Robustness and time-scale hierarchy in biological systems

BioSystems 50 (1999) 71 – 82

Robustness and time-scale hierarchy in biological systems Igor Rojdestvenski a,*, Michael Cottam b, Youn-Il Park a, Gunnar O8 quist a b

a Department of Plant Physiology, Umea˚ Uni6ersity, Umea˚ 90187, Sweden Department of Physics and Astronomy, Uni6ersity of Western Ontario, London Ont. N6A 3K7, Canada

Received 15 May 1998; received in revised form 27 October 1998; accepted 2 November 1998

Abstract This study addresses the issue of robustness of biological systems with respect to microscopic parameters, especially the emergence of robustness as a consequence of time-scale hierarchy, applying naı¨ve thermodynamic and dynamic assumptions. Theoretical considerations of how the time-scale hierarchy can decouple physiological regulatory mechanisms are illustrated by two model systems involving the photosynthetic apparatus of green plants. © 1999 Elsevier Science Ireland Ltd. All rights reserved. Keywords: Green plants; Photosynthetic apparatus; Robustness; Time scale-hierachy

1. Introduction The phenomenon of plant life is associated with systems that efficiently utilize solar energy to sustain their structure, to grow and to develop. This high level of efficiency is achieved through substantial complexity of the plant organization involving a multitude of metabolic pathways and associated regulatory mechanisms. There is, however, an apparent contradiction between the complexity of the system’s organization and the adaptability of the system. Indeed, the more complex the system becomes, the more

* Corresponding author.

efficient it generally is in the stable conditions, to which it is adapted. This is achieved via simultaneous fine tuning of all the regulatory mechanisms to the parameters of the environment. In many situations, however, fine tuning would lower the flexibility of the system with respect to adjustment to variability in environmental parameters. This matter has recently been addressed in Barkai et al. (1997). The above considerations can be illustrated by the following thermodynamic description. Suppose the environment is variable, with its parameters {x1,...,xn } being distributed according to a probability density: Penv({x1, ..., xn })

0303-2647/99/$ - see front matter © 1999 Elsevier Science Ireland Ltd. All rights reserved. PII: S 0 3 0 3 - 2 6 4 7 ( 9 8 ) 0 0 0 9 2 - 6

(1)

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Here the set X of variables {x1, ..., xn } may stand for the temperature, atmospheric pressure, humidity, light irradiance, mineral content of the soil, and other relevant physico-chemical and biological factors. The distribution (Eq. (1)) has a certain entropy contribution, associated with it: Senv = −k % Penv(X) ln[Penv(X)]

(2)

X

The greater the environmental variability, the greater is this entropy contribution (see, for example, Landau et al., 1979). On the other hand, ‘fine tuning’ of the regulatory mechanisms implies that the internal parameters concerned are regulated to match precisely the environmental parameters. So, the more regulatory mechanisms are involved in optimization, the less is the entropy associated with the internal parameter values. Indeed, let us introduce a set of internal system parameters, Y = {y1, ..., yn }. We may then define, similarly to Eqs. (1) and (2), a system probability distribution Ps({y1, ..., yn }) associated with this set, characterizing how precise the tuning of the parameters is, and the corresponding system entropy Ss = − k Y Ps(Y)ln[Ps(Y)] . The more parameters that describe the system and the better their values are defined, the narrower is the distribution Ps and hence the smaller is the entropy Ss. In this case if the environmental variability is high enough, there is an entropy gradient, DS, between the environment and the system. To cope with this entropy gradient, a certain amount of energy, DE TDS

(3)

is needed to be dissipated, lowering the overall energy utilization efficiency. If, however, the regulation is not too finely tuned, this entropy gradient is less and the dissipation energy Eq. (3) is also lower (see Appendix A). In other words, in a variable environment the cost of maintaining a complex regulatory system is not accompanied any more by the corresponding efficiency gain. As time progresses, the system, which will be finely tuned for a certain set of parameters, is no longer optimal when the initial set is changed, over time, to another. This

conclusion stresses the importance of using combined entropy-information-energy approaches rather than simple energy balance calculations (see, for example, Torres et al., 1990). An interesting facet of these phenomena is how the system can, in principle, combine robustness and stability, as observed in living systems, with the elaborate regulation allowing high efficiency. The metabolic pathways are seemingly so interwoven that a change in the parameters of one regulatory mechanism should inevitably influence any other regulatory activities, swinging the whole system away from equilibrium. The importance of accounting for the interconnections and reciprocal influences of different levels of regulation, from quantum mechanical interactions to macroscopic adaptation, for energy and information processing are discussed in Conrad et al. (1996). However, to provide for necessary robustness, a certain mechanism for ‘decoupling’ different regulatory mechanisms must exist. One possible candidacy for such a decoupling mechanism is spatial compartmentalization. If a system, A, can be divided into several spatially separated subsystems (labelled a1, ….., ak) in such a way that the interactions between the subsystems are substantially weaker than the interactions inside the subsystems, then changes in one subsystem will not significantly affect the other. Thus the required decoupling might be achieved. To a degree, this is the case in plants, in which different metabolic reactions occur in different cell compartments and plant organs. There are, however, cell subsystems in which many metabolic reactions occur in the same place. Primary photosynthetic processes may serve as an example. Within the stroma and lumen of the thylakoid membranes in chloroplasts, light reactions (excitation trapping, charge separation and stabilization, ATP and NADPH synthesis), and the dark reactions comprising the Calvin cycle, occur simultaneously and spatially close to each other. In this case, there must be some other decoupling scheme allowing for different regulatory mechanisms to have a large degree of independence from each other.

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Table 1 Characteristic time scales of primary photosynthetic processes Process

Characteristic time (Dau et al., 1994)

(Govindjee et al., 1995)

Transfer of an excitation from one pigment molecule to another Exciton radiative decay

100 fs 3 ns

B1 ps 3 ns

Charge separation P680*Pheo“ P680+Pheo−

3 ps

3 ps

Charge transport to QA P680+Pheo−QA “ P680+PheoQ− A

300 ps

200 ps

Electron donation from TyrZ ZP680+Pheo“Z+P680Pheo

20 }300 ns

20 } 200 ns

Charge transport from QAto QB n− (n+1)− Q− A QB “ QAQB

200 }400 ms

200 }400 ms

State transitions S0Z+ “S1Z S1Z+ “S2Z S2Z+ “S3Z S3Z+ “S0Z

30 ms 100 ms 350 ms 1.5 ms

– 600 ms 800 ms –

A prominent property of biological systems is that the metabolic reactions comprise processes with significantly different time scales. Continuing with the example of the photosynthetic apparatus of green plants and algae, it is instructive to compare the time scales of different light and dark reactions (Table 1). Even at a first glance it can be seen that the characteristic times of different processes span over nine orders of magnitude. The aim of this paper is to discuss possible roles of time-scale hierarchy in decoupling regulatory mechanisms. First we formulate the concept of time-scale hierarchy. Then we discuss how the time-scale hierarchy contributes to the robustness of the system with respect to changes in its internal parameters. Two independent model examples, exciton propagation and trapping in photosystem II (PSII) and CO2 assimilation in Chlamydomonas reinhardtii, are presented to illustrate the concepts discussed. We should note that the mathematical treatment given here is illustrative rather than rigorous since the mathematical tools are employed for the sake of description rather than for precise, formal derivation.

2. Robustness and time-scale hierarchy Suppose there are n processes in the system, and their characteristic times, ti, form a timescale hierarchy such that: t1  t2  t3  ..... tn (4) Here we use the term ‘process’ for any set of metabolic reactions that operate at approximately the same time scale. Let us choose one such process (or group of processes), k, its characteristic time being tk. With respect to tk the other characteristic times t1, …, tn can be divided into two groups. The first, group G1 of processes for tI  tk (the ‘slow’ processes). For the corresponding times we may safely assume that, relative to tk, we have effectively: ti = (for all iG1) (5) The kinetic parameters of the processes characterized by the G1 group come as static boundary conditions with respect to the process k.The second, group G2 of processes for tI  tk (the ‘fast’ processes). For the corresponding times we may safely assume that, relative to tk, we have effectively:

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ti = 0

(for all iG2)

(6)

The kinetic parameters of the processes characterized by the G2 group are encompassed in the thermodynamic boundary conditions with respect to the process k. These thermodynamic boundary conditions are formulated in terms of equilibrium probability distributions with respect to a given set of values for intermediate and slow kinetic variables. The above procedure means, in effect, that if we seek an appropriate description for the process k, we should build the model in such a way that it includes the parameters of the G2 processes as (quasi)-equilibrium values. They are averaged over a time scale that is substantially greater than the G1 times, while still being less than tk. By contrast, the parameters of the G1 processes should be treated as ‘frozen’ at their instantaneous values. The term ‘robustness’ in the given context means that evolution is independent of certain parameters in the wide range of their variation. Suppose now that the fast processes from the G2 group are described by the following formal set of dynamic equations: L2 (y1, ...., ys, x1,..., xr,ys + 1, ..., yk,t ) = 0

(7)

Here, as previously, the y variables stand for the internal parameters of the system (metabolite concentrations, rates of processes, etc.) and the x variables represent the parameters of the environment, while L2 is the time evolution operator. We attribute the first s of the y variables to the parameters of the fast processes, whilst the s + 1 to k of the y variables describe the parameters of the slow processes and, hence, enter into Eq. (7) as static parameters. Suppose also that the Eq. (9) has an equilibrium solution corresponding to: (0) {y (0) 1 , ...., y s }

(8)

at given values of parameters x1,…,xr, ys + 1, …, yk. It is important to stress that the solution Eq. (8) does not necessarily depend on all the parameters of Eq. (7). Some of the parameters may affect only the kinetics, i.e. the pathway which the system chooses to approach the equilibrium, rather the equilibrium state itself. A good

example of such a situation might be a simple two-metabolite reaction equation: dCA =CAKAB − CBKBA dt

(9)

in which the rates KAB and KBA determine the kinetics of the reaction, while the equilibrium concentrations CA and CB depend not on the rates KAB and KBA separately, but only on their ratio. We may say that the equilibrium state of the system in Eq. (9) is robust against such changes in KAB and KBA that maintain their ratio. A formal mathematical discussion of time-scale decoupling, as applied to metabolic pathways control equations, was given in Delgado et al. (1995). The above arguments can be easily generalized for the system represented by Eq. (7). It is important that the kinetics of all the processes slower than those from the G1 group depend on equilibrium solution Eq. (8), rather than on the complete solution of Eq. (7). Hence, the effective number of parameters describing the slower processes may be reduced to only those that determine the equilibrium solution. The slower kinetics are insensitive to the other parameters, or, in other words, are robust against changes in them. It is interesting to look at the situation from the point of view of the entropy balance discussed in the Introduction. Indeed, the entropy of the system depends on the level of description, namely, on what constraints are put into the maximal entropy variational principle. If we study experimentally the kinetics of the slow processes, we cannot gain information about some of the parameters determining the kinetics of the fast processes. This is because the slow processes are insensitive to their values and these values cannot serve as constraints. The resulting arbitrariness produces an extra entropy contribution. Our reference to different level of description might seem subjective. However, it becomes pertinent if one realizes that the description itself is physically actualized in how slow processes interact with (or ‘see, measure, observe, describe’) the fast ones and, hence, does not involve presence of a subjective ‘observer’. We further discuss the consequences of the time-scale hierarchy by taking the examples of

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two systems, that model different situations in light and dark reactions of photosynthesis. The two models are completely independent, but have in common a substantial span of rates of different processes, which leads to emergence of robustness as a consequence in both cases.

2.1. (Example 1) Exciton propagation and trapping in photosystem II dimers: robustness with respect to recombination probability Recently we employed a Monte-Carlo method to study a simple model of exciton propagation and trapping in a simple model of PSII dimeric arrangements comprising one common peripheral antenna, two core pigment antennae and two reaction centers. The spatial model and the kinetic scheme are presented in Figs. 1 and 2, respectively. The dependences of the exciton lifetimes and photochemical trapping efficiency were calculated as functions of exciton kinetics parameters. These parameters were: 1. hopping probability, Phop, i.e. the probability for an exciton to ‘hop’ from the peripheral antenna (A) onto one of the core antennae

75

(B1, 2) and from B1, 2 onto the corresponding reaction center, RC1, 2. Hopping is a self-inverse process; 2. charge separation probability, f, i.e. the probability for an exciton, when in one of the reaction centers, to cause charge separation with a P680 reaction center donating an electron to pheophytin; 3. charge recombination probability, Prec, i.e. the probability of a charge separated state of the reaction centers to decay with production of an exciton in the same reaction center. This process is the inverse of charge separation; 4. charge stabilization probability, Pst, i.e. the probability of a charge separated state of a reaction center to undergo irreversible charge stabilization with pheophytin further donating its extra electron to quinon A (QA). This is a terminal state, which can only be achieved in the open reaction center; 5. dissipation probability, Pdiss, i.e. the probability of an exciton on A, B1, 2, RC1, 2 or a charge separated state of one of the reaction centers to decay irreversibly with its energy dissipating as heat. The exciton lifetime was calculated as the number of ‘Monte-Carlo’ time units (steps) from the creation of an exciton to achieving one of the terminal states, averaged over several thousand runs of the modeling routine. The charge stabilization yields for the CC (both reaction centers initially closed), OO (both reaction centers initially open) and CO (one reaction centre open and one closed) situations were calculated as the proportion of runs ending in the charge stabilization terminal state. The values of the above probabilities can be estimated from Table 1. For example, if we scale the time in single exciton hopping units, then: Pdiss  0.001;

f 0.1−1;

Pst  0.001−0.1 (10)

Fig. 1. A simplified model of a PSII dimer. (A) Common antenna for the PSII dimer. (B) Core pigments, RC1, RC2 reaction centers.

It is important to note that the time of opening of a PSII is from 100 ms to 1 ms in orders of magnitude. This time can be considered as effectively infinite with respect to the primary processes studied here, and we may consider a closed reaction center as staying closed throughout all

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Fig. 2. Kinetic scheme for the model. Bidirectional arrows denote reversible transitions, unidirectional arrows denote irreversible transitions and thin arrows denote dissipation as heat. The corresponding probabilities for the processes are written inside arrows. (A) State with excitation localized in A. Pigments B1, B2 excitation localized in core antenna in the vicinity of RC1 and RC2, respectively; RC1, 2 excitation localized in one of the reaction centers CS1, CS2. Charge separated states P680 + Pheo − in reaction centers 1 and 2, respectively. PhCh1, 2, terminal charge stabilization states.

the lifetime of excitation in PSII, hence the above termination by charge stabilization. As to the value of the recombination probability Prec, it is not easy to estimate from the experimental data. To study the effect of different Prec values on the kinetic properties of the system we plotted the dependences of excitation lifetimes and charge stabilization yields (CO-configuration) as functions of Prec (Fig. 3a, b, respectively). The data show that for the above dimer arrangement to be more photochemically efficient than the monomeric arrangement a small but nonzero charge recombination probability is required. Also the recombination probability affects the photochemical efficiency in the CO situation in such a system in a trigger-like manner, that is,

an onset of a very small recombination probability increases the photochemical efficiency by some 80%, that changes very slowly with further increase of recombination probability. The same can be concluded for the Prec dependence of excitation lifetimes. The presented results mean that the characteristics of the model are robust with respect to variations in recombination probability. Mathematically the discussed robustness can be explained as follows. Two terminal processes, thermal decay and charge stabilization, are competing for the excitons. The overall probability of charge stabilization depends on how intense the recombination processes are. If recombination is too slow (Prec 5 Pdiss), the excitation has a chance

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Fig. 3. Lifetimes (a) and charge stabilization yields (b) for a dimer with both centers open (OO, diamonds), one open and one closed (CO, squares) and both centers closed (CC, triangles) in the case of low value of charge separation parameter f and low charge stabilization probability. Pdiss = 10 − 3, f= 0.1, Pst = 10 − 3. Dashed line corresponds to a situation of two separate centers, one closed and one open (CO-monomers).

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to stay in the closed reaction center in charge separated state for a time long enough to decay thermally. This lowers the charge stabilization yield and increases the overall excitation lifetime. On the other hand if the recombination happens too often (Prec $f ) then the excitation would escape the trap having little chance for charge stabilization, which also lowers the charge stabilization yield and increases the overall excitation lifetime. These are the only effects of Prec. Hence, for intermediate values of Prec that lie within the interval: Pdiss Prec f (11) the precise value Prec does not matter, which accounts for the observed robustness. This feature provides an explanation for the stability of the photosynthetic machinery against variations in the environment, such as slight changes in mutual arrangements and orientations of the antenna, core and reaction center pigments that change the recombination probability. Our model predicts that these changes will not significantly affect the kinetics of the primary exciton propagation and trapping (lifetime), as well as its quasi-equilibrium characteristics (such as charge stabilization yield). If these changes appear as a result of adaptation to changes in environmental conditions, then this adaptation will not disturb the adaptation of further (and slower) photochemical processes, thus providing the necessary decoupling of regulatory mechanisms.

2.2. (Example 2) Carbonic anhydrase-dri6en pump in Chlamydomonas reinhardtii We now turn to another example of time-scale dependent robustness. In Park et al. (1998), Rojdestvenski et al. (1998) we studied the bicarbonate (BC) depletion effects in the high-CO2 grown C. reinhardtii cell wall-less mutants with (cw92) or without (cia3/cw15) active carbonic anhydrase (CA) associated with PSII. We observed the shutdown of oxygen evolution in cw92 that we attributed to CO2 depletion of the Calvin cycle. We suggested the role of CA being to pump CO2 from thylakoid lumen into thylakoid stroma, as a result of a low pH value in the lumen due to PSII-ATPase regulation.

The mechanism, in our opinion, operates in the following way. As it is known, the steady-state values of the pH are approximately 7.8 in stroma and 5.3 in lumen. High value stromal pH facilitates activation of Rubisco, the primary enzyme of the Calvin cycle. However, high stromal pH shifts the BC/CO2 balance in the stroma towards overwhelming prevalence of BC, thus leaving the Calvin cycle virtually without its ‘fuel’, CO2. On the other hand, in the lumen low pH provides for CO2 dominance. CA, by its nature, facilitates pH-determined balance of the BC and CO2 concentrations; the equilibration time in the presence of CA becomes some 105 –106 faster than without it. Admitting the (possibly PSII-activated) BC diffusion into lumen, quick CA-facilitated CO2 production from the incoming BC and CO2 diffusion into stroma, we achieve a sort of ‘engine’, which converts stromal BC into stromal CO2. The backward conversion of CO2 into BC in stroma in the absence of CA is slow enough for the Calvin cycle to compete for CO2, leading to inorganic carbon depletion in the system. In the case of the mutant cia93, which has no active CA in the lumen, the depletion happens much more slowly and its effects are less pronounced. To back up our understanding of the actual CO2 depletion mechanism in the above system we suggested a simple model which was capable of explaining the results of in vivo experiments, as well as the differences between cw92 and cia3. The model is presented in Fig. 4. There are two pools of BC, lumenal and stromal, and there are also two pools of CO2. There are four reversible processes: 1. BC diffusion from stroma to lumen and back; 2. CO2 diffusion from stroma to lumen and back; 3. uncatalyzed BC conversion to CO2 and back in stroma; 4. BC conversion to CO2 and back in lumen, facilitated by CA. There is also one irreversible process, that is, CO2 consumption by the Calvin cycle. The rates of the processes have been calculated, according to Spalding et al. (1985), Raven et al. (1997), Park et al. (1998), Rojdestvenski et al. (1998). In Fig. 5 we present our modeling results for the time dependences of the BC and CO2 concen-

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trations in the lumen and in the stroma. A characteristic feature of the presented data is that after a short induction period (Fig. 5a) the concentrations at all times remain proportional to each other. This fact is observed best in Fig. 5b, where the logarithmic curves are clearly equidistant. The above result hints at the onset of a certain partial equilibrium reached during the induction period. Also, we checked our results against the variations in the diffusion and conversion rates. Our data showed (results not presented) that even an order-of-magnitude change in the diffusion and conversion rates affected only the duration of the induction period, having no influence on the results after the induction is over. On the other hand, changes in the Calvin cycle parameters Vm and Km had almost no influence on the induction period, significantly changing the further evolution of the system (results not presented). Thus, the overall evolution of the model proved robust against substantial variations in four parameters out of six, and the induction period proved robust against variations in the remaining two parameters. Clearly in this model there is a time scale decoupling of regulatory mechanisms, because the induction (characteristic time tind 10 s) and the steady state time evolution (tss 10 min) represent quite different time scales. Also the first four processes in Table 2 seem to ‘rule’ over

the ‘fast’ induction kinetics and the Calvin cycle parameters, respectively, determine the ‘slow’ steady state kinetics. To obtain an explanation on how this timescale hierarchy emerges in the above model, we calculated certain ‘equilibration values’, namely: EBC =

[BC]str for BC diffusion; [BC]lum

ECO2 =

[CO2]str for CO2 diffusion; [CO2]lum

EC/B = f(pHL)

[CO2]lum for CO2 [BC]lum

conversion in the lumen;

(12)

where f(pHL)= 10pHL − 6.25 with pHL standing for the pH value in the lumen. It is easy to prove that each of these values would have been equal to 1, had the corresponding process been the only one in the system. The time dependences of the quantities in Eq. (12) are presented in Fig. 5c. The EBC, ECO2 and EC/B values tend to stabilize after the induction period and the saturation values differ from 1 at most by approximately 13%. This fact means that the above processes are virtually independent of each other and of Calvin cycle. Hence any regulatory mechanism involving changes in diffusion properties of thylakoid membrane would hardly affect either CA-facilitated BC/CO2 conversion in the lumen or the Calvin cycle, and vice versa. Mathematically this robustness is a consequence of the fact that the diffusion rates, conversion rates and Calvin cycle rates represent actually three different time scales, each differing by 1 or 2 orders of magnitude. Then, with respect to ‘slow’ steady state kinetics we can simplify our model, stating: lum KD = K − h = ;

Fig. 4. A model of CA-driven CO2 pump in C. reinhardtii. See text for further details

79

str K− h=0

(13)

This led to presented in Rojdestvenski et al., (1998) ‘quasi-equilibrium theory with only two parameters Vmax and Km of the Calvin cycle. The results of this theory, when compared to the computer modeling results, showed good correspondence (not shown, see Rojdestvenski et al.,

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Fig. 5. (a-b) Time dependences of the concentrations (in the C0 units) of bicarbonate (squares) and CO2 (triangles) in stroma (open symbols) and lumen (solid symbols), as well as total inroganic carbon content (bold solid line). pHS =7.8, pHL =5.3. (a) concentrations, induction period; (b) logarithms of concentrations, longer times; (c) equilibration factors EBC (triangles); ECO2(open squares); EC/B(solid squares) at a) pHS = 7.8, pHL = 5.3; (d) depletion times vs. added BC concentration, quasi-equilibrium theory compared to experimental data.

1998). They also showed a remarkable correspondence with the experiment (for the times of Ci depletion, see Fig. 5d) outside the low-times area, where the depletion time is of the same order of magnitude as tind and the theory becomes transcendental.

3. Conclusions In this paper we discussed the time-scale hierarchy induced robustness and showed, by using two model systems as examples, how the timescale hierarchy leads to the decoupling of regula-

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Table 2 Rates for the model of the CA-driven CO2 pump in Chlamydomonas reinhardii a Process

Rate function

Rate constants values

CO2 diffusion from/to stroma BC diffusion from/to stroma BC/CO2 conversion in stroma (no CA)

lum 6 CO /Ci−[CO2]str/Ci) d = KD([CO2] 2 6dBC = KD([BC]lum/Ci−[BC]str/Ci) str 6h 6−h = ([BC]str/Ci)K−h

KD =3×10−14 mol/cell per s KD =3×10−14 mol/cell per s str K−h =1.5×10−17 mol/cell per s

str = ([CO2]str/Ci)K−h exp{pH−6.25} lum 6−h = ([BC]lum/Ci)K−h 6h

lum K−h =1.5×10−12 mol/cell per s

BC/CO2 conversion in lumen (CA present) CO2 assimilation by Calvin cycle

lum = ([CO2]lum/Ci)K−h exp{pH−6.25} Vmax[CO2]str/([CO2]str+Km)

Vmax =0.9×10−16 mol CO2/s per cell Km =1×10−14 mol/cell

a Abreviations: [CO2]str and [CO2]lum stand for CO2 concentrations in the stroma and in the lumen respectively, [BC]str and [BC]lum stand for BC concentrations in the stroma and in the lumen respectively, Ci stands for the total inorganic carbon content (i.e. quantity of added bicarbonate), and subscripts h and −h refer to hydration (CO2 to BC) and dehydration (BC to CO2) processes.

tory mechanisms and the emergence of robustness. Several important comments have to be made. First, a model is not a real system, but rather a formalized description of our understanding of how the real system operates in given experimental conditions with respect to certain measuring methods. Hence, the properties of the model differ from the properties of the real system. This has to be born in mind when the modeling results are interpreted. Second, an adequate model has to have the properties relevant to the experiment similar to those of the real system. For example, if we study the kinetics of exciton propagation and trapping in PSII, the kinetic parameters of the process, as determined independently (e.g. via absorption spectra studies, see, e.g., Dau et al. (1994), Govindjee et al. (1995) and references therein), have to be incorporated in the model. That the model then becomes capable of explaining some of the features of other experiments (such as the picosecond fluorescence curve sigmoidicity) proves its relevance with respect to these experiments. The more that such ‘emerging’ (as a result of modeling) features can be mapped onto the corresponding experimental data, the more adequate the model is. Robustness as such is one of these emerging features. Third, the robustness with respect to certain parameters, displayed by a model and a real system in a given experiment makes it difficult to utilize a tempting procedure of adjusting the char-

acteristics of the model as to achieve best fit to the experimental results. In this respect robustness would provide for a significant uncertainty in those parameters, often hidden in the user-friendliness of various statistical software packages. A good example of such a situation is given in Hopkins et al. (1996). Acknowledgement This work has been supported by grant no. I-AG/WB 04830-334 from the Swedish National Research Council. Appendix A. Entropy and information in adaptation The aim of this Appendix is to give a justification of the expressions Eqs. (1)–(3) based on a crude model for adaptation mechanisms. Suppose we have an environment which is described by a microcanonic ensemble. If the statistical weight of it is W, then the Boltzmann entropy associated with this system would be Senv = − k ln W (A1) k being Boltzmann’s constant. Suppose we construct an adaptable system, which has to have such a set of adaptation mechanisms that for each state in W a different finely tuned adaptive mechanism exists.

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Suppose also that each mechanism consists of ‘details’ (for example, enzymes). In fact, we refer here to, precisely, how many different ‘details’ have to be encoded in the genome of our adaptive system for the system to be adaptable to every state out of W. If we have N details (and, hence, N corresponding ‘genes’), then the maximal number of mechanisms that can be constructed out of them is the same as the number of bits you need to innumerate all the W states: 2N =W

or

N=

ln W ln 2

The genes encoding details are of molecular size. So, to activate a detail, at least a kT energy has to be applied. If the state changes every, say, t seconds, we have to apply energy of: (A3)

to cope with the change, m being a number of details in the state, or, subsequently, a number of unities in the binary representation of the state’s number. The average energy spending per time t will be, hence: E( 1 = N kT 2 =



N

% mC m N

m=0



=

1 2N



N

% m

m=0

N 2

N! m!(N −m)!



(A4)

or, if we substitute Eq. (2) for N, E( =

kT ln W :0.72TSenv 2 ln 2

(A5)

which is sufficiently close to the thermodynamic expression for the entropy contribution to the free energy for such a crude approximation. Now let us assume that the tuning is more rough, that is, each adaptive mechanism is tuned not precisely to one state out of W, but, more loosely, to W % states out of W. This loose tuning has a certain entropy associated with it, namely: Ss = − k ln W %

2N = W/W %

or

N=

(ln W− ln W %) ln 2

(A7)

and the necessary maintenance energy spending per time t is: E( =

kT(ln W− ln W) :0.72 TDS 2 ln 2

(A8)

where DS = Senv − Ss, which is the same as Eq. (3).

(A2)

E =kTm

In this case the number of details, instead of Eq. (2), obeys:

(A6)

.

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