Reaction kinetics of poikilotherm development

,I. theor. Biol. (1977) 64, 649-670

Reaction Kinetics of Poikilotherm Development PETER J. H. SHARPER AND DON W. DEMICHELE$ Biosystems Research Division, Department of Industrial Engineering, Texas A and h4 University, College Station, Texas 77843, U.S.A. (Received 29 December 1975, and in revised form 3 1 March 1976) A stochastic thermodynamic model of poikilotherm development has been derived from the Eyring equation assuming multiple activity states of the underlying developmental control enzymes. This analysis brings together into a general model the day-degrees concept and the Arrhenius hypothesis as interpreted by Eyring. The compensating effect of enzyme inactivation at high and low temperatures incorporated into the model has the following consequences. (i) It demonstrates the validity of the linear approximation (day-degree concept) in the mid-temperature region for some organisms. (ii) It effectively establishes a low-temperature threshold for develop-

ment. (iii, It reduces the rate of development at higher temperatures, thereby establishing both an optimum and upper threshold for development. The resulting equation has been found applicable to a wide range of organisms. 1. Introduction

With the advent of computer simulation to the applied biological sciences, there has been increased interest in mathematical models that describe organism growth and development as a function of temperature. A review of the effects of temperature on organism development is given by Precht, Christophersen, Hensel & Larcher (1973), particularly the subsection written by Laudien (1973). As is evident from these and other reviews, there are numerous empirical and physiochemical formulations of development,

two of which are relevant to this analysis. These are the day-degree or temperature summation rule put forward by Candolle (1855) and Reibisch

(1902) and the non-linear temperature

inhibition

model derived by Johnson

& Lewin (1946). The day-degree summation rule is widely used in applied zoology, entomology and agronomy due to its ease of application and the fact that within certain temperature limits it approximates observed values t Assistant Professor of Bioengineering. $ Associate Professor of Bioengineering. 649

650

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J. H.

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W. DEMICHELE

(Peairs, 1927). The day-degree concept assumes that the rate of development is proportional to temperature : k = b(T- Ta) (1) where k is the rate of development, b is a constant, Tis the absolute temperature and Tu is the temperature at the so-called developmental zero. Figure 1

Region of validity day-degree concept ‘E 2 B aI 5 73 % (Y z n

\ \ \

Temperature

(*Cl

FIG. 1. Typical relationship between temperature and rate of development (Wigglesworth, 1965). “Ta” represents the theoretical “developmental zero”.

shows the typical S-shaped curve of development, including the approximately linear region of the development curve where the day-degree concept is valid. Problems with the day-degree model include: (a) the developmental zero, “Tu”, is not the true threshold for development: and (b) gross errors in predicted development rates occur at the temperature extremes (Stinner, Gutierrez & Butler, 1974). The results of this study provide a functional form and conceptual basis whereby (a) and (b) are eliminated. Johnson & Lewin (1946) developed a non-linear mathematical function to describe the inhibition of microbial growth at high temperatures. A similar derivation for low temperature inhibition of enzymatic processes has been formulated by Hultin (1955). Each of these equations describes the inhibiting effect of either high or low temperatures on organism development. However, there is no mechanistically based formulation of poikilotherm development which results in a linear response in mid-temperatures coupled with non-linear temperature inhibition at both high and low temperatures. The extent of the temperature inhibition can be shown by an Arrhenius plot

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(Ingraham, 1973). Typical Arrhenius plots of organism development are shown in Fig. 2, in which it can be seen that both high and low temperature inhibition of process rates are significant factors. This paper describes the derivation of a general model of organism development, based on biological process rates incorporating both high and low temperature inactivation of control enzymes. The analysis identifies six thermodynamic constants which appear to characterize many organisms’ enzymatic response to temperature. Further, it suggests a possible underlying cause for the linear response to temperature in the mid-temperature region upon which the degree-day rule depends. Some of the assumptions of the model are similar to those of Johnson & Lewin (1946) and Hultin (1955) ; however, these authors did not clearly state all the underlying assumptions 3,o -

..L---&-d 30

31

32

33

(I/T

.Reciprocol

absolute

34 x 104)

temperature

35

36

37

(“Km’i

2. Arrhenius plots of specific development rates of a variety of micro-organisms. represents a psychrophilic pseudomonal; ‘3” represents a mesophilic strain of Escherichiu co/i and “C” a thermophilic strain of Bacillus circdurs (Farrell & Rose, 1967). FIG.

“A”

T.B.

43

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nor attempt to identify those conditions under which the assumptions may be invalid. In this analysis, the derivation is different in that it is stochastic, while the derivations of Johnson & Lewin (1946) were based upon chemical kinetics. 2. Development of Kinetic Model From a biochemical viewpoint, the development process is a complex series of reactions involving numerous enzyme systems. Although the biochemical complexity is overwhelming at first glance, evolution has ensured that the organism’s development is appropriate to its environment, particularly its thermal environment. Biochemical control is effected at various critical points in the biochemical pathways involved. As pointed out by Reiner (1968), these control points are found at either the beginning or branch points of metabolic pathways. The enzymes located at these strategic points may be termed control enzymes. Due to the series arrangement of enzymes in most biochemical pathways, the reaction rate of the control enzyme will regulate the rate of the overall metabolic process. For convergent parallel pathways, biochemical control will generally be co-ordinated to ensure sufficient metabolism, and the overall process will resemble a single control enzyme process. Thus, the single control enzyme concept, while not necessarily literally correct, has great conceptual simplicity and general plausibility. The three underlying assumptions of the analysis are: (a) Development is regulated by a single control enzyme whose reaction rate determines the development rate of the organism. (b) The development rate is proportional to the product of the concentration of active enzymes and their rate constant (which in itself is temperature dependent). (c) The control enzyme can exist in two temperature dependent inactivation states as well as an active state. The first assumption was put forward by Johnson & Lewin (1946). At this point Johnson & Lewin’s assumption will be deemed valid and applicable to all poikilotherm development. The consequences of this assumption being invalid are discussed in detail later in the paper. The second assumption provides a mechanism for temperature inhibition of development, while the third assumption provides a basis for the unification of the high and low temperature inhibition of development in one model. The first step in the analysis is to calculate the probability that the control enzyme is in an active state. To calculate this probability, a number of assumptions relating to the rate controlling enzyme have been made. These are:

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(i) The control enzyme can exist in one of three energy states. State 1 predominates at low temperatures; energy state 2, which represents the active enzyme configuration, predominates over the mid-temperature range; and energy state 3 predominates at high temperatures. The three energy states and their possible transitions between states are shown in Fig. 3.

0

Energy state 2

0 0

I%. 3. Three energy states of the control enzyme and their reversible transitions.

(ii) Enzyme conformations associated with states I and 3 are catalytically inactive while conformation state 2 is catalytically active. Enzyme structure and activity is associated to a large extent with the specific dimensions and geometry of the catalytic sites of the enzyme molecule. At high and low energies associated with high and low temperatures, the enzyme undergoes a conformational transition which renders the enzyme inactive. (iii) Transitions between energy states are completely reversible. Huang & Cabib (1973) indicate that about 20 enzymes have been reported to exhibit reversible low temperature inactivation. Reversible inactivation has also been observed at higher temperatures, provided the elevated temperature is not maintained for too long (Sizer, 1943) or the relative enzyme concentration is not too high (Brandts, 1967). Alexandrov (1964) has shown that reversibility of thermal inhibition of most biological processes extended up to 8’C above the high temperature inactivation threshold. Trypsin is an example of an enzyme showing complete thermal reversibility of denaturation (Northrup, 1939).

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(iv) An individual enzyme molecule within the population cannot simultaneously exist in more than one form. At any temperature the probability of being in any specific form is less than one, and the cumulative probability of being in energy state 1, 2 or 3 is equal to one. (v) The probability that the enzyme molecule at any given temperature 1 can move from state 1 directly to state 3 is negligible; therefore, it can be assumed that to move from state 1 to 3 and vice versa, the enzyme must pass through energy state 2 (see Fig. 3). (vi) Enzyme energy state transitions are unimolecular: that is, they can occur without the aid of other catalysts or substrates. Sizer (1943) found that reversible inactivation of enzymes by heat was unimolecular, occurring in the presence or absence of substrate. Scrutton & Utter (1965) found that cold inactivation of pyruvate carboxylase was accompanied by dissociation of the enzyme protein subunits which. when heat treated, would reassociate into the active form of the enzyme. Huang & Cabib (1973) found that cold inactivation of glycogen synthetase resulted from a change in configuration rather than a dissociation into subunits. (vii) Transitions between states are randomly distributed with a mean transition rate specified by the Eyring equation where the transition rate constants ki take the general form of equation: ki

=

KT h

_._.

e’ASif

-Alli~/T,/R

(2)

where K is Boltzmann constant, T is absolute temperature, h is Planck’s constant, AS# is entropy of activation, AHf is enthalpy of activation, and R is the gas constant. Implicit in the use of this equation is the assumption that AS! and AH: are temperature independent. The Eyring equation has been found applicable to high temperature enzyme inactivation by Eyring & Stearn (1939) and Sizer (1943). It has been found applicable to low temperature inactivation by Kavanau (1950) and Hultin (1955). (viii) Transitions between energy states have reached steady-state. Enzymes vary in the time required to reach a new steady-state activity given a sudden temperature change. For cold inactivation, Scrutton & Utter (1965), Bergersen (1971) and Huang & Cabib (1973) report inactivation times of 30 to 300 min and reactivation times of 5 to 15 min.? t The authors recognize that some inactivation transition times may result in nonsteady-state conditions under certain circumstances (such as prolonged low temperatures followed by a rapid temperature rise). However, for the purposes of this analysis, it is assumed that enzyme inactivation reaches steady-state equilibrium within a time period of less than 60 min. Alternate formulations are necessary to describe those processes which reach quasi-steady-state within longer observation periods.

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High temperature inactivation and reactivation times will probably be considerably faster as these unimolecular transitions occur at high reactivation energies due to the higher thermal energies of the protein molecules. (ix) If no transition between states of the enzyme has occurred during time interval At, the probability of a transition occurring in the next time interval At is not changed. This is known as the “forgetfulness” propertq which naturally leads to the result that the time between transitions will bc exponentially distributed. This assumption can be justified in that the alternative is to assume some form of molecular memory which is not consistenr with our understanding of the mechanisms of biochemical reactions. Thercfore, the probability density of transition times can be described as f(ti) = ki eekiri ti > 0. (3) The exponential distribution has the convenient property that the probability of a transition occurring during a very short time interval At is equal to kiA.r. As the time interval becomes infinitely small (At -+ 0), the probability o! more than one transition occurring during the time interval becomes infinitely small. Therefore, the probability of no transitions occurring during At is approximately equal to { 1 - k,At). Using simple probability relationships of the exponential distribution, the probability of the enzyme being in energy state 1 at time t +At can bc calculated if the probability of being in each of the states at time t is known : P,(t+At) = P,(t)(l -k,At)+P2(t)k2Af. The rate of change can be assessed over the time interval At: Pt(t+At)-PI(t) -__ = -k,P,(t)+k2P3(t). -,t Taking the limit as At --, 0: @l(t) __

(4)

(6)

dt = -k,P,(t)+k,f’,(r). Similarly, the rate of change of the probability can be calculated:

of being in states 2 and 3

dP,(t) --=dt k,P,(t)-(kz+k,)P,(t)+k,P,(t) dP,(t) dt = k3P2(t)-k4P3(t).

(S)

-___

Solving for steady-state conditions

i.e. dp$

P, = l/(1 + k,/k,

= 0 yields: 1

f k,/kJ.

(9)

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Equation (9) shows that the probability of the developmental control enzyme being in the active state is solely a function of the rale constants controlling inactivation and reactivation. Substituting equation (2) for i = 1, 2, 3, 4 yields

If equilibrium at a given temperature has been reached, then only the differences in entropy of activation and enthalpy of activation need be considered. Therefore, equation (10) can be simplified: p,

=

l/[l

+e(AS~-AfMTJ/R

+e(~S~r~Wl’)l~]

(11)

where ASL represents the difference in entropy of activation between states 1 and 2, AH, represents the enthalpy of activation difference between states 1 and 2 at equilibrium, AS, represents the entropy of activation difference between states 2 and 3, and AH, represents the enthalpy of activation difference between states 2 and 3 at equilibrium. Equation (11) represents the probability that the developmental enzyme will be in the active state thereby catalyzing the developmental process. It can be seen from this equation that the only variable is 7’, the absolute temperature. Each of the other parameters are constants independent of temperature. The concentration of the active control enzyme, c,, is given by: c, = P7E,

(121

where E, is the total concentration of control enzyme. The development rate itself can be described in terms of the control enzyme-substrate interaction. If the substrate level is S, then the development process controlled by the rate limiting enzyme can be described by a typical enzyme kinetic equation which can be written stoichiometrically : S+E+C~P+E/ 1

(13a)

and Ef + c = E, (13b) where E/ is the concentration of free, active enzyme, S is the concentration of substrate, C is the concentration of enzyme-substrate complex, P is the product concentration formed by the reaction and k:, k?,, and li: are rate coefficients for the reaction which also obey the Eyring transition state hypothesis.

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Equating the rate of development

DEVELOPMENT

657

R, to the rate of change of either S or

P, yields (Reiner, 1969): _dP = R

=

’

dt

k;SE&P, ---

(14)

k;Sifk’_,+k;’

For limiting substrate conditions, equation (14) will hold where k;, k’- 1, k; and P, are complex exponential functions of temperature. However, for non-limiting substrate conditions, where k’,S $ li’-, +k;, equation (14) reduces to RD = E,k;P, (15) where k, is the rate constant for development assuming no enzyme inactivation and is described by the Eyring equation [equation (2)]. In this analysis, only conditions of non-limiting substrate supply will be considered. Limiting substrate, which might arise due to poor food quality, quantity or possibly disease will be considered in a subsequent article. Substituting for k; and P, in equation (I 5) KT R, = ____-. ’ h E ._

e(AS,:-AH~r/T,/R

1 + e(AS’--AH~/T’/R

-

----~

+ e(AS~

- AHfr/T)/R

(16)

E, and S: are unknown thermodynamic constants of the organism, which can be summarized by 4, where: 4 = AS: +ln (K&,/h). If we assume E, is relative concentration, then C#Ihas units “K-’ s-‘. Thus, the final equation for development under non-limiting substrate conditions is : R,=--

T ,WA&lR 1 +e(~~-AH~!T”R+e’AS~-AHt/T’/R

(17)

Equation (17) has only one variabIe T, the absolute temperature of the organism. The values of K, h and R are physical constants which are independent of the organism. The values of Cp, AH:, AS,, AH,, AS, and AH, are constants which reflect the individual thermodynamic characteristics of the organism’s control enzyme system which is assumed to control development. 3. Determination

of Organism Constants

The differences in entropies and enthalpies of thermal inactivation of the developmental control enzyme for a particular organism must be inferred from the temperature dependent development data. The analysis presented in this study is based upon the assumption that these thermodynamic

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constants represent the underlying enzymes controlling development. Entropies and enthalpies of thermal inactivation have been measured for numerous enzymes. These values provide an estimate of values one would expect to find for the overall development process. However, it should not be surprising to find that AH:, AS, and AH, are higher than average, and that AHL and ASL are more negative than average. This would result from the fact that the rate controlling enzymes are those enzymes with the most extreme constants and/or lowest concentration. This is why they are rate controlling. Sizer (1943) has documented the activation energies and entropy changes accompanying enzyme inactivation at high temperatures for over 50 enzymesubstrate systems. Enthalpies of inactivation at high temperature varied from 10 000 to 198 000 cal/mol, with the majority in the 40 000 to 100 000 cal/mol range. Entropies of inactivation also showed a wide variation from 24 to 537 cal/(mol-“K). However, over 50% of the reported entropies of inactivation were greater than 1 IO cal/(mol-“K). Johnson & Lewin (1946) report values of ASH and AH, for bacterial growth (E. coli) of 476.46 cal/mol-“K and 150 000 cal/mol respectively. It is apparent from the data presented by Sizer (1943) that the high entropies of inactivation accompany large enthalpies of inactivation. From equation (I 1) it can be seen that it is the difference in these two terms that controls the argument of the exponent which results in reasonably rapid inactivation rates. There is considerably less data available on the reverse reaction; i.c. the reactivation of the enzyme. One enzyme for which these values have been collected is trypsin. Sizer (1943) reports that the reactivation reaction has a negative enthalpy of reactivation of -27 440 cal/mol which is balanced by a high loss in entropy of reactivation of - 168.4 cal/(mol-“K). Thermodynamic constants for low temperature inactivation and reactivation have been calculated by Hultin (1955) from the data of Sizer & Josephson (1942) Enthalpy differences AH, between the cold inactive enzyme and the active enzyme were -27 000 cal/mol for lipase, - 51 000 cal/mol for saccharase and - 53 000 cal/mol for trypsin. Hultin did not calculate the entropy changes associated with the cold inactivation and reactivation transition; however, the entropy changes ASL can be calculated from the T. values yielding AS, equal to - 194 cal/moJ-“K for lipase, - 376 cal/mol-‘K for saccharase and - 386 cal/mol-“K for trypsin. The influence of temperature on enzyme activity can be seen from Fig. 4. In this figure, the probability of the enzyme being in the active state is plotted as a function of temperature for four different sets of enzyme thermodynamic constants. Table 1 gives thermodynamic parameters typical of those presented by Sizer for the curves 1, 2, 3 and 4. It can be seen that, except in

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Temperature

DEVFI

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OPMENT

(OC)

FIG. 4. Probability of the developmental enzyme being in the active state for various thermodynamic constants. The numerical values of the constants relating to lines 1. 2, 3 and 4 are listed in Table 1. TABLE

1

Thermodynamic constants to show efect of temperature on developmental enzyme activity Case 1 2 3 4

ASH (caI/mol-“K 51.9 160.0 210.5 375.3

AHH (cal/mol)

A& (cal/mol-“K)

17000 50000 65 000 115000

-89.5 --177.0 -227.0 - 177.0

AHL (cal/mol) --25 ooo --50 000 -65000 --50000

case I, 1:he probability of being in the active state is small at both high and low temperatures, reaching a maximum at temperatures in the mid range. To test the general applicability of the thermodynamic model, a range ot representative organisms and their respective development rates, as functions of temperature, have been chosen from the literature. These organisms include: Escherichia co/i k-l 1-27 (bacteria); Pseudatomoscelisseriatus Reuter (cotton fleahopper: insect) ; Drosophila melanogaster (fruit fly : insect) : Tetrahymena pyriformis (protozoon) ; Pinus taedu (loblolly pine : root development) ; Zea Mffys (corn : coleoptile development) ; Chiorella sorokiniana (algae): and Gossypium hirsutum L. (cotton: time to first flower-bud). These organisms represent a wide range of poikilotherms for which developmental data as a function of temperature is available. References for these data sets are given in the following section.

may.7 taeda hirsutum

nlelanogaster

cob pyriformis sorokiniana

Maturation rate Maturation rate Maturation rate Prepupal development rate Coleoptile development rate Root extension rate Development rate to first flower bud Nymphal development rate

Processes

j’ Low temperature data point used twice.

P. seriatirs

E. T. C. D. 2. P. G.

Organism

16.15 5564

20.03 IO.65 32.69 19.43 19.49 Il.82

As,

10 033 21 600

215.31 177.1

cal/mol cal/mol-‘K __ -.. 9 630 432.8 9 258 670.5 12 939 180.2 10490 2266 9170 211.8 5 913 538,5

Ai% 660 800 977 113 203 630

65 363 54 671

135 204 56 69 65 163

cal/mol

AHH

Developrnerrt comtarm for a range of poikilotllemic

TABLE 2

~~85.8 -589

cal+t~ol-‘K --. -181% -189.5 ~- 184.4 -156.9 -95.0 -216.6

ASI.

AH,

556 768 632 373 194 603 -25 016 - 168 143

-52 - 54 -54 -44 - 27 61

caljmol

orgnlrisms

2.68 9.06 4.31 158 2.67 1.45

L’ 10-l ,s’ lo-’ :’ lo-’ I, lo-‘, ~ lo- I ‘\ lo- 4

Chi square test x2

t5.t 9.33 lo- *( 7 5.23 /: lo--

8 13 7 16 ‘1 7

Number data points

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Constants for each of the various organisms listed above were estimated using equation (17) with a non-linear regression technique. Table 2 shows the values of the constants for each organism as well as the chi-square estimate of the goodness-of-fit between equation (17) and the relevant published data points. The values of these constants are of the same order of magnitude as the enzymatic thermodynamic constants found by Eyring & Stearn (1939). Sizer (1943), Johnson & Lewin (1946), and Hultin (1955). The high values of AH, and AS, are indicative of high temperature denaturation of proteins, while large negative values of AH, and ASL are indicative of low temperature denaturation of proteins.

4. Comparison of Thermodynamic Model with Experimental Observations Biological development in its broadest sense involves the organism’s process of growth and differentiation. Rates of development are determined from the time required for a developmental process to move from one state to anothler. This can involve times between events, such as consecutive unicellular divisions of micro-organisms, times between instars of insects, or time to flower bud-initiation for plants. Alternatively, development times can be calculated for growing meristematic organs such as coleoptiles or roots in terms of growth per unit time. Arrhenius plots are a convenient method for representing reaction rates as a function of temperature, as deviations from Arrhenius or Eyring equation are readily apparent. Therefore, each of the comparisons of the model with observed data will be shown with log rate plotted against the reciprocal of the absolute temperature. Enzyme inactivation results in deviations from a linear response. O’Donovan, Kearney & Ingraham (1965) gave a number of Arrhenius plots of parents and mutants of Escherichiu cob subject to various growth conditions. Type K-11-27 grown on nutrient broth was chosen as this set of data appeared to represent a typical temperature response of this organism. Figure 5(a) shows a comparison of the thermodynamic model, utilizing the constants listed in Table 2 with the data points of O’Donovan et af. (1965). The effect of temperature on development of cotton fleahopper, Pseudatomoscelis seriutus, was measured by Gaylor & Sterling (1975). Figure 5(b) shows an Arrhenius plot of the data points and a comparison with thermodynamic model. In this figure, the extremely high and low temperature data points were estimated from instars I to IV. One of the largest data sets (26 points) relating development time to temperature is that of Bliss (1926) for the prepupal development of Drosopkib

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1.----

2

0.2 t

I=

Oi P seriafus

0.5 -

005 I

0.2 -

002-

04 =

0.01 -

0.05 -

0.005

-

H E

0.02-

0.002

-

aI 5

O,Ol

sl E E 5 2 n

p”

;,

5 d melanogasfer

T pyriformi.

2 I

0.2 04

32

33

34 Reclprocol

FIG. 5. Arrhenius represent observed (a) Escherichiu coli Gaylor & Sterling, hymena

pyriformis

35

36 absolute

32 temperature

33 (I/T

34

x IO‘”

35

3'

‘Km’1

plots of development of poikilotherm organisms, where data points development rates and line represents constants listed in Table 2. (data of O’Donovan et al., 1965), (b) Pscudatotnoscelis seriatus (data of 1973, (c) Drosophila melanogaster (data of Bliss, 1926), and (d) T&u(data of Brandts, 1967).

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663

melanogaster.Bliss, in attempting to thermodynamically describe D. melanoRaster development, divided t!lc thermal region, 12 to 3O”C, into three

sections. For each of his arbitrary thermal regions, Bliss assigned an activation energy: 33 310 cal/mol for the temperature range 12-16°C; 16 850 cal/mol for 16-25°C and 7100 cal/mol for the range 25-30°C. Above 30°C neither Bliss nor Kavanau (1950) could fit the data with a suitable activation energy. Arbitrary assignment of activation energies to various temperature regions is still a common practice and will be discussed in the following section. Figure 5(c) shows a comparison of the observed development rates of D. melanogasterwith that of the model. An Arrhenius plot for the protozoon Tetrahymena pyriformis is presented by Brandts (1967). The data points for the organism growing in pure water were chosen. Figure 5(d) shows a comparison of the data points with the thermodynamic model. The growth and development of plant roots and coleoptiles as a function of temperature were given by Barney (1951) and Lehenbauer (1914) respectively. Figure 6(a) and (b) shows a comparison between the thermodynamic model and observed development rates for these two plant organs. Similarly Fig. 6(c) shows a comparison of the growth rate of Chlorella sorokiniana (Fogg, 1969) with the model. Finally, Fig. 6(d) shows a comparison of the development rates from planting to first flower bud appearance for cotton (Hesketh, Baker & Duncan, 1972). 5. Discussion

It can be concluded from the examples above that equation (17) has a wide range of application from micro-organisms to insects and higher plants. The accuracy of the determination of the thermodynamic constants is dependent upon the number of data points collected, particularly those at the temperature extremes. It is at the temperature extremes that ecophysiological adaptations will become most evident and biotypes are most likely to be distinguished. It is also at the temperature extremes where the interactions between developmental constants, diet and survival will most likely be observed. Therefore, to realize the potential of the thermodynamic approach to development, experimenters need to consider measurements of development rates over wider temperature ranges with greater emphasis on the temperature extremes. It is still rather common practice to assign arbitrary activation energies to various temperature regions (Lyons & Raison, 1970; Hall & Bjorkman, 1975 ; Nishiyama, 1975 ; etc.). The resulting discontinuities are difficult to explain, leading to the postulation of sudden changes in membrane properties

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Reciprocal

35

AiSD

36

absolute

D.

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DEMICHELII

I

temperature

(i/T

x 10e4

OK-‘)

FIG. 6. Arrhenius plots of development for other organisms. Comparison of model with constants from Table 2 and experimental data. (a) Pinus tuedu (data of Barney, 1951), (b) Zeu mays (data of Lehenbauer, 1914), (c) Chlorellu sorokinianu (data of Fogg, 1969), and (d) Gossypium hirsutum (data of Hesketh et al., 1972).

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rather than of a continuous change in either membrane or enzyme configurations. This article should help in explaining apparent sudden changes in development activation energies with temperature (Bliss, 1926). The analysis also explains the underlying cause of the linear region of the development curve. It is this linear region upon which the empirical day-degree summation method is based. Inactivation of enzymes at the temperature extremes linearized the exponential rate function expected from the Eyring equation. This effect can be seen from two examples already presented, Chlorellu development and P. taedu root development rates. Figure 7 shows a comparison between the Eyring equation as described in equation (2) and that presented in equation (17) for Chlorellu. It is clear that enzyme inactivation results in the linearization of the exponential function in the mid-temperature range as well as reducing significantly the rate of development at low temperatures. The inactivation of the enzymes therefore establishes an effective

I

-1

I

4-

I C sorokiniana

/ I

1 Eyring equahon

J

3-

I

aj ;L

...

Thermodynamlf

E a 2 m

:

Cl

5

IO

15

20

25

Temperature

30

35

40

45

50

(“Cl

FIG. 7. Linearization of Arrhenius or Eyring function by development enzyme inactiva. .__. represents equation (17) with constants for C. sorokiniuna.

tion.

P. J. H. SHARPE

666

AND

D. 1%'. DEMICHELE

low temperature threshold for development, which is unexpected from applying the Eyring equation by itself. In addition, enzyme inactivation reduces the rate of development at higher temperatures, thereby effectively establishing both an optimum and an upper threshold. On a speculative note, the reversible inactivation of the control enzyme may provide the organism with a useful mechanism to prevent development when conditions are unsuitable. The location of the linear region is strongly dependent upon the thermodynamic constants. It does not necessarily lie adjacent to the theoretical

8-

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7-

equation

/ / / /

- 6< E 2 5i u2 6 4-

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,/, ( !:i;., ,,, ::

0

5

(

:

IO

15 20

25

Temperature

!:i;., ‘..'.. .. ..

30

35

,

,

,

40

45

50

5

(“Cl

FIG. 8. Effect of enhanced enzyme inactivation on linearization , . .-. . . represents equation (17) with constants for P. treda,

of Eyring function.

REACTION

KINETICS

OF

DEVELOPMENT

667

chemical reaction rate as suggested by Krafka (1921), nor is it necessarily an approximation in this region to the exponential function. This point is illustrated in Fig. 8, where the linear portion of the development curve is significantly lower than the Eyring equation would suggest. This effect explains the difficulties many authors have found when they attempted to relate the linear portion of the development curve to a corresponding Arrhenius-type function for poikilotherm development. The more severe the inactivation, the more the linear slope differs from that predicted by an Arrhenius-type equation. Thus, the compensating effects of enzyme inactivation on the theoretical chemical reaction rate constants is the reason the daydegree accumulation system is extremely useful within its limits. However, for those researchers interested in describing development over a wide range of temperatures, the day-degree accumulation method results in erroneous answers, and equation (17) should be considered as a possible alternative. As stated in the introductory section of the article, this analysis is based upon the assumption of Johnson & Lewin (1946) that the development rate is determined by a single control enzyme. While this may be true at any given temperature, the possibility exists that at different temperatures :t different enzyme may be rate-limiting, thereby becoming the control enzyme. It is conceivable that numerous different enzymes could each represent the control enzyme over specific temperature ranges. In addition, it is also possible .that during specific developmental stages, different enzymes could sequentiahy become rate-limiting and, therefore. represent the control enzyme. It is extremely difficult to determine from the observed reaction kinetics whether one or more enzymes are rate-limiting over the temperature range considered. As an example, consider the case where enzyme A is rate controlling at low temperatures and enzyme B is rate controlling at high temperatures. The rate functions associated with A and B are shown in Fig. 9(a). The reaction kinetics would follow the curve determined by A, until it intersected the curve given by B, after which the rate of reaction would be determined by B. The data points shown would represent the observed reaction rates for development for this system. From an examination of the data points, it would be difficult to determine whether the sudden change in rate at the intersection of A and B was due to a sudden change in the configuration of a single control enzyme represented in Fig. 9(b), or the transition from one rate controlling enzyme to another as suggested in Fig. 9(a). Additional experimental data and biochemical analysis would be required to differentiate between these two possibilities. Thus, it seems to be essentially impossible to determine with the limited developrnental data generally available whether one or more control enzymes T.B. 44

668

P.

3.

H.

SHARPE

AND

Temperature

D.

W.

DEMICHELE

(“Cl

FIG. 9. Effect of one or more rate-controlling enzymes on development kinetics: (a) represents case where two enzymes A and B are rate controlling. A represents enzyme which would be rate controlling at high temperatures and B represents rate controlling enzyme at lower temperatures. Circles represent hypothetical data points. (b) represents the same hypothetical data points, which could just as easily represent a single rate controlling enzyme.

are operating. The interesting aspect of this analysis is that it is probably applicable to developmental systems involving one or more control enzymes. If more than one enzyme is involved, the thermodynamic constants applicable would represent regression coefficients rather than thermodynamic coefficients to a particular rate-controlling enzyme at the temperature considered. 6. Conclusion

The thermodynamic model provides a fundamental basis upon which to calculate development rates. Development has been shown in this analysis to follow the generally accepted laws of physics and chemistry and deviations

REACTION

KINETICS

669

OF DEVELOPMENT

from the Arrhenius behavior appear to result from structural changes in the enzyme catalysts involved in the developmental process itself at extreme temperatures. These structural changes may be of benefit to the organism as they would seemto prevent development from occurring under unsuitable environmental conditions. This researchwas jointly supported by funds provided by National Science Foundation, Project BMS 7504108and the lnternational Center for Biological Control, University of California through a grant from the National Science Foundation (GB-34718)and the Environmental Protection Agency. The authors are indebted to Dr Guy L. Curry for his interest and support.

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REINER, J. M. (1969). Behaviour of Enzyme Systems. 2nd edn. New York: Van Nostrand Reinhold Co. SCRWITON, M. C. & UITER, M. F. (1965). J. biol. Chem. 240, 1. SIZER, I. W. (1943). Adv. Enzymol. 3, 35. SIZER, I. W. & JOSEPHSON, E. S. (1942). Food Res. 7,201. STINNER, R. E., GUTIERREZ, A. P. &BUTLER, G. D. (1974). Cm. Entomo[. 106, 519. WIGGLESWORTH, B. B. (196.5). The Principles of Insect Physiology. London: Methuen.