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Theoretical equations for describing steady state biological rates and their application in analysing… Borgmann, Uwe 1973

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THEORETICAL EQUATIONS FOR DESCRIBING STEADY STATE BIOLOGICAL RATES AND THEIR APPLICATION IN ANALYSING PHYSIOLOGICAL DIFFERENCES AMONG ANIMALS by UWE BORGMANN A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of ZOOLOGY and INSTITUTE OF OCEANOGRAPHY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March, 1973 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada i ABSTRACT The rates of complex multi-enzyme systems, which may contain d i f f u s i o n processes, can be written i n a standard polynomial form. Approximation of t h i s formula leads to the derivation of equations which provide a theoretical basis f o r the use of log-log plots (for rate-size and rate-substrate rel a t i o n s h i p s ) , and Arrhenius plots (for rate-temperature relationships) i n biology. Sharp discontinui-t i e s or "breaks" on such plots can be explained by summations of simple functions and the power to which these must be raised p r i o r to summation. It has been found unnecessary to have large enthalpies (or activation energies) to produce sharp breaks on Arrhenius plots of the rates of complicated b i o l o g i c a l systems. i i TABLE OF CONTENTS Page I. INTRODUCTION 1 I I . THE THEORETICAL DESCRIPTION OF A BIOLOGICAL RATE 4 Derivation of the Complete Rate Equation ... 4 Simplest Solution of the Rate Equation ... 9 Complex Solutions of the Rate Equation ... 12 Application of the Rate Equation ... 17 I I I . LABORATORY OBSERVATIONS AND ANALYSIS ... 19 A. MATERIALS AND METHODS ... 19 B. RESULTS ... 20 Oxygen Consumption of Calarius g l a c i a l i s ... 20 Heart Beat of EUchaeta japonica ... 25 C. DISCUSSION ... 38 IV. SUMMARY AND CONCLUSIONS ... 41 V. REFERENCES ... 43 LIST OF TABLES TABLE I The Mean Heart Beat of E_. japonica at 10°C TABLE II Values of f and A f o r Heart Beat i n Euchaeta. LIST OF FIGURES FIGURE 1. The plot of: V [ C f 1 C x ) ) q + C f 2 ( x ) ^ ] 1 / c i and 1 1/q 2. The plot of: V. and _ ( f 1 ( x ) ) " q + ( f 2 ( x ) ) " q J [ C f 2 C x ) ) q - C f 1 ( x ) ) q ] 1 / q [C£ 1(x)) q-C£ 2(x)) q] 1 / q 3. Oxygen consumption of C_. g l a c i a l i s as a function of temperature. 4. Oxygen consumption of C_. g l a c i a l i s acclimated to the temperature at which measurements were made 5. Oxygen consumption of C. finmarchicus replotted from Marshall and Orr (1955) and Halcrow (1963). 6. Heart beat of adult female E_. japonica from Indian Arm. 7. Heart beat of adult female E_. japonica from Indian Arm plotted on log rate versus log (temperature + 20) scales. 8. Heart beat of adult (C6) male E_. japonica from Indian Arm and f i f t h copepodite males and fe-males from the S t r a i t of Georgia. 9. Heart beat of P_. b i r o s t r a t a collected offshore. 10. Heart beat of adult female E_. japonica from the S t r a i t of Georgia. 11. Heart beat of adult female E_. japonica from offshore. 12. Heart beat of adult female E_. j aponica from offshore. V LIST OF SYMBOLS V - The rate of a reaction or b i o l o g i c a l process. f(x) - A rate function containing no summations and producing a straight l i n e on a log-log or Arrhenius plot (see page 9 for explanation). w: - Volume, s - Surface area E^ - An enzyme concentration, sometimes subscripted. S - A substrate concentration P - A product concentration k^ - A rate constant, usually subscripted. X - Either a concentration (substrate or product) or 1. L - A li n e a r dimension of the animal or compartment of which the rate i s being measured. T - The absolute temperature (degrees centigrade + 273.15). 3d - The sum of. (see page 7 ) P - The product of. (see page 7 ) K* - The equilibrium constant f o r the formation of an intermediate. AG - The free energy change. AH - The enthalpy change. AS - The entropy change. g - A constant used i n describing the effect of s i z e on a rate. I t i s a function of the number of reactions occur-ing i n solution i n comparison with the number of reactions occuring on, or associated with surfaces, (see page 1 0 ) e - A constant used i n describing the effect of substrate concentration on a rate. I t i s a function of the f r e -quency of occurence of the substrate concentration i n the rate equation (see page 1 0 ) f. - A constant used i n describing the effect of temperature on the rate. It i s a function of euthalpies and i s often subscripted (see page 1 2 ) ? A. - A constant with various meanings R - The gas constant k* - The Boltzmann constant h - Planck's constant p - A pr o b a b i l i t y term, assumed to remain constant^ ,^  c - A c o l l e c t i o n of other constants, c = p(k'/h)e ' . q, n, m, y., and z. - Constants with integer values v i ACKNOWLEDGEMENTS I would l i k e to extend my sincere thanks to the members of my advisory committee and especially to Dr. A.G. Lewis, who supported t h i s project and provided much appreciated en-couragement and c r i t i c i s m . I would also l i k e to thank Dr. T.R. Osborn for his help i n checking the equations which appear i n t h i s paper. I also wish to extend my thanks to the o f f i c e r s and crew of the C.S.S. Vector and C.N.A.V. Laymore without whose assis-tance i t would have been impossible to c o l l e c t the specimens used i n the laboratory studies. 1 I INTRODUCTION One of the major objectives i n biology i s the determination of the way animals react to change either biochemically, p h y s i o l o g i c a l l y , behaviorally, or ecologically. In pursuit of this objective, b i o l o -g i c a l rates have frequently been measured and the resu l t s plotted i n various ways. Two of the most used p l o t s , apart from the simple plot with unmodified axes, are the log-log and Arrhenius p l o t s . The log-log plot i s frequently used to describe the effect of si z e on a rate (often the metabolic r a t e ) , and the Arrhenius plot (log rate versus the inverse of absolute temperature) i s used to describe the effect of temperature on a rate. The Arrhenius plot was o r i g i n a l l y intended for use i n chemical rate analysis, while the log-log plot was developed empirically from experimental data which showed that rate i s a function of body size raised to some power. Both plots often produce surprising-l y straight lines (Johnson et a l . 1954; Zeuthen 1953). The accuracy with which straight lines can be f i t t e d to these plots of b i o l o g i c a l data suggests that there might be a theoretical reason for t h e i r use i n analysing complex b i o l o g i c a l processes. A feature of both the log-log and Arrhenius plots i s the frequent occurrence of breaks (sudden changes i n slope). These breaks have been found on plots of the log of metabolic rate versus the log of body weight and, at the time of t h e i r discovery, could not be adequately interpreted since there was no theoretical basis for the log-log re-lationship (Zeuthen 1953). Theoretical explanations for the existence 2 of regions of curvature between two straight lines on an Arrhenius plot have been made for single enzyme reactions involving more than one form of the enzyme (Han 1972; Johnson et a l . , 1954). However, no good explanations exist for the occurrence of very sharp break points. In f a c t , for this reason, some have c r i t i c i z e d the use of the Arrhenius plot and Arrhenius equation for analysing the rate of complex physiological processes (e.g.: Belehradek 1957). It has been argued that an equation which f i t s the entire set of data with-out necessitating the use of break points would be the most useful formula for describing that rate. However, such an equation might mask what i s i n r e a l i t y a true breakpoint,. thereby hiding an import-ant clue to the understanding of the behaviour of the organism. In t h i s paper, I w i l l attempt to demonstrate that a theoretical reason exists for the use of log-log plots i n analyzing rate-size and rate-substrate relationships, and f o r the use of the Arrhenius plot i n analysis of rate-temperature relationships. Mathematical analysis of a complicated system, containing many enzyme reactions and d i f f u -sion processes, can not only account for the occurrence of breaks, but also explain the degree of sharpness and why the break i s upwards or downwards. This makes i t possible to place a b i o l o g i c a l interpretation on such plots and greatly increases t h e i r usefulness. One of the applications of these equations i s found i n the use the slopes of Arrhenius plots as indicators of the enzymatic, and possibly genetic, makeup of an animal. In an attempt to test t h i s 3 concept, the oxygen consumption of Calanus g l a e i a l i s and the heart beat of Euchaeta japoriica (Pareucheata elbngata) (Crustacea:Copepoda: Calanoida) were measured as. a function of temperature and the re-sult s plotted on Arrhenius graphs. Specimens of Euchaeta were c o l -lected from several areas and the slopes of the Arrhenius plots compared i n an attempt to observe differences between animals from different areas. 4 I I . THE THEORETICAL DESCRIPTION OF A BIOLOGICAL RATE Derivation of the Complete Rate Equation A simple model for an enzyme reaction can diagramatically be represented by: k i k2 E. + S • . ES . i E_ + P f f E^ i s the free enzyme, S i s the substrate, P i s the product, ES i s the enzyme-substrate complex, and the k's are rate constants. The rates for the f i r s t and second parts of th i s reaction are: ~d t ( S ) = kjCS) (E f) - k ^ E S ) and ^ a r - = k 2 ( E S ) " k - 2 ^ E^ i s the concentration of free enzyme (enzyme not i n the form of ES). The t o t a l concentration of enzyme i s : (E) = (E f) + (ES) Under steady state conditions, -d(S)/dt i s equal to d.(P)/dt. The second equation can then be solved for (ES) and substituted into the f i r s t equation giving: d(P) = k 1 k 2 ( E ) f ( S ) - .k^k^CE^CP) dt — k - l + k2 By setting -d(S)/dt equal to d(P)/dt, the f i r s t set of equations can be solved to give (ES) i n terms of (E^). (E) can then be expressed i n terms of I(E^) , and (E^) can be replaced by (E) giving: 5 d(P) =. .k^CEJCS).-.k.^CEDCP) d t ^ ( S ) + k 2 + k_x + k_ 2(P) I f the value of k_2 i s equal to 0, the la s t term i n the numerator and denominator disappears and the Michaelis-Menten equation i s obtained. More complete derivations of rate equations for single enzymes can be obtained i n most biochemistry tests (e.g.: White, Handler, and Smith 1959). The rate of change of the concentration of P i s given by d(P)/dt. The t o t a l rate of production of P can be obtained by multiplying d(P)/dt by the volume of the reaction vessel (w). Total rate = V = (w)d(P)/dt Clearing the denominator of the above equations gives: VkjCS) + Vk 2 + Vk_1*Vk_2CP) - (w)k 1k 2CE)(S) + Cw)k_ 1k_ 2(E) (P) = 0 A l l enzyme reactions can be written, i n this polynomial form. D i f f u -sion rates can be expressed using similar equations i n which w i s replaced by the surface area of the membrane, S and P are replaced by concentrations on either side of the membrane and E i s deleted (unless membrane bound enzymes are considered). I f the theory of absolute reaction rates (tra n s i t i o n state theory) i s applied, i t i s also necessary to include the mean distances between c o l l i s i o n s of the dif f u s i n g molecules on the respective sides of the membrane. (Lakshminarayanaih 1969). The rate constants for d i f f u s i o n can then be treated i n the same way as rate constants for chemical reactions. 6 A rate equation for complex systems such as multiple reactions i n a test tube or physiological reactions i n an animal, should be written i n terms of the dimensions of the region i n which the reac-t i o n i s taking place, the temperature within that region, and the input ( i . e . : i n i t i a l substrates) and output, ( i . e . : f i n a l products). I t i s therefore desirable to eliminate concentration terms of i n t e r -mediates which cannot be measured or controlled. This can be done i f steady state conditions are assumed. For example, the equation for a system of two enzymes of the type described i n the above equations can be obtained i f the rate of the f i r s t reaction i s constant and equal to the rate of the second reaction. I f the i n i t i a l substrate i s S, and the product of the f i r s t enzyme i s P, then the substrate of the se-cond enzyme i s P and i t s product can be designated as ? 2- The rate equation of the second enzyme can then be written i n terms of P.. Vk3(P) + Vk4 + Vk_3 + Vk_4(P2) - (w)K3k4(E2)(P) + (w)k_3k_4(E2)(P2) = 0 p = vk4 + vk_3 + vk_4CP2) > (w)k_3k_4CE2)(P2) This equation can be substituted i n place of P i n the equation for the f i r s t enzyme, thereby eliminating the concentration of P from the f i n a l equation. (w)k 3k 4(E 2) - Vk 3 VkjCS) + Vk 2 + Vk_ 1 - (w)k 1k 2CE)(S) + = 0 7 Clearing the denominator and c o l l e c t i n g similar terms gives: + ( w ) k 1 k 3 k 4 C S ) ( E 2 ) - ( w ) k 1 k 2 k 3 ( S ) ( E ) + ( w ) k 2 k 3 k 4 ( E 2 ) + ( W ) k _ 1 k 3 k 4 ( E 2 ) + ( w ) k _ 1 k _ 2 k 4(E) + ( w D k ^ k ^ k ^ C E ) + ( W ) k _ 1 k _ 2 k _ 4 ( E ) ( P 2 ) - C w ) k _ 2 k _ 3 k _ 4 ( E 2 ) ( P 2 = 0 In order to simplify the above equation, two new symbols w i l l be defined: b ID 3i = Si fa) = the sumo of b number of a's, and a v J b b ?A = P (a) = the product of b number of a's. If the b i s omitted the above symbols w i l l mean the product or sum,, of an undefined number of a's. In contrast to conventional summation and product signs, the above symbols w i l l not indicate which a's are being added or multipl i e d . For example: 2 P k x = the product of 2 k's and 2 X's. The k represents any of the k's i n the above equation and X represents either a concentration or 1. The equation for the two enzyme system can now be written: A t h i r d enzyme could be added by solving the rate equation for that enzyme i n terms of the product ?„ and substituting t h i s i n place V 0 of ?2 i n t n e two enzyme equation. The f i r s t term i n the f i n a l equa-3 t i o n , i f written i n the above shorthand form, w i l l contain a V and 3 3 6 a P, Y and the la s t term w i l l contain a (w) and a P 1 v . S i m i l a r l y , enzyme denaturation and branch points i n enzyme chains can be i n t r o -duced. Diffusion systems would be expressed using s i m i l a r equations but with a term for surface area (s) instead of volume (w). Equations for systems containing both enzyme reactions and diffusions w i l l con-t a i n both w and s. The general form that such equations have i s : v n m(pll+mh (n-1) f , , y l f zl r T,(n+m+l)/"l v^  1 M [ (s) (w) (P^ x  J)j + v'"-2' * [ ( .A-tffVg* 2')] . '• • a [ c s / n (w) z" (F*™h] - o (1), The X's contain concentrations of substrates, enzymes, and i n h i b i t o r s , as well as other chemicals and distances between molecular c o l l i s i o n s (X may also equal 1); n and m are constants. The exponents y and z are integer values and: >1 + z l = X> y2 + z 2 = 2> y + z = n 'n n 9 Simplest Solutions of the Rate Equation It i s possible to solve equation 1 i f , over a given range of temperature, substrate concentration, or s i z e , the t h i r d largest term i s considerably smaller than the second largest term. The t h i r d l a r -gest term and a l l smaller terms can then be ignored. Equation 1 then becomes: v* m [ c s ) y ( w ) z ( P ^ ) ] = m [cs/' (w) z' ( p j n + m + q ) ) ] i / q v = (2) where: (y* + z') - (y + z) = q This equation can be further s i m p l i f i e d i f one of the terms inside the summation i n the numerator and a s i m i l a r term i n the denominator i s s u f f i c i e n t l y larger than the re s t , permitting omission of the sum-mation signs. 1/q V = = f (x) (3) Equation 3 i s the simplest possible solution of equation 1, and such a function, without any summations, w i l l be designated as f ( x ) , with x r e f e r r i n g to temperature, substrate concentration or s i z e . The Rate-size Equation 2 3 Replacing s with L and w with L i n equation 3, where L i s some linear dimension of the reaction vessel or animal being examined, equation 3 can be written: 10 V = A(L g) g • . 3 , . - 2y - 2 , 3 / q • ^ I $ t 3 ( z ' - zj A i s a constant obtained by c o l l e c t i n g a l l the constants i n equation 3 and taking the qth root. A and g can be obtained by p l o t t i n g the logarithm of the rate versus thelogarithm of the siz e . I f a l l parts of an animal increase proportionately i n s i z e , and weight i s proportionate to volume, then L i n the above equation can be replaced by the weight, i f g i s divided by 3. Log rate (especially log oxygen consumption) versus log weight plots are frequently en-countered i n the l i t e r a t u r e . Commonly observed values of g/3 are 0.67, 0.75, and 1.00 (Zeuthen 1953). Rats, house-flys, nematodes, molluscs, crustaceans and other animals have been examined i n t h i s way. The Rate-substrate Equation Since substrate concentrations are contained i n the X's, the rate must be proportionate to the substrate raised to some power. Equation 3 becomes: V = A(S e) A i s a constant (different from the A i n the rate-size equation) and e i s the frequency of occurrence of S i n the numerator, minus the fr e -quency of i t s occurrence i n the denominator, divided by q. S can be the concentration of a chemical or, i n the case of animals and ecosys-tems, can be the concentration of a food organism. A and e can be 11 obtained by p l o t t i n g the logarithm of V versus the logarithm of S. In addition to many chemical reactions, the growth rate of a po-pulation also conforms to the above equation. The rate of increase of the population i s : V = dN/dt = rN Where N i s the number of animals i n the population and r i s the growth rate constant. In this case, e i s equal to 1. The Rate-temperature Equation Temperature affects the rate of a reaction by affecting the rate constants. Using the theory of absolute reaction rates, a rate con-stant can be described by: k = p(k'/h) (T)K* T i s the absolute temperature (degrees centigrade + 273.15), k' i s the Boltzmann constant, h i s Planck's constant, K* i s the equilibrium con-stant f o r the formation of a transient, non-isolatable intermediate and p i s the p r o b a b i l i t y that i t w i l l break down to produce the products (Johnson et a l . , 1954). The equilibrium constant i s equal to: -AG*/RT AS*/R -AH*/RT K* = e = e e AG* i s the free energy change between reactants and the intermediate, AS* i s the entropy change ancUH* i s the enthalpy change. R i s the gas constant. I f a l l constants are replaced with c, the rate constant can be written as: k = c ( T ) e - * H * / R T 12 A product of rate constants, such as appears i n equation 1, can be written as: -(3f" )/RT P £ - C P ? (T n)e Equation 3 can now be written as: V = A(T) Ce" f / R T) Cn+m+q) (n+m) f = AH* AH* A i s a constant. The absolute temperature varies l i t t l e over the range of most.biological reactions and the T outside the exponential can be treated as a constant. The values of A and f can be obtained by p l o t t i n g the logarithm of the rate versus the inverse of the abso-lute temperature, commonly known as the Arrhenius p l o t . Many examples of straight l i n e Arrhenius plots for such rates as heart beat and bioluminescence are available i n the l i t e r a t u r e (Johnson et a l . , 1954). Complex Solutions of the Rate Equation In the la s t section, the simplest solution f o r the rate of a com-plex steady-state system was described as: V = f (x) f(x) = A ( L g ) , A ( S e ) , or A(T) Ce" £ / R T) L i s a linear dimension of the system, S i s a concentration, and T i s the absolute temperature. Each of these equations was obtained from equation 3. I f instead of ignoring both summation signs i n equation 13 2, only the summation sign i n the denominator i s ignored, equation 2 could be written: = [^(£Cx)D q] 1 / q (4) Considering summations only i n the denominator would give: V = 1/q (5) Considering summations i n both the numerator and denominator would give: V = [^Cf(x)) q] _ 1 i/q (6) Equation 6 makes i t possible to analyse complex rates i n terms of simpler functions (f(x)) which can be ea s i l y determined by the use of log-log or Arrhenius plots. The value of q w i l l not affect e s t i -mates of fCx). In f a c t , q need not be known. A log-log or Arrhenius plot of a summation of 2 f(x) terms, as i n equation 4, w i l l produce two straight lines with a region of over-lap where the slope changes from one l i n e to another. Each l i n e ap-proximates the plot of the f(x) function which i s larger, and the intersection occurs i n the region where the two f(x> terms are ap-proximately equal. The intersection i s concave upwards. S i m i l a r l y , a summation of 2 terms i n equation 5 would again produce two straight l i n e s , but with an intersection which i s concave downwards (figure 1). At the point of intersection, both terms i n equation 4 are equal giving: 14 (fOO) V d i f f e r s from either f(x) term by a factor of 2 4 . In equation 5, V d i f f e r s from f(x) by a factor of 2 at the point of i n t e r -section. In either case, as q increases V approaches f(x) and the region of intersection becomes sharper. More complex systems can have higher values of q and consequently can display sharper breaks than simple systems. I f one of the terms being summed i n equation 2 has a minus sign associated with i t , then equations 4 and 5 w i l l contain subtractions instead of additions. For example, equation 4 with a minus sign can be written: V = [ ( f 2 ( x ) ) q " f i « ) q ] V q C 7 ) Forms si m i l a r to equation 7, but with q equal to 1, have been used to describe growth rates of organisms, with f2(X) being the rate of t i s -sue production, and f^(x) the rate of i t s breakdown (von Bertalanffy 1938). Again, the equation produces a plot which i s approximated by two straight l i n e s , but i n contrast to the previous case, one of the lines w i l l be v e r t i c a l . At the intersection where f^(x) = f 2 ( x ) , the rate i s zero and -log(V) i s equal to i n f i n i t y (figure 2). As before, larger values of q w i l l produce sharper break points. I f a subtraction occurs i n the denominator, results s i m i l a r to equation 7 are obtained, except that at f^(x) = t n e rate, and consequently log(V), i s equal to i n f i n i t y . 15 Figure 1 The plot of: V x = [(fjCx)) 0 1 + C f 2 ( x ) ) q ] 1 / q and V2 = (f^x) " q +• (f 2(x) - q V q 16 Figure 2 The plot of: v x = [ ( f 2 ( x ) ) q V 2 = [ ( £ 1 W ) , 1 C^Cx)^] 1 / q and Cf 2(x)j q] 1 / q 17 In order to solve equation 1, two approximations were made. F i r s t , i t was assumed that the t h i r d largest term i n equation 1 was much smaller than the second largest term. Secondly, i t was assumed that the summation signs i n equation 2 could be ignored. In this section I have demonstrated what happens i f the second approximation does not hold for a l l values of L, S or T. It i s also possible that the f i r s t approximation does not hold. Unfortunately, t h i s condition i s more d i f f i c u l t to deal with. The results are the same, however, regardless of where the summations occur. For most data available i t i s impossible to distinguish between the two types of summations and, for p r a c t i c a l purposes, i t i s easier to assume that the f i r s t approximation does hold. Application of-the Rate Equation The equations shown here have a two-fold application. F i r s t l y , they permit the description of a rate as a function of s i z e , substrate concentration, or temperature. Under a given set of conditions, the rate i s controlled by a group of reactions and not just one single reaction. I f the combination of reactions involved i n the control of a rate change, a break may be observed i n the data. Such breaks are clues to a change i n the physiology of an animal and are not ar-t i f a c t s of the log-log or Arrhenius p l o t s . A second application of these equations i s found i n the analysis of the slopes associated with log-log and Arrhenius p l o t s . Although 18 the heights of these plots can be expected to vary from animal to animal, the slopes should remain constant. For example, the slope observed on an Arrhenius plot i s a function of the enthalpies of formation of intermediates i n the various reactions involved i n the control of the physiological rate. Since enthalpy changes are not functions of concentrations or s i z e , the slopes of Arrhenius plots should remain constant for animals with the i d e n t i c a l enzyme makeup, regardless of t h e i r state of acclimation, s i z e , or the concentration of food or chemicals i n the environment. Differences i n the slope are indications of differences i n the co n t r o l l i n g reactions and qu a l i t a t i v e differences i n the enzymes involved i n these reactions. A difference in.slope therefore indicates a possible genetic d i f f e r -ence between animals. In th i s way, slopes of Arrhenius plots can be used to distinguish between enzyme systems i n much the same way as electrophoresis i s used to distinguish between enzymes. 19 I I I . LABORATORY OBSERVATIONS AND ANALYSIS A. MATERIALS AND METHODS The rate of oxygen consumption of the copepod Calanus g l a c i a l i s and the heart rate of the copepod Euchaeta j apohica (Pareuchaeta  elongata) were measured. Oxygen consumption was measured with a con-stant pressure respirometer. Three animals were placed i n a 25 mm diameter glass v i a l with about 10 ml of sea-water. The v i a l was sealed with a s i l i c o n rubber stopper from which hung a small v i a l containing sodium hydroxide (as a carbon dioxide absorber). A glass tube connected the respirometer chamber with a compensation chamber of equal dimensions. The sea-water i n the compensation chamber also served as the manometer f l u i d . As oxygen was consumed, the pressure i n the respirometer decreased. Operation of a p l a s t i c syrings re-turned the pressure to i t s i n i t i a l l evel and the volume of oxygen consumed was recorded. Animals were l e f t i n the respirometer for several days, readings being taken every 24 hours. The entire respiro-meter was held under water i n a constant temperature bath since minor temperature changes i n any part of the apparatus caused s i g n i f i c a n t pressure changes. Twelve such respirometers were used at each tem-perature for each experiment. The heart beat was measured with a flashing l i g h t , obtained by rotating a disc, containing evenly spaced holes, i n front of a micro-scope l i g h t (Bausch and Lomb, No. 31-33-53). The speed of rotation 20 was varied u n t i l i t coincided with the heart beat, making the heart appear as i f i t were standing s t i l l . The disc was rotated by an e l e c t r i c motor, through a "Zero-Max" continuously variable gear t r a i n . The animal was placed on a glass s l i d e with a few drops of water. The temperature was controlled by a cooling unit (Hoke Minifreezer) below the s l i d e , and was measured with a thermister (+0.1°C) placed beside the animal. It took two or three minutes to obtain a reading, after which the animal was returned to i t s storage container. C_. g l a c i a l i s and E_. japonica were collected i n v e r t i c a l hauls with a one meter diameter rin g net. Collections were made i n Indian Arm at IOUBC/Ind 9( 49° 23.5' N, 122° 52.5' W) and i n the S t r a i t of Georgia at IOUBC/GS 1( 49° 17.0' N, 123° 50.5V W) from August to November 1972. E_. japonica and Pareuchaeta bi r d s t r a t a were also c o l -lected about 150 miles seaward from the entrance to the S t r a i t of Juan de Fuca ( 48° 0' N, 128° 0' W) on October 24, 1972. B. RESULTS Oxygen Consumption of Calarius g l a c i a l i s The results of the oxygen consumption experiments are shown.in figures 3 and 4. Figure 3 shows the results obtained for animals which had not been acclimated to the temperature at which t h e i r con-sumption was measured. As expected from the theoretical calculations, the height of the oxygen consumption curve increased when the animals 21 were fed but, within the accuracy of the technique, the slope of the l i n e on an Arrhenius plot did not change. Unfortunately, the standard error (plus and minus two standard error units are indicated i n the figures) i s large, and any minor changes i n slope would not be re-solved. Figure 4 shows the results obtained for animals which had been kept for 11 days at the temperature at which t h e i r r e s p i r a t i o n was mea-sured. A l i n e of the same slope as observed i n figure 3 can be drawn through the 95% confidence l i m i t s . No difference i s therefore detec-table . Figure 5 shows some data replotted from Marshall and Orr (1955) and Halcrow (1963) on the oxygen consumption of Calanus finmarchicus, a related species to C_. g l a c i a l i s . The data from Halcrow i s plotted i n d ifferent units since he converted his values to oxygen consumption per unit dry weight. The s o l i d lines drawn through the points have the same slope as those on figure 3 and 4. To within the accuracy obtainable, one l i n e of varying height but constant slope w i l l account for a l l observed data. The rate of oxygen consumption can be adequately described by: V = A(T) ( e " f / R T ) The value of f i s 15.4 k i l o c a l o r i e s and the value of A varies to account for the s h i f t i n the height of the graph but i s usually about _3 7.9 X 10 . It appears that the enzyme compliment of the two species 22 Figure 3. Oxygen consumption of C_. g l a c i a l i s as a function of temperature. Consumption i s i n mi c r o l i t e r s per hour per Calanus. The top l i n e i s fed and the bottom l i n e unfed animals. (The v e r t i c l e lines represent 95% confidence l i m i t s ) . i Figure 4. Oxygen consumption.of C. g l a c i a l i s kept at the temperature at which measurements were made. a F i n u r e 4 * I 24 Figure 5. Oxygen consumption of C_. finmarchicus replotted from Marshall and Orr (1955) and Halcrow (1963). Marshall and Orr's data i s i n m i c r o l i t e r s per hour per Calanus and Halcrow's data i s i n micro-l i t e r s per hour per milligram, of dry weight times 10*. (The straight lines have the same slope as figures 3 and 4). • Figure 5 25 mentioned i s very s i m i l a r . For t h i s reason, and because of the large standard errors encountered (usually about 10%) population differences were not investigated. Heart Beat of Euchaeta japonica Figure 6 shows the effect of temperature on the heart beat of Euchaeta j aponica adult females. The rate can be described by: V = (T) A ? ( e - f l / R T f l • A q C e - f 2 / R T f l i / q The value of f^ i s 8.05 k i l o c a l o r i e s and of f^ i s 5.49 k i l o c a l o r i e s . i s approximately 5 beats per minute per degree and A^ i s approxi-mately 4 beats per minute per degree. It has been suggested (Belehradek, 1957) that the breaks ob-served on Arrhenius plots are a r t i f a c t s due to the nature of the axes used i n p l o t t i n g , and that Belehradeks function can account for most of the rate temperature curve without requiring a break. To test t h i s p o s s i b i l i t y , plots were made of the logarithm of the rate versus the logarithm of temperature plus a constant for the same data shown i n figure 6. The result (using a b i o l o g i c a l zero of -20°C) i s shown i n figure 7. The data does not f i t this equation as well as the above equation. The points above the l i n e at about 18°C can be made to f i t better i f the value of the b i o l o g i c a l zero i s changed. This w i l l , however, cause other points to f a l l o f f the straight l i n e . The break, therefore, appears to be quite r e a l . Figure 6. Heart beat of adult female E. Japoriica Indian Arm. Heart beat i s i n beats per minute (CPM). 27 Figure 7. Heart beat of adult female E. japoriica from Indian Arm plotted on log rate versus log (temperature + 20) scales. The broken l i n e i s the observed data and the s o l i d l i n e shows the values expected from Belehradek's function. 28 The heart beat of some males and f i f t h copepodite (C5) stages i s shown i n figure 8. As with the oxygen consumption data, there i s no difference i n slope but a d e f i n i t e change i n height. A l l the lines i n figure 8 have the same slope as those i n figure 6. Since the slopes are the same, i t i s possible to extrapolate to any given temperature to compare absolute values. The heart beat for various stages and areas from which the animals were collected, ex-trapolated to 10°C, i s shown i n Table I. Table I includes the means of only those tests i n which animals from both Indian Arm and the S t r a i t of Georgia were tested at the same time. With one exception, males always had higher heart beats than females and adults higher heart beats than C5 copepodites. In a l l seven t e s t s , animals from the S t r a i t of Georgia have higher heart beats than Indian Arm animals of the same stage and sex. This i n d i -cates a difference i n the absolute values of the heart rate i n the two regions s i g n i f i c a n t to the 2% level for a two-tailed test (either the Wilcoxon's signed-ranks test or the simple sign t e s t ) . Males are rare i n some areas and the heart of immature animals i s more d i f f i -c u l t to see than i n adults, so the majority of the data collected was from adult females. Pareuchaeta b i r o s t r a t a , a r e l a t i v e of E_. japonica (Paraeuchaeta  elongata) and probably of the same genus, was collected 150 miles out from the entrance to the S t r a i t of Juan de Fuca. This animal i s not found inshore. The heart beat i s shown i n figure 9. I t i s obvious 29 Figure 8. Heart beat of adult (C6) male E_. j aponica from Indian Arm and f i f t h copepodite males and females from the S t r a i t of Georgia. 30 TABLE I The Mean Heart Beat of E. japonica at 10°C Stage Origin of Specimens Indian Arm S t r a i t of Georgia Adult females 434 (31) 460 (27) Adult males 458 (2) 465 (2) C5 females 390 (4) 457 (3) C5 males 435 (3) 512 (1) (Figures i n parentheses indicate number of animals tested). Figure 9. Heart beat of P. b i r o s t r a t a collected offshore. LOG CPM 35 35.5 36 (1/TEMR °K) X 10 4 36.5 F igu re 9 32 that the slope i s quite different from that of E. j aponica and, consequently, i t s enzyme compliment must be somewhat d i f f e r e n t . This suggests that differences between species can be observed with t h i s technique and stimulates interest i n i n t r a s p e c i f i c examination. Figure 10 shows a more complete graph of the heart beat of adult female E_. japonica from the S t r a i t of Georgia. The values above 8°C are mean values for any one animal, but below t h i s temper-ature each point represents an individual measurement. This i s necessary since the slow beating rate at low temperatures makes measurement with the strobe l i g h t technique d i f f i c u l t . Although the slopes of the lines are constant, the height may vary from animal to animal. This necessitates p l o t t i n g the heart beat of each animal separately and pooling the data for only those animals with s i m i l a r absolute rates. An additional feature of figure 8 i s the one point which i s o f f from the expected curve ( i . e . : the animal with the high-est absolute r a t e ) . The point however, l i e s on the l i n e extrapolated from the other side of the break. This phenomenon was observed on several occasions and suggests a certain i n s t a b i l i t y i n the region close to the break. Specimens of E. japonica were also collected from the offshore station to compare with the inshore data presented so f a r . Some of the results are shown i n figure 11. As before, the slope of the Arrhenius plots remains constant, although the height varie s . Two differences.between inshore and offshore animals are observed. The 33 Figure 10. Heart beat of adult female E_. japbriica from the S t r a i t of Georgia. Only the rates of several selected individuals are shown. The top l i n e i s from one animal and the bottom l i n e includes values obtained from four different animals. TEMPERATURE °C 24 18 12 6 0 33.5 34 34.5 35 35.5 36 36.5 (1 /TEMR °K) X 1 0 4 Figure. 10 34 Figure 11. Heart beat of adult female E_. j apbriica from offshore. The top l i n e includes values from two. animals and the bottom l i n e includes values from four other animals. •0 35 break occurring between 16 and 18 C i n the inshore animals i s not found offshore, and the slope of the l i n e over the lowest tempera-ture range i s shallower i n the offshore animals. In a l l , four different slopes were observed for this species. The equation ac-counting f o r a l l of these can be written: i / q The values of f and A are l i s t e d i n Table I I . In the data presented so f a r , the slope of the l i n e was always the same between approximately 11 and 16°C. This corresponds to an f of 8.05 k i l o c a l o r i e s . This region of the rate-temperature curve was analysed i n d e t a i l for 45 animals, 11 of which were collected o f f -shore. Of these, 43 gave i d e n t i c a l slopes. Two i n d i v i d u a l s , from the 11 collected offshore, gave a different slope. The re s u l t s are shown i n figure 12; f has a value of 6.77 k i l o c a l o r i e s . Since these were the only two different slopes over that region, a regression analysis was done on the data. The p r o b a b i l i t y that either of these two ani-mals has a slope different from the more common one i s 99.9%. The two slopes are therefore s i g n i f i c a n t l y different and the two offshore specimens probably represent a different genotype. i = l (A.(e-V R T);f 36 TABLE II Values of f and A for Heart Beat i n Euchaeta Euchaeta japonica Inshore and Offshore Inshore Offshore £ 1 = 39.6 X 10 3 A x = 20.9 >20.9 f 2 = 17.2 X 10 3 A 2 = >12.2 7.79 (approx.) f 3 = 8.05 X 10 3 A 3 = 4.58-5.70 3.95-4.89 f . = 5.49 X 10 3 A. = 3.86-4.40 >3.55 4 4 — Euchaeta j aponica: the 2 exceptions i n the offshore sample f = 6.77 X 10 3 A = 3.29 (approx.) Pareuchaeta b i r o s t r a t a f = 1.02 X 10 3 (male) A = 8.72 X 1 0 _ 1 (male) f = 0.89 X 10 3 (female) A = 7.17 X 1 0 _ 1 (female) 37 Figure 12. Heart beat of adult female E_. japonica from offshore, The two broken lines are the values for the two animals which gave different slopes than the rest of the E. japonica (The s o l i d l i n e i s the slope obtained for the majority of animals). 38 C. DISCUSSION As predicted from the theoretical equation, the slope of the Arrhenius plots remains constant for physiologically s i m i l a r animals. The same slope was always obtained for the oxygen consumption, even for Marshall and Orr's and Halcrow's data. The same slope for heart rate was obtained for adults and C6 copepodites, males and females, and for animals collected from Indian Arm and the S t r a i t of Georgia. With two exceptions, the heart rate of a l l E_. japbnica can be described with the same equation. Of the eight constants required i n the equation (four f's and four A's) only two need vary, since Aj and A^ might be the same for inshore and offshore animals (see Table I I ) . The var i a t i o n observed among the offshore animals i n figure 9 could be explained i f a concentration of some substance ("C") occurred twice i n A^ and three times i n A^. This would s h i f t the graph to the right whenever i t was shifted upwards. A change i n "C" of 10.7% would account for the observed v a r i a t i o n . For example, some extrapolated data from the two plots i n figure 11 are shown below. Rate at 8.6°C Rate at 12.2°C (below the break) (above the break) Lower Plot 285 400 Upper Plot 385 490 If C_ = 1.107 X C., the predicted values for the upper graph are: 39 At 12.2°C At 8.6°C: V = 285 X C 3 = 285 X (1.107) 3 = 3/87 V = 400 X C 2 = 400 X (1.107) 2 = 490 I f these calculations are accepted, the value of A^ f o r the inshore animals can be calculated from the difference between A^ for the i n -shore and Ag for the offshore animals. The predicted value i s about 12.5, which agrees with the observed data ^ ^ 1 2 . 2 ) . These calcula-tions are approximate and would involve a change i n "C" of about 17%. The v a r i a t i o n i n C alone could therefore account for a l l the observed differences between inshore and offshore animals (omitting the two exceptions which are believed to be a different genotype;)!* This would indicate that differences between inshore and most offshore animals are due to a concentration and not due to a d i f f e r e n t i n en-zyme makeup. There appears to be a considerable difference i n the enzyme makeup of E_. japonica and P. b i r d s t r a t a . This would be expected for d i f f e r -ent species. However, among the E_. japonica, two offshore specimens were also shown to have a slope s i g n i f i c a n t l y different from the majori-ty of animals. No morphological difference could be seen between the two types of animals. The only method of separating them was by use of the rate-temperature curve. This supports the t h e o r e t i c a l l y de-rived hypothesis that the slopes of Arrhenius plots can be used to d i f f e r e n t i a t e between enzyme systems i n a manner si m i l a r to electro-phoresis. One advantage i s that the animal i s not k i l l e d and can be 40 used for subsequent physiological experiments. It should be pointed out that the use of Arrhenius plots for observing differences between animals i s limited by the accuracy of the technique used to measure the rate. For example, i t i s pos-s i b l e that differences i n the slope of the Arrhenius plot of the oxygen consumption exist between Calanus g l a c i a l i s and C_. finmarchicus. However, i n order to detect t h i s , the slopes must be s u f f i c i e n t l y d i f f e r e n t and the standard errors s u f f i c i e n t l y small. This i s ob-viously not the case i n figures 3, 4, and 5. Even i f the same slope i s observed for a given rate measured i n two animals, i t may be possible to observe differences between these animals i f a different rate i s measured. 41 IV. SUMMARY AND CONCLUSIONS The relationship between the rate (V) of a reaction and the value of a factor (x) can be expressed i n the form of a polynomial equation. For complex systems, this equation i s too involved to be used d i r e c t l y and must be approximated. The simplest approximation of V = f ( x ) , where f(x) i s proportionate to the size of substrate concentration raised to some power or, i s proportionate to the ex--f/RT ponential function of temperature (e ). More complex rate functions can be written as summations of simple functions, either by summing (or subtracting) the functions d i r e c t l y , or by taking the inverse of the sum of t h e i r inverses. P r i o r to summation, the functions are raised to some power (q) and after summation are raised to the inverse of that power (1/q). Such summations result i n break points when plotted on log rate versus log substrate, log s i z e , or inverse temperature axes. The break points observed with experimental data are quite, r e a l , and can be very sharp i f a complicated system i s analysed. Between breaks, one function describes the rate. The effects of temperature, substrates, and size have been explained and are m u l t i p l i c i t i v e within this region. Interactions between substrate concentrations, siz e , and temperature are therefore also known. The theoretical rate equations, i n addition to t h e i r obvious application i n simply describing rates, can be used as indicators of 42 physiological differences among animals. Since the Arrhenius plots of the oxygen consumption of Calanus g l a c i a l i s and C_. firimarchicus a l l gave the same slope, i t i s probable that the enzyme makeup asso-ciated with the reactions which control the oxygen consumption i n these animals i s very si m i l a r (although small differences cannot be resolved due to the large standard errors encountered). This conclu-sion i s based on the theoretical calculations which indicate that t h i s slope i s a function of enthalpies only, and not concentrations, s i z e , or other things. Arrhenius plots of the heart beat of Euchaeta j aponica further support t h i s concept. A l l inshore animals, regardless of sex or stage (C5 or adult) gave i d e n t i c a l slopes while a different species (Pareiichaeta birostrata) produced a different slope. The Arrhenius plot and the accuracy with which heart beat can be measured i n these two species makes i t possible to distinguish between them physiologi-c a l l y . Most of the E. japonica from offshore appear to be physiolo-g i c a l l y s i m i l a r to the inshore Euchaeta since a l l differences between heart beats of inshore and offshore animals can be explained by s h i f t -ing the heights, but not the slopes, of the Arrhenius p l o t s . However, two of the specimens collected offshore, l i k e the P_. b i r o s t r a t a , pro-duced a different slope, indicating that they are biochemically and possibly genetically, d i f f e r e n t . 43 V. REFERENCES v ^ Belehradek, J. (1957). Physiological Aspects of Heat and Cold. Ann. Rev. Physiol. 19: 59. von Bertalanffy, L. (1938). A Quantitative Theory of Organic Growth. Human Biology 10: 181-213. Halcrow, K. (1963). Acclimation to Temperature i n the Marine Copepod Calarius finroarchicus (Gunnar). Limnol. and Oceanogr. 8; 1-8. Han, M.H. (1972). Non-linear Arrhenius Plots i n Temperature-dependent Kinetic Studies of Enzyme Reactions. I. Single Transition Processes. J . theor. B i o l . 35: 543-568. Johnson, F.H., H. Eyring, and M.J. Po l i s s a r , (1954). The Kinetic Basis of Molecular Biology. John Wiley and Sons Inc. New York. 874 pp. Lakshminarayanaiah, N. (1969). Transport Phenomena i n Membranes. Academic Press, N.Y. 517 pp. Marshall, S.M. and A.P. Orr (1955). The Biology of a Marine Copepod Calanus finmarchicus (Gunnerus). Oliver and Boyd, Edinburgh. 188 pp. 44 Somero, G.N. and P.W. Hochachka, (1971). Biochemical Adaptation to the Marine Environment. Am. Zoologist 11: 159-167. White, A., P. Handler and E.L. Smith, (1959). Principles of Bio-chemistry. McGraw-Hill, Inc. New York. Zeuthen, E. (1953). Oxygen Uptake as Related to the Body Size i n Organism. Quart. Rev. B i o l . 28: 1-12. 

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