So, the transfer of gaseous components to aqueous solutions is small: 60/2810 - 0.02 or 2%. The reaction enthalpy at 310 K can be estimated from

AHr(310K) -- A H

°

+ Cp,av (T - To ) -- z~kg ° + E piCpi (T - To )

where v is the stoichiometric coefficient. By using the heat capacities from Table B8, we have AH r (310 K)

= AH °

+ Cp,av ( T

- T O) -

-2870 + (0.279)(310 - 298) = -2867 kJ/mol

This value is the same as the one in Table B9, and shows that the temperature correction for the heat of reaction is less than 0.2% and is often negligible. The energy expenditure (E) at a glucose consumption of 390 g/day is 390/180 = 0.025 mmol/s n~ = 24(3600) and =nG ( - A H r ) = 0.025(2867)= 71.7W where the molecular mass of glucose is 180 g/mol.

Example 11.7 Unimolecular isomerization reaction One of the simplest biochemical reactions is a unimolecular isomerization reaction (Qian and Beard, 2005)

S<

kf

kb

>P

where kf and kb are the forward and backward reaction rate constants. A chemical equilibrium is defined by

[P]eq[s]eq_kfkb -

(11.2)

exp (-/x~' -/x~/RT

To simulate a biological metabolic network, consider a special controlling mechanism so that the concentrations of S and P are maintained at prescribed levels. Therefore, the chemical system is at steady state since concentrations remain unchanged with time. However, the system in not at an equilibrium state and the reaction velocity (flow) is not zero: Jr = kfcs - kbCv

= Jrf -Jrb 4:0

We can determine the affinity of the reaction as the driving force for the chemical system

A = -Al~

-

I~p -

tx s -

o

I~p -

o

ix s + RT

Cp Cs

ln~ -

RT

In

[Jrb / ( Jrf )

(11.3)

Introducing chemical potentials for biochemical substrates needs to be done with caution when considering, for example, molecular crowding and signaling molecules with limited copy numbers (Parsegian et al., 2000). This simple chemical system is for cellular metabolic networks, and concentrations replace activities in ideal solutions.

548

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Thermodynamicsand biological systems

Equation (11.3) can be transformed to describe the amount of work done by the controlling mechanism, which pumps reactant S and product P to maintain nonequilibrium steady-state conditions

JrA= RT(Jrf -Jrb)ln(Jrf ]>__O

(11.4)

~,Jrb )

The equality of this equation represents a system at equilibrium where Jr = A = 0. The work done by the controlling system dissipates as heat. This is in line with the first law of thermodynamics. The inequality in Eq. (11.4) represents the second law of thermodynamics. The cyclic chemical reaction in nonequilibrium steady-state conditions balances the work and heat in compliance with the first law and at the same time transforms useful energy into entropy in the surroundings in compliance with the second law. The dissipated heat related to affinity A under these conditions is different from the enthalpy difference M-/° = (O(Atx°/T)/O(1/T)). The enthalpy difference can be positive if the reaction is exothermic or negative if the reaction is endothermic. On the other hand, the A contains the additional energy dissipation associated with removing a P molecule from a solution with concentration Cp and adding an S molecule into a solution with concentration Cs. This simple example may be generalized to biochemical reaction cycles in which there are a number of reactions and boundary flows that add and remove substrates. The controlled concentrations and boundary flows maintain the system in a nonequilibrium steady-state condition. A dynamic equation of entropy change is

T --~ : r

-t-T ---~-]

qdis-t-xIt

(11.5)

This equation shows that in isothermal biochemical reaction cycles, the entropy of the system changes because of the heat dissipation rate qdis exchanged with the surrounding and the rate of free energy dissipation • due to entropy production. This equation also indicates the dissipative character of biochemical cycles. Dynamic equations similar to Eq. (11.5) can also be written for enthalpy and Gibbs free energy changes

where Wnowis the flow work (chemical motive force, Qian and Beard, 2005) determined by the controlling boundary flows and/or concentration. At steady state

we have ~ = !)dis = ~/flow ~ 0, in which the equalities represent the first law of thermodynamics and the isothermal Clausius equation, while the inequality represents the second law of thermodynamics.

11.3

BIOENERGETICS

Energy production, conversion, and storage to maintain the nonequilibrium character of living systems form the basis for bioenergetics. Energy is supplied with the intake of food or with solar radiation. Living systems convert part of this energy to produce electrons and protons. The flow of protons across a specific membrane leads to the production of ATE The hydrolysis of ATP is coupled to synthesizing protein molecules, transporting ions and substrates, producing mechanical work, and other metabolic activity. Some of the internal mechanical work involves the pumping of blood by the heart. Continuous chemical cycles and transport processes maintain a stationary state of chemical nonequilibrium and integrity by the regulated and synchronized production, conversion, and utilization of energy. Bioenergetics provides a quantitative description of the transformation of materials and energy in living systems. Most biochemical reactions occur in pathways, in which other reactions continuously add substrates and remove products. The rate of reactions depends on the properties of the enzymes (large proteins produced in cells) that catalyze the reaction. Substrates bind at the active sites of enzymes, where they are converted to products and later released. Enzymes are highly specific for given substrates and products. Inhibitors of enzymes decrease the rate of reaction.

11.3 Bioenergetics

549

Proteins can bind to enzymes and alter their activities. The clusters of orthologous genes database has identified 210 protein families involved in energy production and conversion; the protein families show complex phylogenetic patterns and exhibit diverse strategies of energy conservation. The organized structures of living systems degrade incoming solar radiation and chemical potential through well-controlled chemical cycles.

11.3.1

Mitochondria

Mitochondria are organelles typically ranging in size from 0.5 to 1 txm in length, found in the cytoplasm of eukaryotic cells. Mitochondria contain inner and outer membranes, separated by a space. Both the inner and outer membranes are constructed with tail-to-tail bilayers ofphospholipids into which mainly hydrophobic proteins are embedded. One portion of the lipid molecule is hydrophilic (water-attracting) and the other portion is hydrophobic (lipid-attracting). The selfassembled lipid bilayer is in a dynamic and liquid-crystalline state. The outer membrane contains proteins and lipids and numerous transport proteins, which shuttle materials in and out of the mitochondrion. The outer membrane is 60-70 thick and permeable to small molecules, including salts, adenine and nicotinamide nucleotides, sugars, and coenzymes. The inner membrane contains all the enzymes and fewer lipids than the outer membrane. The inner membrane is permeable to small neutral molecules such as water, oxygen, and carbon dioxide, while its permeability to charged molecules such as proton and ions is limited. Mitochondrial membranes produce two compartments; one of them is called the intermembrane space (C-side) and the space enclosed by the inner membrane is called matrix (M-side) (Figure 11.1). The intermembrane space is usually 60--80 A in width and contains some enzymes. The matrix, however, is very viscous and rich in proteins, enzymes, and fatty acids. The number of mitochondria in a cell depends on the cell's function. Cells with particularly heavy energy demands, such as muscle cells, have more mitochondria than other cells. The inner membrane houses the electron transport chain and ATP synthesis. The inner membrane has numerous folds called cristae, which have a folded structure that greatly increases the surface area where ATP synthesis occurs (Figure 11. lb). Mitochondria contain deoxyribonucleic acid (DNA) and ribosomes, protein-producing organelles in the cytoplasm. The DNA directs the ribosomes to produce proteins as enzymes (biological catalysts) in ATP production. Outer membrane

(a)

Inner membrane

Matrix

Outer membrane

Cristae

2

Cristae

\ (b)

Inner membrane

Figure 11.1. (a) Structure of the mitochondria and (b) inner membrane structure of the mitochondria.

11. Thermodynamicsand biological systems

550

Mitochondria are involved in the transport and regulation of Ca 2+, protein import, cell death and aging, and obesity. Mitochondria from different organ systems, such as the liver, heart, and brain, display morphological and functional differences. Mitochondria are the major source of reactive oxygen species throughout the respiratory chain. These oxygen radicals may affect the function of the enzyme complexes involved in energy conservation, electron transfer, and oxidative phosphorylation, and play an important role in aging. Experimental evidence shows that mitochondria exhibit anisotropy. Three-dimensional images show that inner membrane involutions (cristae) have narrow and long tubular connections to the intermembrane. These openings may lead to lateral gradients of ions, molecules, and macromolecules between the compartments ofmitochondria. This type of structure may influence the magnitude of local pH gradients produced by chemiosmosis and the internal diffusion of adenine nucleotides. The mitochondria have elongated tubes aligned approximately in parallel and are embedded in a multilamellar stack of endoplasmic reticulum, which could be related to the specific function of the mitochondria (Ovadi and Saks, 2004).

11.3.2

Tricarboxylic Acid Cycle in Mitochondria

The TCA cycle, also called the citric acid cycle or the Krebs cycle, is the major energy-producing pathway and occurs in mitochondria. Foodstuffs enter the cycle as acetyl coenzyme A CoA and are oxidized. The cycle starts with the four-carbon compound oxaloacetate, adds two carbons from acetyl CoA, loses two carbons as CO2, and regenerates the four-carbon compound oxaloacetate. Electrons are transferred to nicotinamide adenine dinucleotide NAD + and flavin adenine dinucleotide FAD, and NADH and FADH2 are produced (Marks, 1999). As the electrons are transferred eventually to oxygen, energy is released, and ATP is produced by the process called oxidative phosphorylation. Intermediate products of the TCA cycle are converted to glucose during the fasting state and to fatty acids during the fed state in the liver. Some intermediate products are synthesized to amino acids (Figure 11.2). The change in free energy or available energy AG to perform work at constant temperature and pressure is A G = M 4 - TAS, where M-/is the change in enthalpy and/kS is the change in entropy. If AG is negative, the reaction proceeds spontaneously and releases energy. If AG is positive, the reaction needs an energy supply to proceed. If AG is zero, the reaction is at chemical equilibrium. The rate of a chemical reaction is not related to its free energy change; a reaction with a large negative free energy change may not lead to a fast reaction. Directions of reactions proceeding near equilibrium can be reversed by small changes in the concentrations of substrates or products. The need for energy by the cell regulates the TCA cycle, which acts in concert with the electron transfer chain and the ATPase to produce ATP in the inner mitochondrial membrane. The cell has limited amounts of ATE ADP,

Intermembrane

/

electrons~------7"xzzEZ 2H+ ~~~_TL-NADH

/

- --el tro.s /

~

/

/ /[

/

I [ Proton /

P

~

2H+

Quinone

/

Cytochrome

/ ~

02

l/m°tlV\ \forcee /

/ ADP+P

cristae

ATP ATP Synthesis

Figure 11.2. Electron transport in phosphorylation.

11.3 Bioenergetics

551

and AMP. When ADP levels are higher than ATR the cell needs energy, and hence NADH is oxidized rapidly and the TCA cycle is accelerated. When the ATP level is higher than ADP, the cell has the energy needed; hence, the electron transport chain slows down, and TCA cycle is inhibited.

11.3.3

Mitochondria and Electron Transfer Chain

Mitochondria organize electron transfer and the associated reactions leading to the ATP synthesis called oxidative phosphorylation (Figure 11.2). Synthesis of ATP is an endothermic reaction, and hence conserves the energy released during biological oxidation-reduction reactions. Electron transfer and associated reactions leading to ATP synthesis are completely membrane-bound. NADH and FADH2 are the reduced cofactors of NAD ÷and the FAD÷. The oxidation of one NADH produces approximately three ATR and the oxidation of one FADH2 produces approximately two ATE The hydrolysis of ATP provides energy for all cellular activity. ATP is transported from the mitochondrial matrix to the cytosol in exchange for ADP through the ATP-ADP antiport system. The photosynthetic energy conversion of light energy into Gibbs free energy of protons takes place in plants, algae, and certain species of bacteria. Photosynthesis, driven by light energy, leads to the production of ATP through electron transfer and photosynthetic phosphorylation. The transmembrane electron transfer process occurs in specialized pigment-protein complexes called photosynthetic reaction centers. Photosynthetic energy conservation takes place in the thylakoid membrane of plant chloroplasts; oxidative phosphorylation takes place in the mitochondrial inner membrane. These membranes facilitate the interactions between the redox system and the synthesis of ATR and are referred to as coupling membranes. Electron transport has three major stages: (1) transfer of electrons from NADH to coenzyme Q, (2) electron transport from coenzyme Q to cytochrome c, and (3) electron transport from cytochrome c to oxygen. These stages are briefly described below. (1) Transfer of electrons from NADH to coenzyme Q NADH is produced by (i) the c~-ketoglutarate dehydrogenase, isocitrate dehydrogenase, and malate dehydrogenase reactions within the TCA cycle, (ii) the pyruvate dehydrogenase reaction, (iii) [3-oxidation of fatty acids, and (iv) other oxidation reactions. NADH passes electrons to the flavin mononucleotide FMN using the dehydrogenase complex. NADH produced in the mitochondrial matrix diffuses to the inner mitochondrial membrane where NADH passes electrons to the FMN. The FMN passes electrons through a series of iron-sulfur complexes to coenzyme Q. Coenzyme Q accepts electrons one at a time and forms semiquinone and ubiquinol. The electron transfers produce energy, which is used to pump protons to the cytosolic side of the inner mitochondrial membrane. The protons flow back into the matrix through pores in the ATP synthase complex, and approximately one ATP is produced for each NADH. (2) Electron transport from coenzyme Q to cytochrome c Coenzyme Q passes electrons through iron-sulfur complexes to cytochromes b and cl, which transfer the electrons to cytochrome c. In the ferric Fe 3+ state, the heme iron can accept one electron and be reduced to the ferrous state Fe2÷. Since the cytochromes carry one electron at a time, two molecules on each cytochrome complex are reduced for every molecule of NADH that is oxidized. The electron transfer from coenzyme Q to cytochrome c produces energy, which pumps protons across the inner mitochondrial membrane. The proton gradient produces one ATP for every coenzyme Q-hydrogen that transfers two electrons to cytochrome c. Electrons from FADH2, produced by reactions such as the oxidation of succinate to fumarate, enter the electron transfer chain at the coenzyme Q level. (3) Electron transport from cytochrome c to oxygen Cytochrome c transfers electrons to the: cytochrome a a 3 complex, which transfers the electrons to molecular oxygen, and the oxygen is reduced to water. Cytochromes a and a 3 contain heme a and two different proteins containing copper. The energy released by the transfer of electrons fi'om cytochrome c to oxygen is used to pump protons across the inner mitochondrial membrane. Every two electrons that are transferred from cytochrome c to oxygen produce one ATR

11.3.4

Oxidative Phosphorylation

Oxidative phosphorylation occurs in the mitochondria of all animal and plant tissues, and is a coupled process between the oxidation of substrates and production of ATE As the TCA cycle runs, hydrogen ions (or electrons) are carried by the two cartier molecules NAD or FAD to the electron transport pumps. Energy released by the electron transfer processes pumps the protons to the intermembrane region, where they accumulate in a high enough concentration to phosphorylate the ADP to ATE The overall process is called oxidative phosphorylation. The cristae have the major coupling factors F1 (a hydrophilic protein) and F0(a hydrophobic lipoprotein complex). F1 and F 0 together comprise the ATPase (also called ATP synthase) complex activated by Mg 2+. F 0 forms a proton translocation pathway and F1

552

11.

Thermodynamicsand biological systems

is a catalytic sector. ATP synthesis by F0F 1 consists of three step: (i) proton translocation through F0, (ii) conformation transmission to F1, and (iii) ATP synthesis in the [3 unit. The rotation of a subunit assembly is an essential feature of the mechanisms of ATP synthesis and can be regarded as a molecular motor (Sambongi et al., 2000). ATPase can catalyze the synthesis and the hydrolysis of ATE depending on the change of electrochemical potential of proton A~H. The ratio of ATP production to oxygen consumption P/O can vary according to various physiological processes: (i) maximizing ATP production, (ii) maximizing the cellular phosphate potential, (iii) minimizing the cost of production, and (iv) a combination of these three processes. The values of P/O change within the range of 1-3, and characteristic of the substrate undergoing oxidation and characteristic of the organ's physiological role. In the case of excess oxygen and inorganic phosphate, the respiratory activity of the mitochondria is controlled by the amount of ADP available. In the controlled state called state 4, the amount of ADP is low. With the addition of ADP, the respiratory rate increases sharply; this active state is called state 3. The ratio of the respiratory rates of state 3 to state 4 is called the respiratory control index. The control of the respiration process and ATP synthesis shifts as the metabolic state of the mitochondria changes. In an isolated mitochondrion, control over the respiration process in state 4 is mainly due to the proton leak through the mitochondrial inner membrane. This type of control decreases from state 4 to state 3, while the control by the adenine nucleotide and the dicarboxylate carriers, cytochrome oxidase, increases. ATP utilizing reactions and transport activities also increase. Therefore, in state 3, most of the control is due to respiratory chain and substrate transport. According to the chemiosmotic coupling hypothesis, ATP synthesis decreases the proton electrochemical gradient and hence stimulates the respiratory chain to pump more protons across the mitochondrial inner membrane and maintain the gradient. However, electron supply to the respiratory chain also affects respiration and ATP synthesis. For example, calcium stimulates mitochondrial matrix dehydrogenase, and increases the electron supply to the respiratory chain and hence the rate of respiration and ATP synthesis.

11.3.5

Glycolysis Pathway

The glycolysis pathway occurs in the cytoplasm outside the mitochondria, and requires no oxygen. During glycolysis, glucose is broken down into pyruvate. The initial reactions of the pathway produce triose phosphate, which produces ATP in the second sequence of reactions. Overall, the glycolysis pathway produces ATP, NADH, and pyruvate. NADH cannot directly enter mitochondria. Pyruvate can enter the mitochondria; it is broken down to acetyl CoA by a special enzyme, and carbon dioxide is released. Acetyl CoA enters the TCA cycle, producing additional ATE Only four ATP molecules can be produced by one molecule of glucose.

11.3.6

Transport Processes and Mitochondria

There are many carrier molecules for electrons: one is called the nicotinamide adenine dinucleotide (NAD +) and another is the flavin adenine dinucleotide FAD+. The reduced cofactors NADH and FADH 2 transfer electrons to the electron transport chain. FMN receives electrons from NADH and passes them to coenzyme Q through Fe-S systems. Coenzyme Q receives electrons from FMN and FADH2 through Fe-S systems. Cytochromes receive electrons from the reduced form of coenzyme Q. Each cytochrome consists of a heme group, and the iron of the heme group is reduced when the cytochrome receives an electron: Fe 3+ ~ Fe z+. At the end of the electron transfer chain, oxygen is reduced to water. Membrane proteins transfer material and information between cells and their environment and between the compartments housing the organelles. Some of these proteins selectively transport specific molecules and ions, and some others are receptors for chemical signals from outside the cell. They act as transducers capable of gathering information, processing it, and delivering a response. Their electrical activities are measurable as an electric potential difference across the membrane. Changes in the membrane permeability would yield a change in the potential difference. In a cotransport system, the movement of one permeant is dependent on the simultaneous movement of a different permeant, either in the same direction (called symport) or in the opposite direction (called antiport). The best-known antiport system is the Na+/K+-ATPase pump that is present in the plasma membrane of all animal cells. The pump transports sodium ions out of the cell and potassium ions into the cell through the lipid bilayer against their electrochemical potential gradients, and operates as an antiport. Transmembrane activities are thermodynamically driven by the gradients of chemical and electrochemical potentials and are able to maintain steady nonequilibrium conditions across cell membranes by generating and controlling the flows of ions or electrons. The cytoplasm houses many metabolic cycles and synthetic pathways, as well as protein synthesis. At the cellular level, communication via the membrane is called signal transduction, and is facilitated by ligands or messengers, such as proteins and peptide hormones. These ligands facilitate communication by directly entering the cell,

11.3 Bioenergetics

553

or interacting with a specific receptor situated on or in the lipid bilayer of the membrane. Insulin stimulates glucose transport into muscle and adipose cells. However, it does not significantly stimulate the transport of glucose into tissues such as the liver, brain, and red blood cells (Marks, 1999).

11.3.7

Formulation of Oxidative Phosphorylation

Theoretical approaches applied to oxidative phosphorylation are the kinetic model, metabolic control analysis, and nonequilibrium thermodynamics. These approaches are helpful for quantitatively describing and understanding the control and regulation of oxidative phosphorylation. The metabolic control theory can provide a quantitative description of microbial growth. A kinetic model of oxidative phosphorylation may not be fully completed, because of several assumptions and simplifications associated with it. A proper kinetic approach, however, allows for a deeper insight into the mechanisms related to the control and regulation of oxidative phosphorylation. It may provide a model and methodological approach for describing the dynamic and stationary properties of energy coupling in membranes. The application of nonequilibrium thermodynamics assumes a linear flow-force relationship between the oxidation and phosphorylation flows. Such a linear dependence has been established by measurements taken during the transition on from state 4 to state 3 as a linear part of a more general sigmodial relationship. Nonequilibrium thermodynamics has proved to be useful in describing the energetics aspects of oxidative ATP production and the transport of substrates coupled to ATP hydrolysis without knowledge of the detailed mechanisms of coupling (Stucki, 1980). However, it is not realistic to assume that simple formulations can lead to a complete description of such complex and coupled biological systems. A starting point in linear nonequilibrium thermodynamic formulations is the representative dissipation function given by altt = J p Ap -+-JHA/~H -+-JoAo

(11.7)

Here, the subscripts R H, and O refer to phosphorylation, the H + flow, and substrate oxidation, respectively, and A/~H is the electrochemical potential difference of protons. We consider only systems at steady state. The dissipation function can be transformed as ~r = j p A ~ x -Jr-JHA/~H 4- JoA(e)x

(11.8)

where A ex is the external affinity. When the interior of the mitochondrion is in a stationary state, it suffices to measure the changes in the external solution only. From Eq. (11.8), the linear phenomenological equations with the resistance coefficients are obtained A~,x -- K p J p + KpHJ H + KpoJ O

(11.9)

~/)'H = KpHJP -+-K H J H -+-KOHJo

(11.10)

A~x -- KpoJ P + KOHJ H + K o J o

(11.11)

If A~ is kept constant and A/2H is not controlled, Jn - 0 in the stationary state, which is also called the static h e a d , and Eqs. (11.9) and (11.11 ) become

11.3.8

,4~x - K p d p + Kpod O

(11.12)

~,4~ -- KpoJ p -t-KoJ o

(11.13)

Degree of Coupling

If no cross-coefficients vanish, then we have three degrees of couplings, qPH, qoH, and qPo. Based on the resistance-type phenomenological coefficients, the degree of coupling qpo is Kpo q--

(KpKo)I/2

(11.14)

554

11.

Thermodynamicsand biological systems

When we have levelflow, the force vanishes, A/2H = 0, and Eqs. (11.9)-(11.1 l) reduce to A~,x = Kp (1 - q2 H )Jp - ( K p K O )1/2 (qPo -k-qPHqOH ) J o

(11.15)

A~)X= _(KpKo )1/2 (qPo + qPHqOH )JP + Ko (1 - q2H)Jo

(11.16)

where q=

11.3.9

qPo+qPHqOU (1-- qZH)(1-- qZOH)

(11.17)

Efficiency of Energy Conversion

With the electroosmotic work JpXp compensated by the chemical work JrA, we can define the effectiveness of energy conversion

JpXp

n = - ~ Jr A

(11.18)

where Xp is the force for proton transportation. It is also useful to consider the force developed per given rate of expenditure of metabolic energy, which is called the efficacy of force

exp =

11.3.10

Xp jr A

(11.19)

Photosynthesis

In photosynthesis, energy-rich organic molecules emerge from simple energy-poor molecules, which absorb solar photons. After charge separation occurs, a proton electrochemical gradient, up to 200 mV, is created. In the purple photosynthetic bacterium Rhodobacter sphaeroids, membrane-bound proteins couple electron transfer to proton release into the periplasmic space of the bacterium. In the case of Halobacterium salinarium, photon-free energy is directly converted into the proton electrochemical gradient by the membrane protein bacteriorhodopsin. Using a flow of material JD and flow of energy Go, the linear nonequilibrium formulations can find a simple relation between the efficiency of photosynthesis and common transport properties of the chloroplast for producing ATE Chloroplasts are found only in plants and have double membranes. Inside a chloroplast's outer membrane, a set of thin membranes called hyaloids contain chlorophyll pigments that absorb solar energy. This is the ultimate source of energy for all the plant's needs and for synthesizing carbohydrates from carbon dioxide and water. The chloroplasts first convert the solar energy into ATP-stored energy, which is then used to synthesize carbohydrates, which can be converted back into ATP when energy is needed. Photosynthesizing bacteria called cyanobacteria use chlorophyll bound to cytoplasmic thylakoids. The maximum available energy of a photon that a chlorophyll at temperature T can utilize is (11.20)

where hv is the photon energy and TR is calculated from the assumed Planck distribution of radiation. Steady-state affinity A of a pigment is

/

(11.21)

where P and P* are the ground and excited chlorophyll states, respectively. A part of photon energy can be used to perform charge separation only if an appropriate branched pathway exists; electron acceptors and donors are located so that charge separation through a branched pathway takes place with high efficiency.

555

11.3 Bioenergetics

The thermodynamic force for light reactions is X L -

(l 1.22)

Area x - A ~ 0

The dependence of XL on photochemistry quantum yield 05 = J/I may be described by

(11.23)

X L --kBT{ln(1-ch)-ln[l+chI]}l+kd

where I is the flow of absorbed photons, and kd is the nonradiative relaxation constant, kd = 108 s-1. The second part inside the curly brackets is small compared with the first term and one needs to know the temperature and the photochemical yield to calculate Xc. Equation (11.23) also relates the flow Jr to the force XL when light intensity is regarded as constant

,{1 exp[xL]IB

(11.24)

The corresponding dissipation function qt due to transmitted free energy A and flow Jr is = AJ r

(11.25)

Using Eqs. (11.20)-(11.23), the dissipation as a function of thermodynamic force XL is q~ = a ( 1 - x)

1-exp(-x/b)

(11.26)

1 + c exp(-x/b)

or as a function of photochemical yield 05, we have = ad~[1 + b ln(1- ~b)- b ln(1 + c~b)] where XL X--

Amax

1 ,

a--

IAmax,

A max

I

- - - , b

c-

k BT

I

+ kd

With the approximation in Eq. (11.24), we have (11.27)

A max= A - kBT ln(1 - 4')

Assuming a small quantum yield, O << 1, or an equivalently small thermodynamic force, XL -- kBT, Eq. (11.27) can be linearized Jr=

X

LI kBT

(11.28)

Equation (11.28) resembles the theory of electrical circuits; Areax is the electromotive force, A is the voltage drop on a load, and XL is the voltage drop through internal resistance R i. External R and internal resistances R i are defined by R - / Amax - XL

--~)

and Ri

kBT

(11.29)

The dissipation in Eq. (11.25) is on the external resistor and obtained by using Eq. (11.28) q~ -- Jr( Ama~ -- XL )= I( Amax - X L )

X L

kBT

= RJ )

.3o)

556

11.

Thermodynamicsand biological systems

The functional and morphological heterogeneity of a lamellar system of chloroplasts indicates that pH values in different compartments (in granal and intergranal thylakoids) differ. This type of structure makes it difficult to measure local pH values at different sites. Therefore, mathematical models taking into account the spatial structure of chloroplasts provide a tool for studying the effect of diffusion restrictions on pH distributions over the thylakoid on the rates of electron transport, proton transport, and ATP synthesis. The rate of ATP synthesis depends on the osmotic properties of a chloroplast-incubation medium and, therefore, on topological factors.

Example 11.8 Efficiency of energy conversion of photosynthesis Consider a model process with an energy exchange between a photon and a composite particle. In this over-simplified model, energy is exchanged through an excited state of the chloroplast by which energy-rich electron/proton pairs from the water react with the carbon dioxide. This produces carbohydrate and oxygen molecules, and heat is dissipated away. The linear nonequilibrium thermodynamics formulations start with the rate of entropy production Vn

VT

n

T

* =--JD - - - J q

(11.31)

where n is the atomic density. Based on the rate of entropy production, the phenomenological equations are

JD = -(nD) --Vn _ VT n LDq T

Jq =--LqD

Vn

n

k VT

ks T

(11.32)

(11.33)

where D is the diffusion coefficient, k is the thermal conductivity, ks is the Boltzmann constant, and LDq and Lqo are the cross-phenomenological coefficients. The flows are related by an approximation (Andriesse and Hollestelle, 2001) (11.34)

Only a small part of the initial free energy of photons is available for photosynthesis, and the rest is dissipated. The efficiency of energy conversion in photosynthesis is low and varies in the range 2.4-7.5% (Andriesse and Hollestelle, 2001). The efficiency of energy conversion is defined by r/-

JD ,~G Jq hp

(11.35)

where iXG is the Gibbs energy per molecule and hv is the energy per photon. Some approximate values for these driving forces are AG--~ 7.95 × 10-19 J per unit carbohydrate, and (by) --~ 2.92 × 10-19 J per solar photon (pertaining to red light with a wavelength of 680 nm, which is best absorbed by chlorophyll-a). These approximate values and Eqs. (11.34) and (11.35) yield an approximate relation for the efficiency of energy conversion

( k B n D ) 1/2 r/= __~qD( 7"95 × 10-19 ) 2.92 × 10-19 = 2.72 k

(11.36)

where n --~ 3.3 × 1028 molecules/m 3, which is a typical value for water and condensed matter in general. Based on the thermal conductivity of water, we have k = 0.607 W/(m K). For a chloroplast that is drenched with water, we may assume that the high frequency of motion of water molecules transports energy. However, the diffusion coefficient is linked to the low frequency motions of molecules through the chloroplast. Assuming that the intercellular diffusion of carbon dioxide could be the limiting process in photosynthesis, we have D--~ 1.95 × 10 -9 mZ/s based on carbon dioxide in water or D--~ 0.67 × 10 -9 mZ/s based on glucose in water. Using these values of k and D in Eq. (11.36), the efficiency of energy conversion is estimated and compared with experimental values in Table 11.2. For any plant growing under ideal conditions, the efficiency is expected to be close to 7%.

11.4 Properpathways

557

Table 11.2 Comparison of predicted efficiencies of photosynthesis with measured values at various values of diffusion coefficients (predicted) 2.4 6.1 7.5

D (m-~/s)

r/(measured)

0.1() x 10 '~ 0.67 x 10 '~ 1.0 ~ 10 ~

For a minimum of D, 4.9 C3-plants ~' 6.2 C4-plants ~' 7 ideal crop, for a maximum of D

~ In practice C4-plants converts approximately 14 Ixg CO2/J of intercepted daylight in CH20, while C3-plants converts approximately 11 txg CO?/J of intercepted daylight in CH20. Source: Andriesse and Hollestelle (2001)

11.4

PROPER PATHWAYS

For a steady state far from global equilibrium, there may be pathways in the vicinity of this state along which a linear flow-force relation holds. The linearity of flows observed in experimental studies of active transport in epithelia suggests the existence of proper pathways where the phenomenological coefficients become nearly constant (Caplan and Essig, 1989). Formulating the relationships between forces and flows leads to understanding the change of affinity of a reaction driving the transepithelial active transport, free energy tissue anisotropy (compartmentalization), and activity. Experiments show that biological processes take place in many steps, each of which is thought to be nearly reversible, and exhibit linear relationships between steady-state flows and conjugate thermodynamic forces, such as transepithelial active Na +and H ~-transports and oxidative phosphorylation in mitochondria. The linear relations between the rate of respiration and the specific growth rate are observed for many microbial systems. The forces can be controlled in various ways to find a proper pathway leading to quasi-linear force-flow relationships so that the theory of linear nonequilibrium thermodynamics can be applied. For a first-order reaction S --, R doubling the concentrations of S and P will double the reaction rate for an ideal system, although the affinity remains the same, and a distinction must be made between thermodynamic and kinetic linearity. Proper pathways are associated with thermodynamic linearity. The rate of a process depends not only on the force but also on the reference state; the flow of a solute across a membrane depends on its chemical potential and on its thermodynamic state on both sides of the membrane. The constancy of phenomenological coefficients L may be maintained by applying appropriate constraints to vary the force X in the relationship J = LX. The values of L reflect the nature of the membrane, and can control the force X. If a thin homogeneous membrane is exposed to the same concentrations at each surface, flow is induced solely by the electric potential difference, and L is constant with the variation of X. However, if X is the chemical potential difference, dependent upon the bath solute concentrations, then L becomes L = KmUClm

(11.37)

~Xz where u is the mobility, Km is the solvent-membrane partition coefficient, qm is the logarithmic mean bath concentration (Ac/Aln c), and zXz is the thickness of the membrane. If a value of qm is chosen and the concentrations are constrained to the locus Ac = (qm)Aln c, then L becomes constant. The logarithmic mean concentration can be used in the linear formulation of membrane transport. If the force is influenced by both the concentrations and the electrical potential difference, then L becomes more complex, yet it is still possible to obtain a constant L by measuring J and X in a suitable experiment. For a first order chemical reaction S --+ R the reaction rate is given by Jr - k f c s - kbCp - kbCp( eA/RT --1)

(11.38)

where kf and kb are the rate constants for forward and backward reactions, respectively. At steady state, far from equilibrium, the reaction rate from • - JrA may be L*A Jr = L A RT

where L* = R T L , and can be evaluated my measuring Jr and A

(11.39)

558

11. Thermodynamicsand biological systems

L*=kbCp( A/RT ) -1 eA/RT_ 1

(11.40)

Equation (11.39) shows that for different values of A at various stationary states, the same values of L* will describe the chemical reaction when appropriate concentrations are chosen. For a specified value ofA, Eqs. (11.38) and (11.39) determine Cap and the ratio of cp/cs, respectively, and a constant L can be found by limiting the cp and Cs to an appropriate locus. As the system approaches equilibrium, A tends to vanish and kbcpapproaches the value L*. This procedure can also be used in more complex reaction systems. Proper pathways can be identified in the neighborhood of a reference steady state far from equilibrium by varying the forces X1 and X2 in such a manner as to lead to the linearity of flows and forces. Highly coupled systems show similarity to a single uncoupled flow, and linear dependencies on conjugate and nonconjugate forces exist. In the vicinity of the static head, where the transmembrane flows are zero, linearity would be expected when the degree of coupling is close to unity. Sometimes, kinetic nonlinearity may occur because of a feedback and not due to large affinities, and the sustained oscillations may occur near equilibrium.

11.4.1

Metabolic Control Analysis

The metabolic control analysis determines quantitatively the effects of various metabolic pathway reactions on flows and on metabolic concentrations. The analysis defines two coefficients: (i) the control coefficients, which characterize the response of the system flows, concentrations, and other variables after parameter perturbations; and (ii) the elasticity coefficients, which quantify the changes of reaction rates after perturbations of substrate concentrations or kinetic parameters under specified conditions. Steady-state metabolic flows Jj depend on the total concentrations of the enzymes Ek, and flow control coefficients Cj,E are defined by

= ( E~:AJj )~ _ E k OJj Cj'E ~Jjz2tE---~ Ek-~o JjOEk

(11.41)

The flow control coefficients relate the fractional changes in the steady-state flows to the changes in the total enzyme concentrations. The partial derivatives of reaction rates Jr, i with respect to the substrate concentrations Sj are called the elasticity coefficients e/;, and given by

Sj aJri

eiJ Jri OSj

(11.42)

The flow or concentration control coefficients are related to elasticity coefficients through the conservation relations and connectivity theorems. Besides the forces controlling the pathways, alternatively, flows controlling a certain pathway are also important. The metabolic control analysis can be used to evaluate flow control within biochemical pathways, and provide information on the regulation of pathway flows. Using the metabolic control analysis from the measurements of flows exchanged through the cell membrane, it is possible to quantify pathway flows and alternative pathways to the same metabolite. This methodology is limited only to the analysis of simple two-step pathways, although larger pathways can be lumped into two overall changes. The influence of the individual reaction rates (enzyme activities) on the overall flow through the pathway is called the flow control coefficients Cr, which are expressed by 0 ln(J)

Cri = ~ '

0 ln(Jri )

(i = 1,...,m)

(11.43)

where Jis the steady-state flux through the pathway, Jri is the rate of ith reaction, and m are the enzymic steps. The flux control coefficients are related to the elasticity coefficients e as follows m

8i,jCr, i i=1

-

0

(j = 2,...,m)

(11.44)

11.4

559

Proper p a t h w a y s

where the elasticity coefficients are defined by Oln(dri ) (je;,i = a ln(c i )

2 .... , m ) ( i - 1 , . . . , m )

(11.45)

The determination of flow control coefficients is difficult, and requires the independent variation of the activity of all the enzymes within the pathway. Based on linear nonequilibrium thermodynamics, the kinetics of enzyme reactions can be described by the linear functions of the change in Gibbs free energy. This yields a direct relation between the elasticity coefficients and the change in Gibbs free energy for the reactions in a simple two-step pathway. The control coefficients can be determined by the linear nonequilibrium thermodynamics formulation. Schuster and Westerhoff (1999) provide a simple example for the coupled processes of oxidative phosphorylation with slipping enzymes, for which a representative dissipative function is

- droAo + drpAp + drlAi

(11.46)

Assuming that the system is in the vicinity of equilibrium, the representative linear flow force relations based on Eq. (11.46) are

dro - - LoAo+ LopAp Jrp

--

LpoAo + LpAp

Jrl

i

(11.47)

LIAp

where Jro, Jrp, and Jrl are the reaction velocities for the respiration, ATP production or hydrolysis, and the load, respectively, and A shows the affinities as driving forces in the system. The forces for ATP production and utilization are the same with different signs. A load process may be a chemical pump powered by the hydrolysis of ATP or a proton leak. The linear flow-force relations in Eq. (11.47) indicate that the cross-coefficients Lol, Lpl, Llo, and Lpl vanish, and Lop = Lpo, according to Onsager's reciprocal rules. The reaction flows in Eq. (11.47) represent a steady state attained by the slipping enzymes, with Ao and Ap being constant. However, for a steady state of the whole system, Ap becomes variable (such as the variable proton concentration within the mitochondria), while Ao is constant. At steady state, Jrl = - J r p , and from Eq. (11.47), we have

AP =

L°PA°

(11.48)

L~ +L p

For the steady-state flows of oxygen, ATE and load, we have Jo - Jro and Jp = Jrp - --Jrl" Substituting Eq. ( 1 1 . 4 8 ) in Eq. (11.47), we find

jo-(L

°_

L2°p ~Li + L

dp-

)

A°

LpLop ]

Lop-~ Ao L1+ Lp

(11.49)

(11.50)

The nonnormalized control coefficients are defined by

OJP/OL°

*

-

Lo

*c@ - OJ°/OLP 0-7-#-;Tp

(11.51)

The control coefficient of the load is

OJp/OL 1 Odrl/OL1 The partial differential terms in the control coefficients are obtained from differentiation Eq. (11.47)

(11.52)

560

11.

Thermodynamicsand biological systems

OJro __ Ao, OJr° - 0 , 0Lp OLo

0Jro

=0

OJ~p _ O, OJrp - Ap, OJ rp _ 0 aLp aL 1 OLo OJrl

(11.53)

-

and Eqs. (11.49) and (11.50) yield 2 LopAo OJo _ Ao, O J o _ OLo aLp (L1 + L p ) 2 OJp_

OJ1 -

OLp

OLp

LopL1Ao (L1 + Lp) 2'

OJo __

O/__q

L2opAo

OJp

_

_

(Z,1 + Lp) 2' OLo

OJp_

OJ 1 _

LpLopAo

0L 1

0L1

(L 1 + Lp) 2

OJ1 - 0 OLo

(11.54)

Substituting Eqs. (11.53) and (11.54) in Eq. (11.51), we have *~o

--

1, *~b;

=_

L°P o , (~1 = L 1 + Lp

Lop L 1 + Lp

L 1 + Lp

L 1 + Lp

(11.55)

Here, the coefficients expressing the control by the load process do not carry an asterisk superscript, since they are not overall control coefficients. Multiplying the nonnormalized control coefficients with the ratio of flows Jo/Jp yields the normalized coefficients *CJ° , which satisfy the usual summation conditions (Schuster and Westerhoff, 1999) *CoJ° --1, *CJ° =

2L2opLI(Ll + L p ) , *C -oJp = 0 , Lop - Lo(L1 +Lp)

1 -*C' -Jp

Lp L 1 +Lp

Since Jp = - J b we have the summation condition: *CJ° +*CpJ° +*C1J° = 1. Metabolic systems usually consist of a number of functional units and metabolic pathways. Modular control analyses are developed to streamline the analysis of control and regulation of metabolic systems. The slipping enzyme may be considered a module catalyzing two reactions of exergonic and endergonic processes, providing a biological energy transudation. Control coefficients related to slipping enzymes can be calculated by the linear nonequilibrium thermodynamics approach. The overall control coefficients in the modular approach describe the control exerted by the particular degrees of freedom of a module on the measurable variables at steady state. Using the degree of coupling q (qop = Lop/x/LpLo) and the phenomenological stoichiometry Z (Z = k/Lp/Lo), the reation flows of Jro and Jrp in Eq. (11.47) become Jro = Lo(Ao + qopZAp)

(11.56)

Jrp = Lo ( qop ZAo +Z2Ap)

(11.57)

Using qop and Z, Eq. (11.48) becomes (11.58)

- q°pZL°A°

Ap =

+ LoZ

As before, we can substitute Eq. (11.58) in Eqs. (11.56) and (11.57), and with Jo = Jro and Jp

[ 22 /

Jo =

qop Z Lo 1 - ~ Z - - ~ L ° LoAo

-

Jrp we get (11.59)

561

11.4 Properpathways

Jp - [ 1 -

Z 2Lo

| qopZLo Ao

(11.60)

L1+ Z2Lo )

The control coefficients in terms of q and Z are obtained from the following matrix definition

* o *aSO) *

bo p

-r-p

_

Oj ° Oq OJp Oq

Oj° OZ OJp OZ

OJro OJro Oq OZ OJrp OJrp Oq OZ

-

1

(11.61)

The partial derivatives are obtained from Eqs. (11.56), (11.57), (11.59), and (11.60), and the matrix in Eq. (11.61) yields the control coefficients qopAo + 2 ZAp *6o°=

*~b; =

qopL1"do

ZAp ( + Loz2 ) qop t qopL1 Z Z( L~ + LoZ2 ) (11.62)

• 6op -

Jp(qopAo + 2ZAp) 2qopZ 2Log L 1 ( L o Z 2 _ LI )

2Lo Zz(LI + LoZ2)

(Lo Z2 - L 1 ) L 1 A ° ZZ2LoA + Lo z 2 ) jp

2Z 2LoAp

The metabolic control analysis can be used to study diseases caused by enzyme functions or dysfunctions, and helps us to understand certain pathways. It may be critical in determining the enzymes with the highest flow control coefficient, in order to inhibit or control enzyme functions. This may lead to the quantification of rate limitations in complex enzyme systems.

11.4.2 Complex Systems in Cell Biology Biological systems often reach stationary states, which may have certain characteristic properties, such as being robust when subjected to internal and external fluctuations, and displaying adoptive behavior. The free energy released by the hydrolysis of ATP is also utilized to achieve structuring by lowering entropy as a result of coupling. Biochemical network systems are complex due to the occurrence of multiple branches and cycles. These networks operate with multiple enzymes that sequentially convert different substrates into various products, and are complicated by regulatory interactions including feedback and feed-forward loops, which can be both activating and inhibiting. Biological cells function at the level of (macro)molecules. The cells are composed of interacting low-molecular-weight molecules (metabolites, such as lactate and pyruvate) and macromolecules (enzymes, protein complexes, DNA, and mRNA). Metabolites include substrates, inhibitors, activators, and products. The magnitudes of metabolite concentrations describe the state of a biochemical network at a given time. The cell has a compartmental structure surrounded by a semipermeable lipid-containing membrane, and is composed of networks of interacting microprocesses. The proteins interact either through direct physical interactions or through the binding of metabolites. The amino acid sequence of the proteins is coded by structural genes on the DNA. The genes are transcribed by RNA polymerases into mRNA strands under specific and regulated conditions, and the transcripts are translated into proteins by ribosomes. These macromolecular phenomena underlie cell behavior. We may model complex systems by top-down or bottom-up approaches. In the top-down approach, we describe the components from the systemic behavior of the actual system. For example, from the flow balance analysis in a steadystate bacterium, we learn the input and output flows, topology of the network, and the rates of many metabolic reactions.

562

11.

Thermodynamics and biological systems

On the other hand, the bottom-up approach describes the system's behavior using information about the properties of parts. We rely on the properties of isolated parts. Using kinetic modeling and measuring the parameters that characterize the rates, we can determine the capacities of the parts. The behavior of the whole system is in part a function of these properties. There are two kinds of properties that characterize the parts: (i) intrinsic properties, which are determined by the part itself, such as mass, or the amino acid sequence of a protein, and (ii) relational properties, which are determined not only by individual parts but also by one or more other parts, such as dissociation constants. In complex biochemical systems, aggregative system properties, such as the mass of a bacterium, are a function of only the intrinsic properties of the parts. However, the flow through a biochemical pathway is a nonlinear function of the concentrations of its constituent enzymes. Therefore, the flow is not an aggregative property. To characterize the system, in terms of state-independent properties, we need to impose initial and boundary conditions, as well as concentrations of nutrients, enzymes, metabolites, mRNA, temperature, and pressure. The statedependent properties include rates of free energy dissipation, rates of heat production, nutrient uptake flows, and growth rates. "System biology" requires quantitative predictions on the degree of coupling, metabolic consequences of gene deletion, attenuation, and overexpression.

Example 11.9 A linear pathway Consider a linear metabolic pathway composed of five consecutive reversible reactions (Boogerd et al., 2005) where each reaction is catalyzed by an enzyme E

Xo < 1

)X 1 < 3 )X 2 < 3 )X 3 < 4 )X 4 ( 5 )X 5

A_

(11.63)

i

Metabolite X4 inhibits the rate of enzyme 1. The metabolites X0 and X5 are maintained constant at all times. The kinetics model for this linear pathway yields

d& dt

= Jrl - Jr2

dX 2 dt

= Jr2 - Jr3

(11.64)

dX 3 dt

= Jr3 - Jr4

dX4 -

Jr4 - Jr5

dt

where Jri is the rate of reaction i as the number of product molecules formed per unit time per unit volume. Enzyme 1 E1 has two binding sites" a substrate (product) binding site for Xo or X1 and an allosteric binding site for X4, and Jr1 is a function of the concentrations of Xo, X1 and X4

Jrl

=

1(Jrlf(yO/Kl,yo)-Jrlb(X1/Kl,y,)) 1 + (X4/KI,x4)

(11.65)

1 + (Xo/Kl,xo) + (X1/KI,x,)

The first term on the right represents the inhibition effect, while the second term is the net reaction excluding the inhibition. Kl,Xl is the equilibrium dissociation constant (in mM), and indicates the ratio (E 1 - X1)/(E1X1) , when the binding relaxes to equilibrium. Here, Xi stands for X0, X1, or X4, and (El - X1) is the concentration of the enzyme-substrate X1 complex, and E1X1 is the product of the concentrations of the free enzyme E1 and the free substrate X 1. The reaction velocities Jrlf and Jrlb are the maximal rates of forward and backward rates of catalysis (in mM/min), respectively. As seen from Eq. (11.65), the rate is a nonlinear function of the concentrations of metabolites.

563

11.4 Properpathways

The equilibrium constant of reaction 1 is defined by

/ X') K1-

(11.66)

~ o eq

The equilibrium constant depends only on the properties of the reactants and temperature. The enzyme shortens the time necessary for the reaction to reach equilibrium and does not affect the equilibrium constant. If the actual ratio of product and substrate is X1

then

(11.67)

Q/K1is the extent of displacement from equilibrium. Using the relation

/rlf'lX1/

(11.68)

JrlbKl,xo Equation (11.65) becomes

I

Jr1 =

1 +(X4/KI,x4 )

( Jrlf(X°/KI'x° )(1-(Q/K1))I 1+(Xo/Kl,x,' )+(X1/KI,x, )

(11.69)

The extent of displacement is directly related to the rate Jrl, which vanishes at equilibrium where Q is equal to Keq. The displacement is directly related to the Gibbs free energy difference or the chemical potential difference

/

(11.70)

Modeling the pathway starts with the experimental determination of the rate and the parameters. Equation (11.64) can be integrated to estimate X, at other times.

Example 11.10 Sensitivity of the rate of the enzymatic reaction to substrate concentration Using the reaction J,-1 network in Eq. (11.63), the sensitivity of this enzyme to the substrate X 0 is defined by the elasticity coefficient exo

e'x"-- ~o) ~

- 1--~1 -1-+-(Xo/Kxo)+(X1/Kx,)

(11.71)

J" quantifies the fractional change in the rate of an enzymatic reaction (OJrl/Jrl) upon fractional change in the The ex,, concentrations of any of its substrates (Xo/OXo)or products. We have similar relations for all enzymes with respect to their substrates and products. The first term in Eq. (11.71) shows the change of sensitivity when the systems is approaching thermodynamic equilibrium, while the second term shows the change of sensitivity as the amount of substrate-bound enzyme increases. The elasticity coefficient is a function of the concentrations X0 and X1 as well as the relational properties of enzyme Kxo,Kxl, and the reactants Keq. The displacement from equilibrium [X1/(XoK1)]also affects the elasticity coefficient. As the reaction approaches equilibrium, the first term approaches infinity. Therefore, the values of Kxo and KXl affect the sensitivity of the enzyme to X0 only if the reaction is displaced from equilibrium. For the metabolic pathway in Eq. (11.63), consider the steady-state flow J after changing the concentration of the first enzyme E~. The rate of the first reaction is a linear function of the enzyme concentration and a nonlinear

564

11.

Thermodynamicsand biological systems

function of the concentrations of metabolites X0, XI, and X4. The fractional change in the steady-state rate through the pathway upon a fractional change in E1 is dlnJrl d In E 1

_

OlnJrl + 0~In Jrl d In X 1 t 0 In Jrl d In X 4 = 1 + Jrl CqYl _+. Jrl e xl e x 4 C x4 0 In E 1 d ln X 1 d l n E 1 d l n X 4 d l n E 1

where Cq is the concentration control coefficient. This relationship shows that the effect of the enzyme on its own rate is eE~Jr, = 0 lnJra/O InE 1 = 1 at the start. The effects ofX1 andX4 are shown by the second and third terms, respectively. Changes in component properties describe the changes of a complex system at steady state.

11.4.3

Multiple Inflection Points

A common intermediate metabolite of two enzymes may catalyze consecutive reactions of a pathway, and diffuse from one active center to the other without dissociation. This is called metabolic channeling, which could lead to a decrease in the steady-state concentration of the intermediate metabolite even at constant flux. The existence of a multidimensional inflection point well outside of the global equilibrium in the force-flow space of enzyme-catalyzed reactions may indicate linear behavior between the logarithm of reactant concentrations and enzyme-catalyzed flows (Rothschild et al., 1980). Thus, enzymes, operating near this multidimensional point and leading one to choose particular state variables, may produce some linear coupled biological systems. This range of kinetic linearity may be far from equilibrium. Disregarding the electrical effects, the conditions for the existence of a multidimensional inflection point are (i) each reactant with varying activity influences the transition rates so that only one state remains; (ii) the kinetics of the reaction involving the given reactant are of fixed order with respect to that reactant; and (iii) for various concentrations of reactants, at least a certain cycle is present. The first condition excludes autocatalytic systems; for many biological energy transducers, however, it may well be satisfied. The fact that local asymptotic stability is supported by local symmetry and experimental evidence of the linear behavior of some coupled biological energy-transducing systems suggest that kinetic linearity may lead to thermodynamic linearity and cause a proper pathway to form. Consider an ensemble of enzyme molecules or membrane proteins in the coupled processes of reactions and vectorial flows. Such systems consist of a set of cycles and subcycles of reactions and transport processes. For a flow in cycle k as Jk (k = a, b,..., h), the first two steady-state flows are given by the following relations

Jl = Ja + Jb + J f

(11.72)

J2 = Ja + Jb + Jc + Jg

(11.73)

Expanding these two flows as functions of their conjugate forces in a Taylor series about some reference steady state, and assuming all other forces as constant, yields the finite differences from the first-order terms

•Ji =

axe

¢~Xi -~-

]

OnE k(j) Jk aJj = OXi 6X i

OXj

OXj

I

6Xj +...

(11.74)

6Xj +...

(11.75)

where k(i) and k(j) show the sets of cycles associated with Ji and Jj, respectively. The expansion of flows in a Taylor series about a multidimensional inflection point yields expressions linear in ln(ci) and ln(cj) if changing concentrations of components i andj are the reactant concentrations. In Eqs. (11.74) and (11.75) the proper conjugate forces, X~ and Xj, appear explicitly. The reference state may not be an inflection point except with respect to ci and cj. Caplan and Essig (1989) provided a simple model of active ion transport, having properties consistent with a multidimensional inflection point when one of the variables was the electrical potential difference across the membrane. A multiple inflection point may not be unique; other conditions may exist where flows ,/1 and J2 simultaneously pass through an inflection point of on variation of X1 at constant X2, and vice versa. It is frequently not possible to vary both

565

11.4 Properpathways

forces independently in biological systems. However, if X1 can be controlled experimentally along a proper pathway, while X2 is kept constant, the response of the flows to change in Xl will permit a thermodynamic characterization of the system. Stucki (1980) demonstrated that in mitochondria, varying of the phosphate potential, while maintaining a constant oxidation potential, yields linear flow-force relationships. Extensive ranges of linearity are found in active sodium transport in epithelial membranes, where the sodium pump operates close to a stationary state with zero net flow. In the vicinity of such a stationary state, kinetic linearity to a limited extent simulates thermodynamic linearity at the multidimensional inflection point. There may be a physiological advantage in the near linearity for a highly coupled energy transducer, since local asymptotic stability is guaranteed by these conditions. Also, the thermodynamic regulation (buffering) of enzymes may be interpreted as an indication that intrinsic linearity would have an energetic advantage and may have emerged as a consequence of evolution.

11.4.4

Reaction Diffusion in Biofilms

Many biochemical signaling processes involve the coupled reaction diffusion of two or more substrates. Metabolic biochemical pathways are mainly multicomponent reaction cycles leading to binding and/or signaling and are coupled to the transport of substrates. A reaction-diffusion model can also describe the diffusion of certain proteins along the bacterium and their transfer between the cytoplasmic membrane and cytoplasm, and the generation of protein oscillation along the bacterium (Wood and Whitaker, 2000). Three dominant processes in the reaction diffusion in biofilms and cellular systems are (1) diffusion in a continuous extracellular phase B, (2) transport of solutes across the membrane, and (3) diffusion and reaction in the intracellular phase A. Consider aerobic growth on a single carbon source. The volume-averaged equations of a substrate S and oxygen O (electron acceptor) transport are

0[Cs(eB +c~ leA)]

= V. (Ds, ~ .Vc s ) _ Jrs,maxe A

at

O[Co(eB + K~qle/, )] at

Cs

(11.76)

c s + chK s Co + KeqK o

Cs

-- ~7 • (I)o,e " ~7Co ) -- Jro,max~A

Co

Co

-- ~:A r i o K e q l c o

(11.77)

c s + ol 1K S c o + Keq K O

where rio is the effective oxygen uptake parameter in m3/(kg s). The concentrations above are equilibrium weighted, and defined by CO = Keq (CO,av,A) -- (CO,av,B)

with a volume average concentration of 1

!CO,av, ) = 7 f

dV

Here, V is the averaging volume and VB(t) is the volume of phase B within the averaging volume. The parameters SA and SB are the volume fractions of the phases A (SA = VA/V) and B (SB = VB/V), respectively. The parameter Di,e is the effective diffusivity tensor of component i, J,,max,~ is the maximum reaction rate, and rio is the first-order endogenous respiration rate coefficient. The cell mass per unit volume of biofilm is defined by 1

f p dv

Under thermodynamic equilibrium, the average concentrations of the substrate are related by (CS,av,B) = Oll(CS,av,A ) and (CO,av,B)= Keq (CO,av,A)

where c~1 is the equilibrium coefficient and Keqis the equilibrium coefficient of oxygen portioning between extracellular phase B and intracellular phase A. Even under nonequilibrium condition, the concentrations are close enough, and the equations above can be reasonable approximations. At local mass equilibrium, the solute concentrations on either side of the cell membrane are equal and the parameter a l is equal to 1.

566

11.

Thermodynamicsand biological systems

The oxygen and substrate transport processes are similar. However, through the cell membrane, the flow of oxygen follows the diffusion or permeability model, while the solute flow involves complex transport mechanisms.

11.4.5

Effective Diffusivity of Cellular Systems

The effective diffusivities in Eqs. (11.76) and (11.77) are defined by

DS'e:(eBDsB+eA0/llDsA)I+DSB I nBAbsBda+DSA f V V aBA ( t )

D°'e =(SBDoB

nABbsAda

(11.78)

aAB ( t )

-4-e'AKeqlDoA)I+D°B f nBAboBda+D°A I V V aBA(t )

nABbOA

da

(11.79)

aAB(t)

where I is the unit tensor, nBA = --nAB is the unit normal vector directed from phase B toward phase A, and bsB is the vector field that maps VCs onto the spatial deviation concentration for substrate S in phase B, while bsA is the vector field that maps Vcs onto the spatial deviation concentration for substrate S in phase A times 0/1. The term aAB is the interfacial area contained within the averaging volume. The definitions of effective diffusivity tensors are key parameters in the solution of the transport equations above. For an isotropic medium, the effective diffusivity is insensitive to the detailed geometric structure, and the volume fraction of the phases A and B influences the effective diffusivity. When the resistance to mass transfer across the cell membrane is negligible, the isotropic effective diffusivity, Ds,e = D~,eI may be obtained from Maxwell's equation D~,e _ 3Ks - 2eB(Ks -- 1) DSB 3+eB(K S --1)

(11.80)

where DsB is the mixture diffusivity of S in phase B, and the dimensionless parameter Ks is KS--

DSA

DSB0/1

Maxwell's solution for permeable spheres with 0/1 = 1 is defined by Ds,e _ 2DsB + DSA -- 2e A (DsB -- DSA) DSB

(11.81)

2DsB + DSA -- e A (DsB -- DSA )

Maxwell's solution for impermeable spheres with Ks = 0 is defined by DS,e_ DSB

2~:B

(11.82)

3 - ~3B

These equations are applied to biofilms and cellular systems. If mass transfer across the cell membrane is important, then the following equation is used D's, e _ 2Ks - 2eB(Ks -- 1) -4-28B~3A1/3 (47r/3) 1/3(ys//) 1/3 1/3

DSB

3 + eB (KS --1) + (3 -- eB )e a

(47r/3)

(11.83)

(YS/1)

where 0/2 -+-0/3CSB,av+ 0/4CSA,av nt-0/C 5 SB,avCSA,av TS -- DSA

0/1Eo

The parameter 1is the characteristic length for a unit cell, E0 is the surface concentration of a carrier protein molecule, and c~2,c~3,cq, c~s are the reaction rate parameters analogous to that half saturation constants. Table 11.3 displays the experimental effective diffusion coefficients and the volume fraction of intracellular phase A. In the first four sets

11.5

567

Coupling in mitochondria

Table 11.3 Experimental data of effective diffusivity System

Cell type

Solute

e

D a,e/D A~

Source

Gel immobilized cells

Mammalian cells (ascites tumor)

Glucose

0.980 0.705 0.590

1.05 0.76 0.64

Chresand et al. (1988)

Fermentation media

S. cerevisiae

Oxygen

0.980 0.961 O.922 0.883 0.844 0.822 0.776 0.700

0.990 0.957 0.895 0.900 0.890 0.848 0.805 0.748

Ho and Ju (1988)

Biofilm (natural) on a hollow fiber filter support

E. coli

Nitrous oxide

0.860 0.660 0.640 0.270 0.080 0.040

1.00 0.62 0.65 0.37 0.29 0.28

Libicki et al. (1988)

Biofilm (artificial) on a hollow fiber filter support

E. coli

Nitrous oxide

0.940 0.910 0.835 0.825 0.815 0.810 0.800 0.690 0.630 0.500 0.470 0.400

1.00 0.85 0.77 0.79 0.75 0.74 0.84 0.64 0.61 0.49 0.51 0.50

Libicki et al. (1988)

Gel immobilized cells

S. cerevisiae

Lactose

0.882 0.879 0.770 0.650 0.530

0.96 0.82 0.69 0.62 0.42

Axelsson and Persson (1988)

Gel immobilized cells

S. cerevisiae

Ethanol

0.882 0.770 0.650 0.530

0.88 0.70 0.47 0.36

Axelsson and Persson (1988)

Gel immobilized cells

Mammalian cells (ascites tumor)

Lactate

0.980 0.805 0.705 0.580

0.96 0.76 0.56 0.47

Chresand et al. (1988)

o f data, the substrate was transported inside the cells (B --+ A); the r e m a i n i n g three sets represent the substrate being transported from the cells (A --+ B). The experimental data was satisfactorily r e p r e s e n t e d by Eqs. ( 1 1 . 8 1 ) - ( 1 1 . 8 3 ) , especially for high values o f eB.

11.5

C O U P L I N G IN M I T O C H O N D R I A

A two-flow coupling implies an interrelation b e t w e e n flow i and flow j, so that a flow occurs w i t h o u t a force or against its conjugate driving force. The e n e r g y level o f a reactant m a y c h a n g e due to the c o u p l i n g effect, while the catalyst effect m a y be limited to the lowering o f the reaction barrier for both the forward and b a c k w a r d reactions (Jin and Bethke, 2002). M a n y biological reactions can take place against their o w n affinities b e c a u s e o f the t h e r m o d y n a m i c coupling effect. For e x a m p l e , m a n y transport systems in bacteria are driven by the proton gradient across

568

11.

Thermodynamicsand biological systems

the plasma membrane. The chemiosmotic theory also describes the role of proton flows in bacterial bioenergetics. Proton and lactose are cotransported into the cell by lactose permease. At the same time, protons are transported out of the cell in connection with electron flow through the respiratory chain. Overall, the cell maintains a nonequilibrium level of pH by keeping its interior at a higher pH than its environment. Eucaryotic cells posses a hierarchy of transport systems to maintain nonequilibrium concentration levels of some substrates within organelles than those in the cell's cytoplasm. Still, the cell controls its complex array of chemical reaction cycles so that the supply and demand for substrates, energy, and electrons are balanced and resources are utilized efficiently. In early experimental work, the interior and exterior cell concentrations of K +, Na+, and C1- ions were measured and compared against those obtained from the Nernst potentials, given by

~J =

z - ~ lnt, aj, o )

(11.84)

where ~j is the equilibrium potential, zj the valence, and aj the activity of speciesj on both sides of the membrane (i and o). The measured potentials were different from the Nernst potentials, indicating that a cell maintained the concentration difference at a steady-state diffusion flow. The basis of active transport in animals is the coupled metabolic reaction to external diffusion, while most of the chloride flow in plant cells depends on photosynthesis. One of the conventional methods for establishing the existence of active transport is to analyze the effects of metabolic inhibitors. The second is to correlate the level or rate of metabolism with the extent of ion flow or the concentration ratio between the interior and exterior of cells. The third is to measure the current needed in a short-circuited system having similar solutions on each side of the membrane; the measured flows contribute to the short-circuited current. Any net flows detected should be due to active transport, since the electrochemical gradients of all ions are zero (A~ - 0, Co = ci). Experiments indicate that the level of sodium ions within the cells is low in comparison with potassium ions. The generalized force of chemical affinity shows the distance from equilibrium of the ith reaction X i =

RTln Ki'eq m

I-I c~..~, j=l

(11.85)

where R is the gas constant, K is the equilibrium constant, cj is the concentration of the jth chemical species, and Pji are the stoichiometric coefficients, negative for reactants and positive for products, for the ith reaction. The phosphate potential in mitochondria is expressed as

Xp--AGp-RTln

[ATP] ) [ADP][Pi ]

(11.86)

Stucki (1980, 1984) applied the linear nonequilibrium thermodynamics theory to oxidative phosphorylation within the practical range of phosphate potentials. The nonvanishing cross-phenomenological coefficients Lo.(i 4=j) reflect the coupling effect. This approach enables one to assess the oxidative phosphorylation with H+pumps as a process driven by respiration by assuming the steady-state transport of ions. A set of representative linear phenomenological relations are given by J1 = LalX1 +La2X2

(11.87)

J2 = I-azX1 + L22X2

(11.88)

where J1 is the net flow of ATE J2 is the net flow of oxygen, X1 is the phosphate potential as given by Eq. (11.86), and X2 is the redox potential, which is the difference in redox potentials between electron-accepting and electronreleasing redox couples.

11.5.1

Degree of Coupling in Oxidative Phosphorylation

The degree of coupling is defined as q:

L12 (Zl 1L22) 1/2

0<[q[
(11.89)

11.5 Couplingin mitochondria

569

and it indicates the extent of overall coupling for the various individual degrees of coupling of the different reactions driven by respiration in the mitochondria. With an angle o~ whose sine is q we have c~ - arcsin(q)

(11.90)

By defining the phenomenological stoichiometry Z (11.91)

L22 and by dividing Eq. (11.87) by Eq. (11.88), we can determine that the reduced flow ratioj = reduced force ratio x = X1Z/X2 and the degree of coupling J _

x + q

J1/(J2Z) varies with the

(11.92)

qx+l Assuming that oxidation drives the phosphorylation process, then Xm < 0 and X2 > 0, and Jl/J2 is the conventional P/O ratio, while )(1/)(2 is the ratio of phosphate potential to the applied redox potential. The following relations are from Stucki (1980). At static head (sh), analogous to an open circuited cell, the net rate of ATP vanishes, and the rate of oxygen consumption and the force (the phosphate potential) are expressed in terms of a as follows

(,J2)sh = L22X2 c0S2 a (Xj)sh - -

X2 sin~

(11.93) (11.94)

Z where L22 is the phenomenological conductance coefficient of the respiratory chain. Therefore, energy is still converted and consumed by the mitochondria. The nonequilibrium phosphate value is distant from the equilibrium value 0(1 = -X2 Z) by a factor of q, and is given by

Y2 (Xl)sh -(Xl)eq = - ~ ( 1 - q )

(11.95)

The dissipation at the static head can be obtained from ~ h - Lll( tan2

a)-l(X1)s2h

(11.96)

This equation shows that the energy needed at the static head is a quadratic function of the phosphate potential. At level flow (lf), analogous to a short-circuited cell, the phosphate potential vanishes. Hence, no net work is performed by the mitochondria, and we have

J1

(11.97)

This equation shows the maximal P/O ratio measurable in mitochondria at a zero phosphate potential. Equation (11.97) also indicates that at level flow, the flow ratio does not yield the phenomenological stoichiometry Z but approaches this value within a factor of q. Therefore, if the degree of coupling q is known, it is then possible to calculate Z from the P/O measurements in a closed-circuited cell. Obviously, in state 3, the phosphate potential is not zero, however, for values of q approaching unity, the dependence of the flow ratio on the force ratio is weak, according to Eq. (11.92). Therefore, state 3 is only an approximation of the level flow at values of q close to unity, and the dissipation function to maintain a level flow is given by q~lf - (Jj)~f 2 q Lll

(11.98)

570

11.5.2

11.

Thermodynamicsand biological systems

Efficiency of Energy Conversion

The efficiency of energy conversion as a ratio of output energy to input energy is related to the degree of coupling as follows r/ .

.

J1X1 _

.

.

x +q

.

JzX2

.

(11.99)

q + (1/x)

The efficiency reaches a maximum value between the static head and the level flow, which is the function of the degree of coupling only, and expressed by

r/°pt

[1 + X/1- q2 ]2 = tan2

(11.100)

tan

(11 101)

The value of x at Tlmaxis given by

Xop t .

.

.

.

l+x/l_q2

The dissipation function ~ can be expressed in terms of force ratio x and degree of coupling q xlt = (X 2 + 2qx + 1)L22 X2

(11.102)

If we assume X2 as constant, the dissipation function is at a minimum at the static head force ratio Xsh = - q

(11.103)

Within the region of validity of linear phenomenological equations, the theorem of minimal entropy generation at steady state is a general stability criterion. The static head is the natural steady state where the net ATP flow vanishes and a minimum of • occurs along the loci of the static-head states XI/'sh = COS2 0lL22X2

The dissipation at the state of optimal efficiency is obtained using

Xopt

COS20L L22 X 2 XItopt = 2 ~ 1+ cos a

(11.104) in Eq. (11.102)

(11.105

)

The dissipation for the level flow is given by XIrlf = L22 X2

(11.106)

Without a load, Stucki (1980) suggested the following order Xltsh < xI/'opt < XI/'lf

(11.107)

This inequality means that the minimum dissipation and the natural steady state do not imply the optimal efficiency of oxidative phosphorylation. At the level flow, there is a load and hence a load conductance corresponding to the state of optimal efficiency between the static head and the level flow. The dissipation of oxidative phosphorylation with a coupled process (load) utilizing ATP is given as follows * c = Jl & + J2 X2 + J3 X3

(11.108)

Assuming that the ATP-utilizing processes are driven by the phosphate potential, X3 = X1, and a linear relation between the net rate of ATP utilization and X1, we have

11.5

571

Coupling in mitochondria

J3 = L33X1

(11.109)

Here, L33 is the phenomenological conductance of the load, and the dissipation function in terms of the force ratio x becomes ~c-

L33/

x" l + ~ l

+2qx+l

1L22

(11.110)

Only if the following equation is satisfied L33 - X / 1 - q 2 - cosoL LI l

(11.111)

then Eq. (11.110) is minimal at Xopt. For mitochondria, L33 is an overall phenomenological coefficient lumping together all the conductances of ATPutilizing processes, while Lll shows the conductance of phosphorylation. If these two coefficients match according to Eq. (11.111), then the natural steady state of oxidative phosphorylation is at the optimal efficiency. Stucki called Eq. (11.111) the condition of conductance matching of oxidative phosphorylation, and presented an experimental verification.

11.5.3

Dissipation with Conductance Matching

Dissipation with conductance matching at the static head is given by (q~rc)sh -- ( cOSOI + cOS20l-- COS3 ~)L22222

(11.112)

The dissipation function at the state of optimal efficiency of oxidative phosphorylation is (XIfc)opt -- COS°LL22X2

(11.113)

At the conductance matching state, the dissipation Eq. (11.110) is minimum at the loci of the optimal efficiency states. At the optimal efficiency, the P/O ratio is given by (11.114) "~2 opt

~2-2 If l+cOsa

Equation (11.114) shows that unless q = 1, a maximal P/O ratio is incompatible with the optimal efficiency, and we have the inequality 1

Jl

()opt

(11.115)

Therefore, the low P/O ratios do not necessarily mean a poor performance of the oxidative phosphorylation. Similarly, the net rate of ATP synthesis at the optimal efficiency is given by (')¢1)opt ~---(J1)lf

cos a 1+ cos a

(11.116)

with the boundaries 1

0 < (Jl)opt < 2(J1 )If

(11.117)

This inequality means that a maximal net rate of ATP production is incompatible with the optimal efficiency. Cellular pathways balance the rate and efficiency of ATP production with respect to the energy needs of the cell. For example, heart and brain mitochondrial systems utilize more oxygen and produce ATP at a faster rate than the

572 .

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.

.

.

.

.

.

.

.

.

.

.

.

11.

.

Thermodynamicsand biological systems

systems in the liver. However, liver mitochondria can produce ATP more efficiently based on a higher P/O ratio and a higher degree of coupling in oxidative phosphorylation.

11.5.4

Variation of Coupling

The degree of coupling depends on the nature of the output required from the energy conversion system in the mitochondria and on metabolic regulation and stability. If the system cannot cope with instabilities, then fluctuations, such as in pH and the pressure of the blood, could irreversibly harm the organism. For an optimal efficiency state, the condition of conductance matching must also be satisfied. Experiments with livers perfused at a metabolic resting state suggest that the conductance matching is satisfied over a time average, and the degree of coupling qec yields an economical oxidative phosphorylation process. Optimization may be based on constraints other than the efficiency, such as the production of thermal energy in the mitochondria, which requires low degrees of coupling. The degrees of coupling are also measured in Na÷transport in epithelial cells, and in growing bacteria where the maximization of net flows are the most important considerations for the system. On the other hand, for a fed rat liver at a metabolic resting state, an economical power output is the priority, while a starved rat liver has to produce glucose, and the maximum ATP production is given priority over energy conservation. In the heart and brain, the experimental value of q for the cellular respiration pathway is close to the value of q~C = 0.953, which suggests that the pathway is optimized to economical ATP for cellular processes. In the brain, the coupling of the acetic acid cycle approaches qp = 0.91, suggesting a maximized cellular energy state. However, in the heart, the acetic acid cycle coupling is 0.786, which is between qp and qf, and consistent with the maximum ATP production necessary for preserving the cellular energy state (Stucki, 1980; Cairns et al., 1998). Optimal flow ratios are also a characteristic of oxidative phosphorylation, and may provide additional information on the relationships between the respiratory response and energy demand stimulation by ADP. Most metabolic processes in living cells are dynamic systems, and the behavior of flows may better reflect complex system mechanisms than do the models dependent on end-point measurements. For example, the ratio of ADP/O describes the state of the end-point capacity of oxidative phosphorylation based on the input flow of ADR Figure 11.3 shows the effect of the degree of coupling on the characteristics of four different output functions f given by f = tanm ( 2 ) c o s ( c 0

0.35

i .........

I

. . . .

I ......

I

.....

I

....

I

(m = 1-4)

I

'

'

I

(11.118)

'

I'

''

I

qf=0.786 0.3

0.25

_

(J1)opt--~ 0.2

0.15

_

q~0=0.953 0.1

0.05

(JlXlq)opt 0

0

0.1

0,2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Degree of Coupling,q

Figure 11.3. Effect of the degree coupling q on the output functions f (Eq. (11.118)) (Demirel and Sandier, 2002).

11.5 Coupling in mitochondria

573

The optimum production functions and the associated constants are described in Table 11.4. The followings are the output functions seen in Figure 11.3: (i)

If the system has to maximize ATP production at optimal efficiency, then f = (J1)opt and the degree of coupling

q f = 0.786. (ii) Conversely, if the system has to maximize power output at optimal efficiency, we have the output function f = (J1X1)opt occurring at the degree of coupling qp = 0.91. (iii) If an efficient ATP synthesis at minimal energy cost is imposed, then the function for economical ATP occurs at the degree of coupling q~C __ 0.953. (iv) Economical power output occurs at the degree of coupling q~C _ 0.972. The optimum output power (J1Xl)opt and the efficiency (J1Xlqq)opt are calculated from the plots of J1X1 vs. x and J1XI~q vs. x, respectively. A transition from qp to q~C causes a 12% drop in output power (J1X1) and a 51% increase in

efficiency. For a favorable ATP production at optimal efficiency of oxidative phosphorylation, we should have q < 1. With the consideration of conductance matching, Stucki (1980) determined four production functions, which are given in Table 11.4. Stucki (1980) analyzed the sensitivity of the force (the phosphate potential) to the fluctuating cellular ATP utilization, and found that the sensitivity is minimal at q - 0.95. This analysis indicates that the phosphate potential is highly buffered with respect to the changing energy demand to maximize kinetic stability and thermodynamic efficiency at the same degree of coupling. For H+-translocating ATPase, the H+/ATP coupling ratio is important for mechanistic, energetic, and kinetic consequences, and a value of 4 has been adopted for the ratio; the standard reaction Gibbs energy of ATP production is obtained as 31.3 kJ/mol at T - 20°C, pH 8.0, pMg 2.5, and 0.08 M ionic strength, and the standard enthalpy of the reaction is 28.1 kJ/mol. 'The differences in the rates of proton pumping, ATPase activities, and degrees of coupling are adjusted by each biological system in order to survive and compete in its environment. Cairns et al. (1998) reported the experimental mean degree of coupling for isolated liver mitochondria as 0.955, which is close to the value of 0.952 found by Soboll and Stucki (1985) using isolated perfused whole livers from fasted rats. Under similar cytoplasmic ATP levels, the ATP utilization for muscular contraction and the ion transport would be much higher in a beating heart than an arrested heart. ADP supply to the mitochondria of the beating heart would lead to higher rates of oxidative phosphorylation and ATP production.

Table 11.4 Production functions with the consideration of conductance matching

Production function

Loci of the optimal efficiency states

1. Optimum rate of ATP production:

From the plot of Jl vs. x:

ZL?gX9 -" -

(11 "123)

_

_

L~X~ .

qp = 0.910

No

(JiX1)op~=tan 2 ~ coso~L22X_~ (11.122)

oz=65.53 °

r/ = constant

From the plot ofJ~r/vs, x:

q~C= 0.953

Yes

(JlT~)opl

. )521.1(

.

=tan3(2)cosc~ZLg~X~ . . . . (11 124)

.

/ ¸) (Y

9

(dlXlq),,pt . . = tan4 ~- cos~L~,X; (11.126)

Reprinted with permission from Elsevier, Demirel and Sandler (2002).

o~=72.38 °

q~C= 0.972

From the plot ofJiX~ Xl vs. x:

4. Optimum output power of oxidative phosphorylation at minimal energy cost: J1Xlrl = - x2(X+qqx +1

r/ = constant

/)

3. Optimum rate of ATP production at minimal energy cost: + q+) 2~ Jl rl = - x ( xxq

c~=51.83 °

(11.120)

cosozZL22X2

From the plot of J1Xi vs. x:

(11.121)

JlXl=x(x+q)Le2X~

No

/)

2. Optimum output power of oxidative phosphorylation:

Energy cost

qt.= 0.786

(Jl)opt =tan ~

(11.119)

JI=(q+x)ZL~,X~

q

c~= 76.34°

Yes

574

11.6

11.

Thermodynamicsand biological systems

REGULATION IN BIOENERGETICS

Regulation implies a physiological outcome as a result of manipulating a mitochondrial function, and hence it is different from control. However, many physiological signals to mitochondria cause large changes in activity associated with control, such as in the shift from state 4 to state 3. Mitochondrial function is regulated by a number of factors over time scales. These include calcium stimulating NADH supply to the respiratory chain, or oxidative phosphorylation complex activities, nitric oxide-inhibiting cytochrome oxidase, and thyroid hormones binding to cytochrome oxidase. The physiological reasons for these regulations of mitochondria are mainly to match ATP supply efficiently to changes in workload, modulate thermogenesis, biogenesis or cell death, as well as respond to external stimuli. According to the chemiosmotic theory, the electrochemical proton gradient across the membrane is one of the important mechanisms for regulating the rate of respiration and ATP synthesis. Respiratory control mainly means stimulation ofmitochondrial respiration by ADP and its decrease because of conversion of ADP to ATP. Various substrates regulate the metabolism of energy; fatty acids may regulate and tune the degree of coupling by inducing uncoupling, leading to optimum efficiency of oxidative phosphorylation. Experiments with incubated rat-liver mitochondria show that the adenylate kinase reaction can buffer the phosphate potential to a value suitable for the optimal efficiency of oxidative phosphorylation in the presence of a very high rate of ATP hydrolysis. Stucki called this class of enzymes, such as adenylate kinase and creatine kinase, thermodynamic buffer enzymes. A fluctuating ATP/ADP ratio and deviations from the optimal efficiency of oxidative phosphorylation are largely overcome by thermodynamic buffering. As a terminal component of the respiratory chain, cytochrome oxidase catalyzes the transfer of electrons from cytochrome c to oxygen, which is coupled with proton pumping. Although the reaction catalyzed by cytochrome oxidase is far from equilibrium, it has been commonly assumed that the flow-force relationship is unique and proportional. The reaction becomes more nonlinear in the thermodynamic branch as the oxygen concentration decreases. This is interpreted as the thermodynamic cost of kinetic regulation. When the oxygen concentration decreases, the cell prefers to optimize the flow through cytochrome oxidase rather than the thermodynamic force of the reaction in order to maintain constant oxygen flow and ATP production. The electrochemical potential of protons may be considered as a universal regulatory factor of cytochrome oxidase. The enzymes are capable of causing certain reaction pathways by catalyzing a conversion of a substance or a coupled reaction. For example, on adding nigericin to a membrane, the system reaches a steady state in which the gradients of H+and K +are balanced. On the other hand, if we add valinomycin and protonophore, both the gradients rapidly dissipate. The mitochondrial creatine kinase is a key enzyme of aerobic energy metabolism, and is involved in buffering, transporting, and reducing the transient nature of the system. This can be achieved: (i) by increasing the enzymatic activities in a pathway, (ii) by the metabolic channeling of substrates, and (iii) by damping oscillations of ATP and ADP flows upon sudden changes in the workload.

11.6.1

Uncoupling

Uncoupling proteins are a subgroup of the mitochondrial anion transporter family, and are identified in prokaryotes, plants, and animal cells. Three mammalian uncoupling proteins are called UCP1, UCP2, and UCP3. The proton electrochemical gradient developed across the inner membrane during electron transport of the respiratory chain is used to phosphorylate ADP to ATP by FoF1-ATP synthase, and hence the respiration is coupled to phosphorylation. However, ATP synthesis is matched to cellular ATP utilization for osmotic work of (downhill and uphill) transport, or mechanical work such as muscle contraction and rotation of bacterial flagellum. The uncoupling of the mitochondrial electron transport chain from the phosphorylation of ADP is physiological and optimizes the efficiency and fine tunes the degree of coupling of oxidative phosphorylation, and prevents reactive oxygen species generation by the respiratory chain in the resting state. Uncontrolled production of reactive oxygen molecules can cause the collapse of mitochondrial energy conservation, loss of membrane integrity, and cell death by necrosis. The respiratory chain is a powerful source of reactive-oxygen molecules, which include oxygen-free radicals, hydroxyl radical hydrogen peroxide, and nitric oxide; they are very reactive and able to damage cellular components and macromolecules, and influence programmed cell death or apoptosis. Cells have developed various strategies to dissipate reactive oxygen molecules and remove their oxidation products. Uncoupling proteins are capable of modulating reactive oxygen molecules. Fatty acids facilitate the net transfer of protons from intermembrane space into the mitochondrial matrix, hence lowering the proton electrochemical potential gradient and mediating weak uncoupling. Uncoupling proteins generally facilitate the dissipation of the transmembrane electrochemical potentials of H +or Na+produced by the respiratory chain, and result in an increase in the H+and Na+permeability of the coupling membranes. They provide adaptive

11.6 Regulation in bioenergetics

575

advantages, both to the organism and to individual cells, and also increase vulnerability to necrosis by compromising the mitochondrial membrane potential. Some uncoupling is favorable for the energy-conserving function of cellular respiration. In oxidative phosphorylation, leaks cause a certain uncoupling of two consecutive pumps, such as electron transport and ATP synthase, and may be described as the membrane potential-driven backflow of protons across the bilayer.

11.6.2

Slippages and Leaks

A slip means a decreased proton/electron stoichiometry of proton pumps. Mainly, slippage results when one of two coupled reactions in a cyclic process proceeds without its counterpart, which is also called intrinsic uncoupling. On the microscopic level, individual enzymes cause slippage by either passing a proton without contributing to ATP synthesis, or hydrolyzing ATP without contributing to proton pumping. On the macroscopic level, the measured degree of coupling may be different from the expected coupling, as these microscopic slips are averaged across a population of enzymes. Slippage is an intrinsic property of the enzyme, and hence is related to an enzyme's mechanism and structure. In terms of the thermodynamic energy conversion, a slip may decrease efficiency; it may, however, allow dynamic control and regulation of the enzyme over the varying ranges of the electrochemical gradient of protons and the chemical potential of ATP in equilibrium with ADP and Pi. Mitochondrial energy metabolism may be regulated by the slippage of proton pumping in cytochrome c oxidase at high proton motive force. It is possible that slips have evolved to enhance the function of particular coupling enzymes in particular conditions. In transportation, leaks can be found in the proton-sugar symport in bacteria where a protein mediates the transport of protons and sugar across the membrane, and adding a protonophore, a parallel pathway occurs, causing a leak in the transport. Leaks and slips may affect the metabolic rate. Schuster and Westerhoff (1999) developed a theory for the metabolic control by enzymes that catalyze two or more incompletely coupled reactions. The control by the coupled reactions is distinguished quantitatively from the control by the extent of slippage using the linear nonequilibrium thermodynamics formulations; here the limits of coupling may be an important parameter, and may be obtained as the ratio of coupled-to-uncoupled rates, which is a function of the binding energy of the substrate and the carrier protein. One other concern in an interconnected biological network is the behavior of a subsystem (e.g., glycolysis), which may become unsteady and chaotic, so that the output of this subsystem (e.g., ATP production) is adversely affected, and becomes external noise for other subsystems, causing inhibition and desynchronization.

11.6.3

Nonequilibrium Thermodynamics Model of a Calcium Pump with Slips

Waldeck et al. (1998a, 1998b) presented a nonequilibrium thermodynamics model for the calcium pump shown in Figure 11.4. This section summarizes this model. During the hydrolysis of ATE a variation of the coupling stoichiometries with the electrochemical gradients is an indication of molecular slips. However, the Ca2+and H+membrane-leak

Ca2+

"~

A

H+ ] H+

Ca2+ • .........

ATP

Membrane

H+--l Ca2+-ATPase

; Ca2+

ADP+Pi+H+

Figure 11.4. Schematics of calcium transport with Ca 2 -ATPase liposome, ionophore (A), and leaks. Ionophore A23187 induces uptake of Ca 2÷iOns. Leaks are shown with dashed arrows. "i" is interior (alkaline) and "e" is exterior (acidic). Scalar flow of ATP hydrolysis drives the uphill transport of Ca 2+and H + .

576

11.

Thermodynamicsand biological systems

conductances may also be a function of their respective gradients. Such leaks yield flow-force relations similar to those that are obtained when the chemical pump slips. Hence, one needs to exercise caution when interpreting data of CaZ+-ATPase mediated flows that display a nonlinear dependence on the electrochemical proton gradient A/2u and/or calcium gradients 2~/2Ca. CaZ+-ATPases exist in the plasma membranes of most cells and in the sarcoplasmic reticulum of myocytes, where they pump Ca 2+out of the cytosol and into the lumen, respectively, while simultaneously counterporting H+ions. Ca2+-ATPase requires MgZ+on the side from which CaZ+is pumped. It is generally established that the Ca2+/ATP stoichiometries for the plasma membrane and sarcoplasmic reticulum are 1 and 2, respectively. Using a nonequilibrium thermodynamics model, the extent of slippage in the plasma membrane Ca 2+-ATPase can be estimated from steadystate H +flow measurements. Molecular slips are intrinsic to the ATPase, while membrane leaks are intrinsic to the membrane proper; thus, slips should be distinguished from leaks. Slips and leaks occur in parallel. Molecular slips in the ATPase may be dependent on the thermodynamic force producing the backpressure, in the absence of the other force. The rate of leakage of the coupling ions depends on the magnitude of the particular thermodynamic force operating within the system in the absence of the other force. In an idealized Ca 2+-ATPase liposome without slips, a representative equation for the dissipation of free energy is ~It -- JpAp - J c a A/d, Ca - JHA/.~,H

(11.127)

where Ap is the affinity of the reaction of the ATP-hydrolyzing activity of the enzyme (-AGp). The Ap, A/.LCa , and A/.~ H are the thermodynamic forces in J/mol. The flow Jp (mol/s) is the rate of reaction of ATP hydrolysis, and Jca and Ju are the respective ATPase mediated transmembrane flow rates for calcium and hydrogen ions. ATP hydrolysis is a highly exergonic (downhill) reaction -AGp. The uphill transport processes of inward Ca2+ (negative flow) and outward H + (positive flow) possess forces with negative signs. Assuming that the system is in the vicinity of equilibrium, the linear phenomenological equations with the conjugate flows and forces based on Eq. (11.127) are Jp = LpAp - LpcaA/~ca - LpHA~H

(11.128)

J c a = Lcap Ap - Lca A/2ca - Lca H A/.LH

(11.129)

J n = LHpAp - L n c a A ~ C a -- LHA/~H

(11.130)

Lii and L~j are the straight and cross-coefficients, respectively. By Onsager's reciprocal rules, we have L O.=

Zji.

The

electrochemical potential differences between internal i and external e regions are defined by A/~ca --/~Cai -/'~Cae = RT in Cai + z F 2 ~

(11.131)

Cae A/'~H -- ~Hi -- ~He = -2.3RTApH + FA~

(11.132)

where subscripts i and e denote interior and exterior, respectively. The Gibbs free energy difference is [ATP] ) AGp = AGp - RT In [ADP][Pi ]([H + ]/10 -p" )

(11.133)

where AG~, is the standard Gibbs free energy for the hydrolysis of ATP at pH 7 ( - 30.5 kJ/mol). Here, R, T, and z are the universal gas constant, absolute temperature, and number of Coulombic charge, respectively. The gradients H ÷and Ca 2÷and the transmembrane voltage may counteract the chemical reaction system of ATP hydrolysis; for example, these forces induce backpressure effects. There are two important conditions with respect to calcium pumping; these are the static head and level flow. At the static head, calcium pumping vanishes (Jca = 0), and at the level flow calcium gradient vanishes (A/2ca = 0). At nearequilibrium and static-head conditions, phenomenological stoichiometry Z and the degree of coupling q are

577

11.6 Regulation in bioenergetics

Zsh = Z H p -

z

qsh -- qup

IL~-~p

(11.134)

LHp N/LHLp

(11.135)

The efficiency of energy conversion becomes JHA/~H = jx ~sh -- TIHP = -- JpAp

(11.136)

In terms of Z and q, the efficiency is T/sh --

T/HP =

Zx( Zx + q ) qZx + 1

(11.137)

where j and x are the flow and force ratio, respectively. For Ca 2+-ATPase, maintaining the thermodynamic force is a priority, and the pumps operate close to static head conditions at close to zero efficiency. During a turnover cycle of the pump, nca calcium ions and nn protons are transported across the membrane per mole of ATP molecule. Therefore, nca and nH represent Ca2+/ATP and H+/ATP coupling stoichiometries, respectively, while nH/nca represents the H+/Ca 2+ stoichiometry. Experimental observations show that both A/XCa and A/2 n can inhibit Jp by exerting a backpressure effect on the rate of AYP hydrolysis for the plasma membranes. Therefore, the actual total thermodynamic force Xp may be Xp - Ap - rtCaA/.ZCa -- r/H A/.~H

Also, Onsager's reciprocal relations suggest that Lpc a - Lca p - ncaLp and LpH = LHp = rtHL p. If AG* denotes the measure of the offset from equilibrium of the Jp - AGp space, an effective driving force may be AGp,ef= AG - AG* ( - -Ap,ef) for the uphill transport of H + and Ca 2+ . As a result, the value of coefficient Lp changes in different proportions with respect to A/Xca and A/2 H, and we need to introduce asymmetry coefficients YCa and VH defined by ~(AGp ).xgc ~,

a(AGp)A/x H

~pCa = 8(~/.~Ca)AGp '

~pH -- 8(A~H)AG p ,

~pCa -- ")/Cap

With the consideration of new forces and the asymmetry coefficients, Eqs. (11.128)-(11.130) reduce to

Jp = Lp (,4 p,~r- nc,a'YpcaA/~ca -

(11.138)

n H ~ p H A]..~H )

Jca - r/Ca Lp ( .4 p,ef-- HCa")/CapA/~Ca -- r/H"YpHA/-~H )

(11.139)

JH -- r/H Lp (.4 p,ef- r/Ca"YpCaA~Ca - r/H'YHpA/'~H )

(11.140)

By assuming, for the sake of simplicity, 7Ca - TH -- 1, we may incorporate ATPase slips and the membrane leaks o f H + and Ca 2+ into the definitions ofphenomenological coefficients Lp,s, Lca,1, and LH,b where the subscripts s and 1 refer to slip and leak, respectively, and we have the following flow-force equations in the matrix form Jp ! Jca,t JH,t

-

Lp + Lp, s

--nca Lp

nHLp

nCa Lp

_[(nca )2 Lp + Lca,1 ]

-- nCa nH Lp

A/~Ca

n H Lp

- n Hnca Lp

_[(nil )2 Lp + LH,1]

A~H

(11.141)

where Lp,s is the slip rate coefficient that does not take into account the kinetic properties of molecular slips, while Lp is the strictly mechanistically coupled ATPase rate coefficient. Jca,1 and Jn,1 are the leak flows of Ca 2+ and H+, respectively,

578

11.

Thermodynamicsand biological systems

while the pump-induced flow plus leaks yields the total value of the flows Ca 2+and H +

Jca,t = Jca 4- Jca,1 JH,t -- JH + JH,1 Equation (11.141) is only a simplified representative model of the ATPase without the leaks of bulk ions Na + and K +. However, it can separate ATPase slips from membrane leaks. In a simple application of Eq. (11.141), Waldeck et al. (1998a, 1998b) estimated nH with and without the ionophore A23187 (see Figure 11.10) from steady-state NMR data using the relation nu

_

Ju

_

Jn,e--JH,s +Ju,1

Jp

(11.142)

Jp

An expression of nH measured in the presence of the ionophore A23187 is

nH

_JH_ __ Z _ _ {JH,e--JH,s Jp

/-

nil,Ca

(11.143)

with

/ pHe/

JH,e = - - / ~ e -

6t

~

ApH e

At

/

JH,s -- g/H,sJp where the subscript e denotes external, fie is the buffering capacity of the extraliposomal medium, and nil,Ca is the number of H + counterported inwards per Ca 2+ extruded by A23187 as a result of Ca 2+ pumping. The leak flow of hydrogen can be estimated from an independent measurement of LH,1and ZX/2H,ss,where subscript ss denotes steady state

JH,1 -- LH,lZ~/-~'H,ss-- LH,1(FA~ss - RTApHss)

(11.144)

From Eqs. (11.142) and (11.143), the estimated value is nH= 1.9 _+0.3 (Waldeck et al., 1998a). Using Eq. (11.141), the extent of slippage Lp,s/L p can be estimated in the plasma membrane CaZ+-ATPase. This is done close to the static head without the ionophore (-), and close to the level flow with the ionophore (+). Using the equation for Jp, the control ratio of the ATPase is obtained from

Jp,+ Lp )

(11.145)

Estimated approximate values of Ap,_ and Ap,+ are 61 and 52kJ/mol, respectively (Waldeck et al., 1998b). Thus, Lp,s/L p <- 0.4, which means for every five coupled turnovers leading to Ca 2+ and H + translocation, there may be as

many as two uncoupled ones or slips. In the case of slips, the energy associated with the hydrolysis of ATP would be dissipated as heat. 11.6.4

Potassium Channels

A class of cardiac potassium channels operates in smooth and skeletal muscle, brain, and pancreatic cells. Potassium channels are activated when intracellular ATP levels decrease, and are an important link between the cellular excitability and the metabolic status of the cell. The ratio of ATP/ADE pH, lactate, and divalent cations determines and modulates the channel activity. The opening of the potassium channels leads to membrane hyperpolarization and a potential decrease as the potassium ions flow out of the cell. Since phosphorylation changes the activity of potassium channels, it modulates cellular excitability.

579

11.6 Regulation in bioenergetics

Potassium channels play an important role in the control of insulin secretion in [3-pancreatic cells. In a resting [3-pancreatic cell, the membrane potential is maintained below the threshold for insulin secretion by an effiux of potassium ions through the open potassium channels. As glucose levels rise within the cell, ATP production increases, and the change in the APT/ADP ratio leads to the closing of potassium channels. This causes the calcium channels to open, and entering calcium ions signal insulin secretion. Factors that modulate the [3-pancreatic potassium channels can fine-tune the insulin secretion.

11.6.5

Aging and Biochemical Cycle Deficiencies

Aging is mainly characterized by a general decline in mitochondrial function and damage to the oxidative chain. Various models and theories on aging are based on the study of aging in a population, an organism, or a single cell, and a large number of parameters changing with aging. The study of aging may produce the survival curve of a genetically uniform population in a controlled environment. Many studies established a correlation between aging and the accumulation of reactive oxygen species-modified molecules such as lipofuscins in various organisms from fungi to humans. Adducts caused by lipid peroximation also accumulate, and may damage DNA and proteins, causing loss o f - S H groups and protein carbonylation. Manipulating the expressions of genes encoding reactive oxygen species scavenging enzymes may help determine the importance of mitochondrial oxidative stress in aging. It is established that mitochondrial genomes accumulate alterations, deletions, rearrangements, or point mutations with age in humans, monkeys, and rats. Reactive oxygen species are produced by the respiration cycle and metabolic activity. Quantifying reactive oxygen species is difficult because of their short life span, low concentrations, and the existence of cellular scavenging systems (Dufour and Larsson, 2004). Mitochondrial uncoupling keeps the proton potential at a lower level and reduces the production of reactive oxygen species. Hence, the regulation of reactive oxygen species production depends on the degree of coupling of oxidative phosphorylation and the efficiency of energy conversion (Lionetti et al., 2004). Reactive oxygen species production can damage the respiratory cycle proteins and may lead to even more production of reactive oxygen species. At least 1% of the total mitochondrial respiration using pyruvate leads to reactive oxygen species and hydrogen peroxide (H202) production. The main sites for producing reactive oxygen species are in the electron transfer chain.

Example 11.11 Approximate analysis of transport processes in a biological cell A typical biological cell and its surroundings are characterized by the following concentrations (Garby and Larsen, 1995) Components

Outer (o) (mmol/L) Inner (i) (mmol/L)

K+

Na+

C1-

HCO3-

P-

3.7 139

145 12

118 -v4

24 -0.8

-1 136

In this table, P- represents anions of protein and organic phosphate. The membrane is permeable to the group represented by P-. The mean values ofthe charge on P- are - 6 . 7 and - 1.08 for the interior and the exterior of the cell, respectively. An electrical potential difference of Atp = Oi - 4'0 = 90 mV is measured, i and o denote the intracellular and extracellular, respectively. The activity coefficients of components inside and outside the cell are assumed to be the same, and pressure and temperature are 1 atm and 310 K. Assume that the diffusion flows in from the surroundings are positive and the diffusion flows out are negative. Using tracers, the unidirectional flows are determined as follows: Components

10l° (J+) (mol/(m2 s)) 10l° (J-) (mol/(m2 s))

K~

Na+

C1-

HCO3-

P-

722

250 -

14.2 -

3.6 -

0 0

Using these approximate flows, we may estimate permeabilities using Jk = --EkPk

% - Cko exp(--Ek)

1 - exp(-Ek)

where E k = z k F A d / R T and the permeability coefficient is

(a)

580

11.

Thermodynamicsand biological systems

DkKm Pk-

where D is the diffusion coefficient and Km is the distribution coefficient defined by Km = c k (x = 0) = c k (x = L) Cko

Cki

The permeability coefficient depends on the characteristics of the membrane and solute, and can vary considerably for various solutes. For example, p = 10 -21 naJs for sucrose and 10 -4 m/s for water in the human red blood cell membrane. Equation (1) may be generalized by including the effect of pressure gradient z ~ m - - P ( 0 ) - P(L), and we have Ek =

zkFAO + gkz~m ~ RT

RT

To use this relation, we need to relate pressure difference in the membrane, z~dgm = P ( 0 ) - P(L), and over the membrane, zkP = P i - Po- This requires the introduction of the concept of osmotic pressure in a nonequilibrium membrane system. From Eq. (a) we obtain the value of E and the permeability for sodium E(Na +) = (+ 1)(96500)(-0.090) = - 3 . 3 7 8.314(310)

p(Na +) =

J + ( N a +){1- e x p [ - E ( N a +)]}

2.5 × 10-811- exp(3.37)]

E(Na + )c o (Na + ) exp [ - E ( N a + )]

( - 3.37)(145)(exp(3.37))

= 4.94 x l 0 -11 m/s

E(Na +) = E(K +) = -3.37. As co(K +) = 0 for tracer, we have P(K+) = 7.22×10-8tl_exp~3.37~j r ~ al = 4.33×10 -9 m/s (-3.37)(145) The net diffusion flows of sodium ions in and potassium ions out can be calculated by J ( N a +) = 3.37(4,94 x 1 0 -11)

1 2 - (145) exp(3.37)

= 2.49 X 10 -8 mol/(m 2 s)

1 - exp(3.37) J ( K +) = 3.37(4.33 ×10 -9)

1 3 9 - (3.7) exp(3.37)

= - 1.63 × 10 -8 mol/(m 2 s)

1 - exp(3.37) The flow of potassium depends on the outside concentration of potassium ions. Under steady-state conditions, the chemical pump creates an active flow ofNa + out of the cell and simultaneously an active flow of K + into the cell: J ( N a + ) = - 2 . 4 9 × 10 -8 mol/(m 2 s) J ( K +) = 1.63 × 10 -8 mol/(m 2 s) The pumping ratio Na+/K + = 2.49/1.63 = 1.53 --~ 3:2. After performing similar estimations, the following values for permeability coefficients and diffusion flows of all components are obtained Components

101°(Pk)(m/s) 101°(Jk) (mol/(m2 s))

K+

Na+

C1-

43.3 163

0.494 249

1.0 0.2

HCO 3-

1.25 0.1

P0

0

11.7

581

Exergy use in bioenergetics

As the estimations above display, the net flows of chloride and bicarbonate ions are negligible, and the transport of ions is passive. Using the van't Hoff equation, the osmotic pressure difference across the membrane is estimated by

( I n a system with one single nonpermeating component dissolved in water, II = P i - P o = RT(cki-Cko).)

k = 2,

and this equation reduces to

P~ - Po = 0.0827(310)[(139 + 12 + 4 +0.8 + 136)-(3.7 + 145 + 118 + 24 + 1)] × 10.3 = 0.0025 atm This result corresponds to an osmotic activity difference of 0.1 mmol/L. The total electrical current density (C/m 2 s) through the membrane is I= ZJ~zk = J(Na+)+Jp(Na+)+J(K+)+Jp(K+)=O

Here, Jk consists of passive diffusion flows and active flows Jp due to chemical pumps, which compensate for the nonequilibrium concentrations of cations and the total charge current becomes zero. The transference number of an ionic component tk

--

Jk Zk and ~_,~-"t k = 1 I

is usually used to describe the passive diffusion of charge. This representative example illustrates transport processes in biological cells using a highly simplified analysis. Biological cells also operate hydrogen and calcium pumps. Some of the concentrations also represent only approximate values.

11.7

EXERGY USE IN BIOENERGETICS

Biological systems extract useful energy from the outside, convert it, store it, and use it for muscular contraction, substrate transport, protein synthesis, and other energy-demanding processes. This useful energy is called exergy, which is lost in every irreversible process because of entropy production. The ATP produced through oxidative phosphorylation is the form of exergy that originates due to oxidation of reduced equivalents of nutrients. A living cell uses ATP for all energy-demanding activities and maintains nonvanishing thermodynamic forces, such as electrochemical potential gradients. However, mitochondria cannot maximize ATP production, and exergetic efficiency at the same time.

11.7.1

Exergy Management

Michaelis-Menten equation shows that the enzyme reactions in certain regions can be approximated by linear kinetics. Stucki (1984) demonstrated that variation of the phosphate potential at constant oxidation potential yields linear flow-force relationships in the mitochondria. Through linear flow-force relationships, cells may optimize their free energy production and utilization by lowering their entropy production and hence exergy losses at stationary states. The second law of thermodynamics states that entropy production or exergy loss as a consequence of irreversibility is always positive. A representative overall dissipation function for oxidative phosphorylation is dS -- it"--i7: __ JoXo +JpYp - i n p u t work+output work -> 0 dt

(11.146)

where the input force Xo is the redox potential of oxidizable substrates, and Xp is the output force representing the affinity A, or the phosphate potential expressed by Eq. (11.86)

- X p = AGp + R T l n

[ATP] ) [ADP][Pi ]

582

11.

Thermodynamics and biological systems

where AGp is the Gibbs free energy at standard conditions. The associated input flow Jo is the net oxygen consumption, and the outflow Jp is the net rate of ATP production. For the sum to be positive in Eq. (11.146), we can have either JoXo > 0 and JpXp > 0, or JoXo >> 0 and JpXp < 0, which requires coupling. The sequence of coupling is controlled at switch points where the mobility, state, and catalysis of the coupling protein can be altered in specific ways, such as sifted equilibria or regulated rates of conversion between one protein state and another. Prigogine showed that the total exergy destruction reaches a minimum in a stationary state, which is a stability criterion. Optimal performance regimes of biological systems are associated with minimum entropy production.

11.7.2 Exergy Efficiency The exergetic efficiency rt is defined as the ratio of dissipations due to output and input powers in oxidative phosphorylation, and from Eq. (11.146) we have output work input work

Jp Xp JoXo

(11.147)

This equation suggests that the efficiency is a function of the state of the system, as both the forces and flows are state dependent. For a coupled system (q < 1), the efficiency is zero at the static head (Jo = 0) and at the level flow (Xo = 0). Therefore, as the process progresses from the level flow to the static head, the phosphorylation, as a linear energy converter, passes through a state of maximal efficiency ~max defined by

~max =

q l+x/1-q 2

(11 "148)

where q is called the degree of coupling which is expressed in terms of the phenomenological cross-coefficients L O. q=

Lop

(LpLo) 1/2

(11.149)

Equation (11.148) shows that 'qmax depends only on the degree of coupling. Here, q is a lump sum quantity for the degrees of coupling of various processes of oxidative phosphorylation driven by respiration. Absolute values of q vary from zero for completely uncoupled systems to unity for completely coupled systems: 0 < ql < 1. It is customary to use the following simplified, representative linear phenomenological equations for the overall oxidative phosphorylation Jp -- Lp Xp + Lpo Y o

( 11.150)

nt- LoX o

( 11.151)

Jo = LopX p

The matrix of the phenomenological coefficients must be positive definite; for example, for a two-flow system, we have L o > 0, Lp > 0, and LoLp-LopLpo > 0. L o shows the influence of substrate availability on oxygen consumption (flow), and

Lp is the feedback of the phosphate potential on ATP production (flow). The cross-coupling coefficient Lop shows the phosphate influence on oxygen flow, while Lpo shows the substrate dependency of ATP production. Experiments show that Onsagers's reciprocal relations hold for oxidative phosphorylation, and we have Lop = Lpo. By dividing Eq. (11.150) by Eq. (11.151), and further dividing the numerator and denominator by Xo(LoLp)1/2, we obtain

"q-ix=-

x+q q+ l/x

(11.152)

where j = (Jp/JoZ), x - (XpZ/Xo). Here, Z is called the phenomenological stoichiometry defined by (11.153) Equation (11.152) shows the exergetic efficiency r/in terms of the force ratio x and the degree of coupling q. The ratio Jp/Jo is the conventional phosphate to oxygen consumption ratio: P/O. Figure 11.5 shows the change of

11.7

1

,

583

Exergy use in bioenergetics

,

,

,

,

,

'

0.9 0.8 0.7

qopt= 0.6195

~~...

~- 0.6

~_q~C

0

" 0.5

/

.m 0

0.4

0.3 0.2 0.1

/o,:/ -0.9

Figure 11.5.

-0.8

. . . . . . -0.7

-0.6 -0.5 -0.4 force ratio, x

-0.3

-0.2

-0.1

0

The change of efficiencies 7, given in Eq. (11.153), in terms of flow ratio x and for the degrees of coupling qf, qp, q~C, and q~C.

efficiencies r/in terms of flow ratio x between-I and 0, and for the particular degrees of coupling qf, q p , q~C, and q~C.As Eq. (11.148) shows, the optimum efficiency values are dependent only on the degrees of coupling, and increase with increasing values of q.

11.7.3

Exergy Losses

For the oxidative phosphorylation described by Eqs. (l 1.151) and (l 1.152), the exergy loss can be obtained from Eq. (11.146) in terms of the force ratio x and the degree of coupling q, and is given by - (x 2 + 2qx + 1)LoXo2

(11 154)

Minimum exergy loss or minimum entropy production at stationary state provides a general stability criterion. There are two important steady states identified in the cell: static head (sh) and level flow (lf). At the static head, where ATP production is zero since Jp =--0, the coupling between the respiratory chain and oxidative phosphorylation maintains a phosphate potential Xp, which can be obtained from Eq. (11.151) as (Xp)sh = --qXo/Z, and the static head force ratio Xsh becomes X~h= --q. The oxygen flow Jo at the static head is obtained from Eqs. (11.151) and (11.152) (Jo)sh

=

LoXo(1-q 2)

(11.155)

where Lo may be interpreted as the phenomenological conductance coefficient of the respiratory chain. If an uncoupling agent, such as dinitrophenol, is used, the ATP production vanishes and hence Xp = 0; then, Eq. (11.151) becomes (Jo)unc -- LoXo

(11.156)

(do)sh -- (do)unc ( 1 - q2)

(11.157)

Combining Eqs. (11.155) and (11.156), we obtain

Using the experimentally attainable static head condition (state 4 in mitochondria) and the uncoupled oxygen flow (Jo)unc, we can determine the degree of coupling q

1/2 (Jo)sh q--

1-- (jo)un

(11.158)

c

At constant Xo, Eq. (11.154) yields the minimum value of exergy loss at x (q~)~h - (q~)mi. - (1- q2 )LoX2o

-q

(11.159)

11.

584

Thermodynamicsand biological systems

1

0.8

0.6 470.4

0.2 ..._...

.

-().9

-0.

8

-().7

-().6

-0'.5 -0'.4 -0'.3 -().2

- .O' 1

0

force ratio,x Figure

11.6. The change of ratio 4'1, given in Eq. (11.160), in terms of flow ratio x and for the degrees of coupling qf, qp, q~C, and q~C.

The ratio of dissipations expressed in Eqs. (11.159) and (11.154) depends only on the force ratio and the degree of coupling, and becomes an exergy distribution ratio over the values ofx ¢hl -

1- q2

a'Itsh __

(11.160)

x 2 + 2qx + 1

Figure 11.6 shows that the values of ¢hl reach unity at various values of x, and the economical degrees of coupling q~C and q~C yield lower values of 4'1 than those obtained for qf and qp. The exergy loss at the static head is relatively lower at the degrees of coupling corresponding to economical ATP production and power output. At level flow, the phosphate potential vanishes, and no net work is performed by the mitochondria, and the flow ratio becomes Jlf -

Lpo Lo

- qZ

(11.161)

Combining Eqs. (11.160) and (11.161) yields an expression for estimating the phenomenological stoichiometry Z from measured Jp/Jo = P/O ratios at level flow Z=

P/O

(11.162)

41-(Jo)sh/(Jo)unc The efficiency expressed in Eq. (11.153) is zero at both the static head and level flow, due to vanishing power at these states. Between the static head and level flow, efficiency passes through an optimum, which is given in Eq. (11.149). The force ratio at optimal efficiency is expressed by _

q

Xopt----Cpt--l+41--q

2

(11.163)

The rate of optimal efficiency of oxidative phosphorylation is not characterized by the exergy decrease, and the exergy loss at optimal efficiency is given by

Xi/,opt __ x 2 )2 _ (1l+x-_------------~LoX2

(11.164)

17.7

585

Exergy use in bioenergetics

1.8

/,///~

1.6 1.4 1.2 O4

e-

1 0.8 0.6 0.4 0.2 0

t._

-0.9

-0.8

i

-0.7

I

I

-0.6 -0.5 -0.4 force ratio, x

I

I

I

-0.3

-0.2

-0.1

Figure 11.7. The change of ratio ~h2,given in Eq. (11.165), in terms of flow ratio x and for the degrees of coupling qf, %, q~C,and q~C.

The ratio of dissipations ~opt and ~ , given in Eqs. ( 1 1.164) and (11.154), respectively, shows the effect of optimal operation on the exergy loss in terms of the force ratio x and the degree of coupling q ~)2 --

opt _

qf

(I-- v2) 2 " (1 + x 2 )(x 2 + 2qx + 1)

(11.165)

Figure 11.7 shows that the values of 052 reach peak values higher than unity, and the exergy loss is not minimized at optimal efficiency of oxidative phosphorylation. The exergy loss is the lowest at the degree of coupling corresponding to the economical power output. For the optimal efficiency to occur at steady state, oxidative phosphorylation progresses with a load. Such a load JL is an ATP-utilizing process in the cell, such as the transport of substrates. A load, which will make the steady state the optimal efficiency state, can be identified through the total exergy loss q~c q/c -- J p X p -Jr-J o X o

+

JLX

(ii.166)

Here, Jk is the net rate of ATP consumed and X is the driving force. If we assume that the ATP-utilizing process is driven by the phosphate potential Xp, and Jc is linearly related to Xp, then we have JL - LXp. Here, L is a phenomenological conductance coefficient. The dissipation caused by the load is JkXp = LX2p, and the total exergy loss becomes

(11.167)

Using Eq. (11.163), and from the extremum of Eq. (11.167), Stucki (1980, 1984) found the condition L _ x/l_q2

(11.168)

Lp

which is called the conductance matching of oxidative phosphorylation. After combining Eqs. (11.23) and (11.168) when the conductance matching is satisfied with a percentage/3, we obtain qZc _ ix211 + fiX/1_ q2 )+ 2qx + 1)LoX,~

(11.169)

The exergy lost at the static head with conductance matching, (Xlrc)sh , and at the state of optimal efficiency, (Xlrc)opt, are expressed by Stucki (1984) as follows

11. Thermodynamicsand biological systems

586

(attc)sh = 4 I - - x 2 (X2 + 4 I - - X 2 )LoX2o

(11.170)

x2 i + x 2 LoX2o

1 -

(atrc)op t

u

I t is now possible to estimate the ratios of exergies 4~3 and q by using Eqs. (11.169)-(11.171)

4'3-

(11.171)

~4 in terms of the force ratio x and the degree of coupling

,,/1- x2 (x2 + x/1- x2 )

(XIYc)sh _ _

(11.172)

x2 (l + ~x/l- q2 )+ 2qx + l

"~II"c

(XItc)opt __

I--X 2

/)4 ~ - ~

4

i

i

I

I

I

I

I

i

2.5 g

2

0.5 0

g

I

-1

I

-0.9-0.8-0.7-0.6-0'.5-o'.4-d.a-d.e

-.1 o'

0

force ratio, x Figure 11.8. The change of ratio (b3, given in Eq. (11.172), in terms of flow ratio x and for the degree of couplings qf, qp, q~C, and q~C with load, and/3 - 1 and/3 = 0.9 conductance matching.

i 1.7

Exergy use in bioenergetics

587

Equations (11.172) and (11.173 ) are analogous to Eqs. (11.160) and (11.165) and show the ratios of exergy with and without the load. Figure 11.8 shows the change of ~3 when a load is attached with the values/3 = 1 and/3 = 0.9 at the static head. With the load, the exergy destruction increases considerably. This increase is larger with decreasing values of/3. Figure 11.9 shows the values of &4 with/3 = 1 and/3 = 0.9. At optimal efficiency, exergy destruction is the lowest at qf, corresponding to the maximum production of ATR while the exergy destruction is relatively higher at the economical power output with a minimal effect of/3. The comparison of energy conversions in linear and nonlinear regions requires a combination of thermodynamic and kinetic considerations to express the exergetic efficiencies of nonlinear TinI and linear r h modes

/ AoX

tin'--

--v-' and TII----

-7-

1

(11.174)

"p o

Denoting y as a measure of the relative binding affinity of a substrate H + on the either side of the membrane, the following inequalities are obtained for y >- 1

i

i

i

!

i

I

i

i

i

i

9=1 _

2.5

e- 1.5

qf 0.5

-1

I

I

i

I

I

I

I

I

I

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

force ratio, x 3 p=0.9 2.5 P

2-

1.5

0.5

0

-1

-0.9

1

1

-0.8

-0.7

I

I

I

-0.6 -0.5 -0.4 force ratio, x

I

I

I

-0.3

-0.2

-0.1

Figure 11.9. The change of ratio ~h4,given in Eq. (11.173), in terms of flow ratio x and for the degrees of coupling qf, q> qp, and q~C with load and/3 = 1 and/3 = 0.9 conductance matching.

588

....

11.

Thermodynamicsand biological systems

~n] > ~7~ for ~'-

T/n1 < T]I

for ~"-

1

(11.175)

q 1

(11.176)

q

where ~"is the reduced stoichiometry Z/n, which is subjected to kinetic limitation (l/q) > ( >_q, and n is the mechanistic stoichiometry. Inequalities (11.175) and (11.176) suggest that at a specified value of (, the efficiencies of nonlinear and linear modes become equal to each other, and there exist values for ~"where the energy converter operates more efficiently. The ratio of efficiencies (called as gain ratio) at linear and nonlinear modes is 'r/1

r/r -

(11.177)

Tlnl

This shows a measure for the efficiency gain in linear mode operation. The efficiency in linear modes depends on only q (Eq. (11.149)), while the efficiency in nonlinear modes depends on input force Xo besides q. In nonlinear regions, the efficiency decreases at high values of input force, and the force ratio at optimum operation Xopt.nl is shifted towards the level flow where x = 0. In oxidative phosphorylation, the input force is the redox potential of the oxidizable substrates and the output force is the phosphate potential. If these two forces are balanced, the system operates close to reversible equilibrium. Experiments show that in mitochondria, q < 1, and the input force is well above 50RT. For a fully coupled system in the nonlinear region of a single force, the phosphate potential Xp would be very small. However, a dissipative structure can only be maintained with a considerable Xp. On the other hand, in the linear mode of operation, optimum force ratio Xop t does not depend on the input force (Eq. (11.163)). Gain ratio T/r can be calculated at a reference force ratio, such as Xopt, which is a natural steady-state force ratio of oxidative phosphorylation. This is seen as a result of the adaptation of oxidative phosphorylation to various metabolic conditions and also as a result of the thermodynamic buffeting of reactions catalyzed by enzymes. The experimentally observed linearity of several energy converters operating far from equilibrium may be due to enzymatic feedback regulations with an evolutionary drive towards higher efficiency. A living organism requires q] < 1. The particular value of the degree of coupling depends on the nature of the output required from the energy converter. For example, fatty acids decrease the degree of coupling and act as uncoupler. Uncoupling is not restricted to thermoregulation; some uncoupling activity is favorable for the performance of the metabolic and even the energy-conserving functions of cellular respiration. Mitochondria can regulate their degree of coupling of oxidative phosphorylation depending on the energy demand of the cell. For example, for fed rats, oxidative phosphorylation operates very close to the conductance matching, i.e., at the state of optimal efficiency with an economical degree of coupling. The load in a living cell fluctuates and compromises the optimal efficiency of oxidative phosphorylation. Some enzymes operate as sensitive thermodynamic buffering to decrease deviations from optimal efficiency. ATP-utilizing reactions act as a load as well as thermodynamic buffers. This regulatory mechanism allows oxidative phosphorylation to operate with an optimal use of the exergy contained in the nutrients. Every reversible ATP-utilizing reaction can, in principle, act as a thermodynamic buffer. For example, adenylate kinase can buffer the phosphate potential Xp to the value permitting optimal efficiency of oxidative phosphorylation in the presence of too high loads. The adenylate kinase reaction is reversible, and acts as a buffer. If this buffer is treated as another load with a conductance La, the overall load conductance L* becomes E =LaO+L

(11.178)

where

0=

c~+ RT ln(1 + e x p - (6 + Xp/RT))

Xp

-1

11.7

Exergy use in bioenergetics

m

589

[AMP] [ATP] + [ADP] + [AMP]

z

6 = kGp - RT ln[Pi ]

Stucki (1984) expressed the dissipation function with buffering from Eq. (11.167) in terms of L*

('qY'c)b-- X2 1%"

+2qx+l

Lp

Lo X2

(11.179)

Dividing Eq. (11.179) with Eq. (11.167) shows the effect of thermodynamic buffering on the exergy loss

!

!

'

'--1--

'

'

,

i

i

!

i

|

1 L=0.9 0.95 0.9 0.85 0.8 -C- 0.75 0.7

qo

0.65 0.6

<__q~c 0.55 0 . 5

i

-1

,

-0.9

i

1

i

1. . . .

-0.8

i

-0.7

i

i,

,, i

i

,,,

-0.6 -0.5 -0.4 force ratio,x i

r

1

i

i

-0.3

-0.2

-0.1

'1

|

......

i

0

i

L=0.5

0.9 0.8 0.7 0.6 ~-

0.5 0.4 0.3

0.2 0.1 0

I ........

-1

i,,

-0.9

I

,I

I

I

I

-0.8

-0.7

-0.6

-0.5

-0.4

,,

I

-0.3

,,

I

I

-0.2

-0.1

0

force ratio,x Figure 11.10. The change of ratio 65, given in Eq. (11.180), in terms of the flow ratio x and for the degrees of coupling qf, qp, q~C,and q~C. The plots were normalized by Xp = (x/Z)Xo with Xo = 209,200 J/moi, Z = 3, 2~G°k = 630/mol, 2~G~ = 35560J/mol, M = 0.005, Pi = 8 mM, T - 310 K with load conductance of L = 0.9 and L = 0.5D.

590

11.

Thermodynamicsand biological systems

4,5 -

(XI/'c)b

(11.18o)

q~c Figure 11.10 shows the change of 4'5 in terms ofx between 0 and-1 for the degrees of coupling qf, qp, q~Cand q;C with the load conductance L - 0.5 and L = 0.9. The values of 4~5are lowest at the economical power output and highest at the maximum output of ATE Therefore, the exergy loss is relatively lower in the economical power output than in maximum ATP flow. Within the framework of the theory of dissipative structures, thermodynamic buffering represents a new bioenergetics regulatory principle for the maintenance of a nonequilibrium conditions. Due to the ATP production in oxidative phosphorylation, the phosphate potential is shifted far from equilibrium. Since hydrolysis of ATP drives many processes in the cell, the shift in Xp to far from equilibrium results in a shift of all the other potentials into the far from equilibrium regime.

Example 11.12 Exergy efficiency Assume that the oxidation of glucose (G) in living cells produces 38 mol of ATP per mole of glucose (Garby and Larsen, 1995). Estimate the maximum theoretical production and the exergy efficiency. C6H1206 +602(g)~6CO2 +6H20 The steady composition of the mixture contains 0.01 mol/L glucose and the partial pressures of carbon dioxide and oxygen are 0.07 and 0.21 atm, respectively. The state of the mixture is characterized by the activity ratios (ai/ai°) = 0.07, 1.0, 0.01, and 0.21 for carbon dioxide, water, glucose, and oxygen, respectively. The entropy of reaction for oxidation of glucose at standard conditions is obtained from Table B8 AS~°G = 6(121) + 6(70) -- 151-- 6(115) = 305 J/(mol K) The entropy of reaction at actual state should be estimated. Assuming that the entropy of mixing is negligible, we have (a)

For a reaction at 310 K, entropy may be obtained from

Si(T,Po)--Si(To,Po)+Cp

In (~-0-o)

Here, Cp is assumed to be a constant. Substituting Eqs. (a) and (b) into the relation of reaction entropy ASr = ~ we have

(b)

viSi,

(c)

The heat capacities from Table B8 show that Cp = 0.037, 0.075, 0.219, and 0.029kJ/(mol K) for carbon dioxide, water, glucose, and oxygen, respectively. At 310 K, this equation with the data from the Appendix and above becomes

: 30' + (0

× 1.06 ] 1000'1hi310)' 3141n[0.076 0.01×0.216

= 305 + 11 - 16.5 = 299.5 J/(mol K) On the other hand, the heat of reaction is obtained from

z~J--/r,G -- AHr°G at-(Z piCp i ) ( T - To) -- -2870 + 0.279(310- 298) = -2867 kJ/mol

11.7

591

Exergy use in bioenergetics

The Gibbs energy of the reaction at 310 K is obtained from - 2 8 6 7 ( 1 0 0 0 ) - 310(299.5) = - 2 9 6 0

AGr, G = A H r , G - T ~ S r , G -

kJ/mol

For the reaction ATP ~ ADP + Pi at the standard state and each component at a concentration of 1 mol/L, at 310 K Table B9 provides the heat of reaction and the Gibbs energy of reaction AHr°ATP = --

20 kJ/mol ATP, and A Gr°ATP -- -- 31 kJ/mol ATP

With the values above, we find (-20)-(-31) ASr°,ATP =

= 0.0355 kJ/(mol ATP K)

310

With concentrations ofATR ADR and Pi of 4, l, and 10 mmol/L, respectively, in the aqueous solution, we find

AGr,ATp=AGr°ATp+RTEPiln( °a--L) a

AGrATP = - - 3 1 + 8.314(310) × 10-3 lnI10-3 ' 4 × (0"01) 10 .3 ] = --46 kJ/mol ATP The maximum yield is AGr,c~

-2960

AGr,ATP

--46

- 64.3 mol ATP mol glucose

Therefore, the process has an approximate efficiency of 38/64.3 --~ 0.6 to transfer useful energy. The total dissipation, ~ = T~, is the summation of the product reaction rate and affinity (the Gibbs energy of reaction), and then we have q.r = _ ( j r A Gr ) a -- ( J r A Gr ) ATP -- --( 1 )(--

2960) - (38)(46) = 1212 kJ/mol glucose

From the energy balance, we have 0 = (JrZ2tHr)a + (JrAHr)ATP = --2867 + 38(20) = --2107 kJ/mol glucose Assuming that

qG/£/ATP

--

1, we have qG __ qATP __

de

do

We can find the chemical work of glucose

1 2 ( - 2107) = - 1053.5 kJ/mol glucose oxidation

f~/ch from

7ch _ 6)G

- - -- AHr, G - - 1 0 5 3 . 5 - ( - 2867) - 1814 kJ/mol glucose

JG

JG

ch alt G = - - J G A G r , G . . . .

JG

- - ( 1 ) - (2960) - 1814 - 1146 kJ/mol glucose

For the ATP synthesis, we have

XIrATP = --JATPZ~Gr ATP -+'

Wch ....

JG

38(-46) + 1814 = 66 kJ/mol glucose

592

11. Thermodynamicsand biological systems

The exergy efficiency is ~rch -- ~ A T P __ 1 ~/rch

'0 --

66 _ 0.96 1814

This result shows somewhat high exergy efficiency.

Example 11.13 Approximate exergy balances in a representative active transport Consider the representative active transport in Example 10.9. Estimate the distribution of exergy and the exergy efficiency of the active transport system shown in Figure 10.3. The dissipation equations for control volumes are -- ~rch1

XI/"1 -- - J r l ~ G r G

for cvl

XI~2 -- Jr2Z~Gr,ATP -1-mch 1

for cv2 for cv3

XI)'3 -- --Jr3Aar,ATP -- Wch 2

• 4=

i +Wch

for cv4

for cv5 The total exergy loss is related to the oxidation of glucose XIf -- --JrGAGr,G

The exergy efficiencies are defined by J'/t/chl " 0 1 - XI$1 _+. ~rch 1

'02 =

~rchl--XIt2 mchl ' '03

J/Vch2 * 3 "t-~rch 2

'04 --

~rch2 -- XI)'4 ~rch 2

For the control volume cv5, there is no yield and no efficiency. For the total control volume, the yield is measured in terms of the capacity of the system to maintain its nonequilibrium character in concentration and electrical potential across the membrane. The maintenance of nonequilibrium conditions is expressed quantitatively as the product of a generalized potential difference and generalized flow represented by ~5. Taking the values from Example 10.9, and for Jr3 -- Jr2 and Jr5 = Jr4 = 1 mmol/min (stationary state), we have xIg'l = - J r l A G r ,

a -}Vchl

=0.360×10-3(2960) -0"653=0"413w

XI)"2 -- Jr2AGr,ATP nt-~fch 1 -- 13.7×10-3(-46)+0.653 = 0 . 0 2 3 W

xIt3 = --Jr3AGr,ATP

-- ~'~ch2 -

0.629 - 0.261 = 0.368 W

for c v l

for c v 2

for cv3

• 4

....

(0.0011[ 60

8.314(310) In

r,4 1 k, 12

]

+ 1(96500)(0.090) + 0.261 = 0.009 W

for cv4

11.9

q.r 5 = J r s R T l n

593

Molecular machines

Ce + J,:sz(Na+)F(Oe - q t i ) = 0.252W

for cv5

The total dissipation is = --JrGAGr,G -- --0.360 × 10-3 (--2960) = 1.066W The total exergy efficiency becomes .

xP5 0.252 . . . q~ 1.066

0.24

The other efficiencies are • Wchl

=

0.61, "112

T]I = XI'rl -Jr- I/Vchl

11.8

];~7chl -- ~ 2 __

=

--

~l/chl

__

~'~rch2

0.44, r/3 -

.

XP3 -+- Wch2

__

- 0.41, 774

__ ~/ch2 -- ~J'4 --

.

=0.96

Wch2

MOLECULAR EVOLUTION

Living systems utilize a set of genetic instructions and develop physical characteristics. Quantitative theories for describing the information transfer generally assume that the organization and transfer of information, while constrained by the principles of chemistry and physics, may not necessarily be a consequence of these principles. Proteins are synthesized as linear polymers with the covalent attachments of successive amino acids, and many of them fold into a three-dimensional structure defined by the information contained within the characteristic sequence. The folding results largely from an entropic balance between hydrophobic interactions and configurational constraints. The information content of a protein structure is essentially equivalent to the configurational thermodynamic entropy of the protein relating the shared information between sequence and structure. From the perspective of the fluctuation-dissipation approach, Dewey (1996) proposed that the time evolution of a protein depends on the shared information entropy S between sequence and structure, which can be described with a nonequilibrium thermodynamics theory of sequence-structure evolution. The sequence complexity follows the minimal entropy production resulting from a steady nonequilibrium state 0 (_~t)= 0 OXj

(11.181)

A statistical mechanical model of thermodynamic entropy production in a sequence-structure system suggests that the shared thermodynamic entropy is the probability function that weighs any sequence average. The sequence information is defined as the length of the shortest string that encodes the sequence. The connection between sequence evolution and nonequilibrium thermodynamics is that the minimal length encoding of specific amino acids will have the same dependence on sequence as the shared thermodynamic entropy. Dewey and Donne (1998) considered the entropy production of the protein sequence-structure system based on linear nonequilibrium thermodynamics. The change of composition with time is taken as the flow, while the sequence information change with the composition is treated as the thermodynamic affinity, which can be interpreted as the chemical potential of the sequence composition. Since the change of entropy with time is a positive quadratic expression in forces, Eq. (11.181 ) shows the regions of the sequence that are conserved; the rest of the sequence is driven to a minimum entropy production, hence toward the lower complexity seen in the protein sequence, creating a stable state away from equilibrium with a specific arrow of time. At steady state, a system decreases its entropy production and loses minimal amounts of free energy (the concept of least dissipation). A restoring and regulating force acting in any fluctuation from steady state may be one of the principles in the evolution of biological systems.

11.9

MOLECULAR MACHINES

Either the hydrolysis of ATP or the draining of an ion gradient is converted into an osmotic work by a pump or a mechanical work by a motor protein. Substrate binding sites in pumps or motor proteins occur in two sections: (i) one of them binds the substrate and (ii) the other converges on the bound substrate molecule, and the protein changes into an

594

11.

Thermodynamicsand biological systems

altered conformation. Some important biological processes resemble macroscopic machines governed by the action of molecular complexes. Pumps are commonly used for the transport of ions and molecules across biological membranes, while the word "motor" is used for transducing chemical energy into mechanical work in the form of rotatory or translationary by proteins or protein complexes. Some identified motor proteins such as kinesins and dyneins move along tubulin filaments, and myosin moves along filaments. These motor protein families play a major role in muscle contraction, cell division, and the transport of substrates in and out of the cell. Molecular motors are isothermal and hence the Carnot efficiency concept is not applicable. The internal states are in local equilibrium; they operate with a generalized force for the motor/filament system. This may be the external mechanical forcefext applied to the motor and the affinity A, which measures the free-energy change per ATP molecule consumed ATP.~ ADP + Pi

(11.182)

A = -A/~ = ~ATP --/J'ADP -- ~Pi

(11.183)

The external forces may be optical tweezers, microneedles, or the viscous load of the substance that is carried. These generalized forces create motion, characterized by an average velocity v, and average rate of ATP consumption Jr. Molecular motors mostly operate far from equilibrium, and the velocity and rate of ATP consumptions are not linear functions of the forces. However, in the vicinity of the linear region, where A << kBT, linear relations hold xp = Vfext + Jr A > 0

(11.184)

v = ~lfext +L12A

(11.185)

Jr = L21fext + L22A

(11.186)

Here, L 11 is the mobility coefficient, while L22 is a generalized mobility relating ATP consumption and the chemical potential difference, and L12 and L21 are the mechano-chemical coupling coefficients. A given motor/filament system can work in different regimes, and in a regime where the work is performed by the motor, efficiency is defined by r / = - ~ Vfext

(11.187)

Jr A

For nonlinear motors operating at far from equilibrium, velocity reversal allows direction reversal without a change in the microscopic mechanism. Molecular motors are classified in two groups depending on whether they operate in groups or individually. Those that operate in groups are relevant to muscle contraction. In principle, muscle fibers can oscillate in appropriate conditions. Skeletal muscle myofibrils can oscillate spontaneously; for example, spontaneous oscillations of asynchronous muscles are common in the wings of many insects. The relationship between the Gibbs energy of ATP hydrolysis and the performance of muscle is sigmoidal, and the normal operating domain is in the quasi-linear region of the curve. This corresponds with the magnetic resonance spectroscopy results obtained from studies of the finger flexor muscle. Chemiosmotic potentials are coupled to rotation in multiprotein subunit systems of bacterial flagellar motors and FoFl-ATPases. Rotation of a subunit assembly of the ATP synthase is considered an essential feature of the ATPase enzyme mechanism and of FoF1 as a molecular motor generating a torque. The bacterial flagellar motor and FoF1 are macromolecular assemblies and utilize six to eight distinct protein components to affect chemiosmotic energy transduction. Both assemblies have integral membrane modules and extensive cytoplasmic modules. With these systems, the work is accomplished outside the membrane, whereas the chemiosmotic pumps and transports take place within the membrane. The flagellar motor rotates an external filament, and according to the intracellular signals, it modulates the direction of rotation. FoF1 system activity is modified by the intracellular phosphate potential. The energy transduction reactions are reversible for FoF1 and partly reversible for the flagellar motor. (The tightly coupled and efficient operation of these machines may be achieved by alternating access of protonable residues to the adjacent bulk phases linked to the conformational motions, which generate force.) ATP is synthesized by mitochondrial oxidative phosphorylation (Jp,op) and glycolysis. The latter contribution may be disregarded for a muscle engaging little or moderate exercise. Exerted power by a muscle P is directly related to the rate ofATP hydrolyzed by the myosin ATPases Jp, and we have P - aJp

(11.188)

11.10

595

Evolutionary criterion

where a is a constant and is independent of the muscle operation. Some of the energy released by ATP hydrolysis is not associated with the muscle operation Jp,1, and may be considered a leak. For the ATP balance for steady-state muscle operation, Eq. (11.188) yields

P = °t(-Jp,op - Jp,1)

(11.189)

The quasi-linear variation of power with ATP hydrolysis is observed experimentally, as the contraction is being activated at the level of actinomyocin activity. The kinetic approach suggests that the muscle power output varies hyperbolically with the ADP concentration. Both the ADP control and the Gibbs energy of ATP hydrolysis control are similar, and when muscle power is varied voluntarily, muscle energetics may be represented by the linear flow-force relationships. 11.10.

EVOLUTIONARY CRITERION

Tellegen's theorem can be used in an evolving network, where the forces are allowed to change with time, and after a time interval dt, the forces become XI + dX,/dt. Since, according to Tellegen's theory, the flows and forces lie in the orthogonal spaces, we have (11.190)

~_~ J; dX; _ 0 dt

Since dt is an arbitrary time interval, we get

"

d[

--

~_~

J i - - ~ irrev

+E

Ji

dt )rev

=0

(ll.191)

This equation comprises both the reversible and irreversible contributions, and also reduces to

dX i ] = - - E Ji--~-]rev

dX i

The reversible part obeys the constitutive relations C i - dNi and Ji - Ci d l~i >- 0 d#; dt

(11.192)

Since the capacity of an ideal capacitor C; >> 0, we have ci dX; _ J~ >_0 dt C;

(11.193)

Therefore, the first term of Eq. (11.191 ) is negative definite

Ji dt

)irrev~ 0

(11.194)

This equation is an evolutionary criterion. The change of dissipation function with time yields d~ dt

dXi ] (Xi dJi -- E (Ji --~)irrev -+-E -~)irrev

(11.195)

In the linear region of the thermodynamic branch and with constant phenomenological coefficients, we have

dX i

dJ

(11.196)

596

11.

Thermodynamicsand biological systems

Combining Eqs. (11.195) and (11.196), we have

d*-2~_~(j i dXi dt

TJirrev

(11.197)

From Eqs. (11.194) and (11.197), we obtain

d~ dt

~-<0

(11.198)

This equation is valid only in the linear region, which may be rare in biology. Equation (11.194) may be used for the evolution of all biological networks, which can be characterized by thermodynamic considerations. Equation (11.194) is valid for both linear and nonlinear constitutive relations, and can be used for quasi-equilibrium and far-from-equilibrium regions of the thermodynamic branch.

PROBLEMS 11.1

In living systems, ions in the intracellular phase and the extracellular phase produce a potential difference of about 85 mV between the two phases. The intracellular phase potential is negative. Determine the difference in electrical potential energy per mole positive monovalent ion, e.g., Na+, between the two phases.

11.2

Calculate the change in the enthalpy of blood when it is subjected to an isothermal increase in pressure of 20 kPa.

11.3

The enthalpy change of a blood stream flow at 5.2 L/rain has an isothermal increase of pressure of 21 kPa. Calculate the work added to the blood as useful energy.

11.4

A small organism has a heat loss o f - q = 1.65 W and performs external work W = 0.025 N m/s. Calculate that part of the total energy expenditure that originates from its internal circulation, which involves the pumping of 122 mL/min of fluid against a pressure drop of 3.4 kPa with a net chemo-mechanical efficiency of 12%.

11.5

Estimate the energy expenditure for a steady process involving oxidation of 425 g/day glucose at 310 K and 1 atm. C6H1206 (aq) 4- 602 (g)~6CO2 (g) + 6H20(1 )

11.6

An adult male has an oxygen uptake of about 21.16 mol over 24 h, and the associated elimination of carbon dioxide and nitrogen is 16.95 mol and 5.76 g, respectively. The male has performed 0.12 MJ of external work over the same period and his energy expenditure at rest is E0 = 70 W. Estimate his energy expenditure, heat loss, and net efficiency for the external work.

11.7

An amphipod with a body weight of 9 txg consumes 3.5 × 10 - 9 mol oxygen every hour at steady state and eliminates 3.5 × 10 -9 mol carbon dioxide, 0.4 x 10 -9 mol N (as ammonia), and 0.1 × 10 -9 mol lactic acid. The external work power is 47 x 10 -9 W. Estimate the heat loss of the animal when the following four net reactions contribute to the energy expenditure. C6H120 6 4- 602(g)--*6CO 2 + 6H20

C6H1206---.2C3H603 C 5 5 H 1 0 4 0 6 4-

7802 ~

- 2870 kJ/mol

- 100kJ/mol

5 5 C 0 2 4-

52H20

- 34300 kJ/mol

C32H48010N8 4- 330 2 --. 32CO 2 4- 8NH 3 4-12H20

- 14744 kJ/mol

597

References

11.8

A small organism has a heat loss of 1.52 W and performs external work of 1.2 N m/min. Calculate that part of the total energy expenditure that originates from its internal circulation, which involves the pumping of 120 mL/min of fluid against a pressure drop of 25 mmHg (3.34 kPa) with a net chemo-mechanical efficiency of 10%.

11.9

The intensity of the sun's radiation on a clear day is observed to be 50 mW/cm 2. Calculate the accumulation of glucose during 8 h of exposure in a green leaf. Its surface area is 10 cm 2 and the heat loss from the leaf is 49 mW/cm 2. It is assumed that the temperature of the leaf is steady at 25°C and that only the net process proceeds (the reaction enthalpy is 478 kJ/mol CO2 at 25°C).

11.10

Consider the total energy available from the oxidation of acetate. What percentage is transferred through the TCA cycle to NADH, FADH2, and GTP? Acetate + 202 -~ 2CO 2 + 2 H 2 0

- 243 kJ/mol

1 N A D H + H +- + - 02 ~ NAD + + H 20 2

1

FADH2 + 2- 02 --* FAD + H 2 0 Z

11.11

GTP---, GDP + Pi

- 8 kJ/mol

ATP ~ ADP + Pi

- 8 kJ/mol

- 41 kJ/mol

- 41 kJ/mol

Consider the oxidation of acetate to produce ATE What percentage of the energy is available from the oxidation? Use the chemical reactions in Problem 11.10.

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