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A Thermodynamic Approach to the Temperature Response of Biological Systems as Demonstrated by Low Level Luminescence of Cucumber Seedlings
A Thermodynamic Approach to the Temperature Response of Biological Systems as Demonstrated by Low Level Luminescence of Cucumber Seedlings
1) Laboratory of Biophotons, LudwigstraBe 31, 6520 Worms, F.R.G. 2) Institute of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing, People's Republic of China.
3) Department of Biology, The University, P. O. Box 3049,6750 Kaiserslautern, F.R.G. Received August 31,1983 . Accepted October 6,1983
Summary The temperature response of biological systems (QlO) is described in terms of the physical model of dissipative structures of thermodynamically open systems, which are far away from thermal equilibrium. The theory is shown to be consistent with experimental results on physiological temperature responses of cucumber seedlings.
Key words: Cucurbita pepo, biophotons, luminescence, temperature response, thermodynamics.
Introduction The response of biological systems to alterations of the temperature of their environment was carefully investigated (Precht et al., 1973). For its characterization, commonly the parameters QlO and )( are used. The QlO value is the factor, by which the biological effect under study (in terms of a measurable quantity )() increases for a rise of 10 degrees of the temperature T: (1)
)( corresponds to a «rate constant» of the biological phenomenon, which is rather abstract for practical purposes. It is assigned to a hypothetical Arrhenius equation (2)
where R is the gas constant. )(0 describes a hypothetical response at T- 00. Obviously, E describes the temperature characteristics of the biological process in terms of an activation energy of an equilibrium system. The parameters QlO and E have their origins in the projection of the biological ef*) To whom reprint requests should be addressed.
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FRITZ-ALBERT POPP, KE-SHUE
LI and WALTER NAGL
fect to biochemical reactions at about thermal equilibrium, which are referred to as the proper regulators of biological temperature response. Despite of the advantage of such a simple approach, a general criticism of its validity is justified for several reasons. For instance: 1. Even in the case of simplest systems (microorganisms), there appear nonlinear phenomena in the Arrhenius plot, which point to a regulation of temperature that originates not alone from enzymatic activities, but from much more complex overall-interactions within the system (Davey et al., 1966). 2. Interference of the temperature response of biological systems with DNA activities are well established, for instance in the case of mutations (Hoffmann and Ingraham, 1970), activation of heatshock genes (Key et al., 1981; Burke and Ish-Horowicz, 1982), growth control (Ryan and Kiritani, 1959), and adaptation to new temperature environments (Olsen and Metcalf, 1968). 3. Since Prigogine introduced the theory of dissipative structures (Prigogine et al., 1972), it appears to be very unlikely that the living state is governed by thermal equilibrium. The latter assumption, however, underlies the concept of usual treatment of biological response to temperature. The model of dissipative structures points to a general mechanism serving as the common regulator of temperature response, irrespectively of the particular enzymatic pattern or other individual configurations of genetic expression. This new concept is based on thermodynamic laws far away from thermal equilibrium, considered to govern the overall-regulation of biological systems. The temperature response of a definite biological system has to be expressed in such a model in terms of special modifications of the general equation of temperature responses. Hence, we are primarily concerned with the problem of finding a common nominator of temperature response which can serve as the regulator of physiological processes. Its adaption to the specific constraints and effects within the systems under study is then the topic of further experimental research. Let us, therefore, exceptionally start with a purely theoretical investigation of the temperature response of open systems in the range of phase transitions. In fact, such phenomena have often been suggested to underly the temperature response of biological systems (Precht et al., 1973). The second part of this study deals with the application to biological systems, in particular to the thermoluminescence of cucumber seedlings, which was experimentally investigated by our group (Popp et al., 1981). The results can serve as a paradigm by which the theoretical predictions can be examined.
Properties 0/ an Ideal Open System Coupled to a Heat Bath The coupling of (1) an ideal open system 1, representing the active regulator of physiological functions to (2), a two level system 2 with energy difference 1', corresponding to the spectral thermodynamic properties of the passive parts of a cell popZ. Pjlanzenphysiol. Bd. 114. S. 1-13. 1984.
Thermodynamic approach to the temperature response of biological systems
3
- - 1. Environment ,r--+
Fig. 1: Model of photon interaction of a cell population and its environment. I represents the external world as a heat bath of temperature T. The cell population is not in thermal equilibrium. Therefore, it is represented by a transition zone II of «excitation» temperature e(>.), and an active photon store III (e.g. DNA) with excitation temperature 6(>'). In contrast to T, e and 6 depend on the interaction energy E, corresponding to the wavelength >..
ulation (including nutrition), and to (3) a heat bath of temperature T, representing the chaotic part of the external world (Popp, 1979), can deal as a useful simple thermodynamic model of a biological system (Fig. I}. The «ideal» open system is defined as an ensemble with a variable (quasi-}particle number No reflecting the degrees of freedom. No shall be a function of the energy content EI (and the volume). In contrast to a closed system, which is under the constraints of energy conservation, an ideal open system is completely open, this means that it exists without any constraints. The thermodynamics of such a system can be obtained from the grand canonical ensemble under special conditions (Appendix). To get an idea of an ideal open system, we may imagine, for instance, the open exciplex system of DNA bases (Li et al., 1983), which couple and decouple due to their energy content, thus changing their particle number, and at the same time the degrees of freedom. The entropy 51 of an ideal open system (Appendix) becomes a function of No(EI}: S,
=
-k . In No,
(3)
where k represents the Boltzmann constant. The system 2, which is coupled to systems 1 and 3, takes the entropy 52
=
k . (p1lnpl + P2Inp2).
(4)
Ph P2 are the occupation probabilities of the lower and upper energy levels, respectively. 5ince PI +P2 = 1, we should be aware that 52 is a function of PI only (or P2 = 1 - PI). The entropy 53 of the external heat bath is written in the well-known form (5)
where E3 and F3 represent the internal and free energies, respectively. Coupling these three sub-systems to one isolated system, we obtain, by combining the energy conservation law (~~OEi = O) and the condition of maximum entropy Z. Pjlanzenphysiol. Bd. 114. s. 1-13. 1984.
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KE-SHUE
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(which, according to Jaynes, 1957, can be extented to any system, namely ElliSj = 0), the formula (6): (6)
liE2 has been substituted by - (liEI +liE3). liS2 = -k ·In(pdp2)· lipi (according to (4) and PI +P2 = 1) is the change of entropy in system 2. For liE2 = 0 (liEI = -OE 3 ), the system 1 becomes unstable, since in this case we have a coupling only between the ideal open system and the heat bath. This can be deduced from (6): for which we obtain:
No > o. (aNo) aE, kT
(7)
=
As is shown in appendix I, system 1 remains stable only for
<0 (aNO) aE, ' thereby diminishing its degrees of freedom by uptake of energy. Dissipative structures satisfy just this condition. We can conclude that system 2 is necessary for stabiliza~ion of system 1 by means of nutrition (oE2 < 0). Of course, even in the case of liE3 = 0, the ideal open system can be stabilized for
in
fEl) ~(aNo) No ilE,
\P2
=
:0:;;
o.
(8)
This means that the buffer system 2 has to be overpopulated (P2 > pI), or at least to be at threshold (P2 = pI). If system 2 does not transport energy to system 1 or take up energy from system 1, it would decay into the Boltzmann distribution (9), involving destabilization of system 1, too. This results also for liEI = 0 from (6):
;=exp(~;)
(9)
The general solution of (6) takes then the form
=-exp --[-€ (1- + -1(aNo))] p, 1 + aE kT N() aE, n2
(10)
I
aEz
For a more profound understanding, we may consider system 2 as a heat pump which removes heat (liE3) from a lower-temperature reservoir at T, delivering it to a higher one with 8( > T), while absorbing work (liE,), and vice versa. Z Pjlanzenphysiol. Ed. 114.
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As is well-known, the coefficient of performance is OE3=~ oEI 8-T
(11)
In order to evaluate (10) in terms of e, we have to introduce the real physical mechanism of the ideal open system, represented by the term e~;). It describes the energy which is necessary for an alteration of the degrees of freedom. We confine ourselves to a system, which can be derived from a grand canonical ensemble (Appendix). This means that the transition from the equilibrium state into the actual state far away from thermal equilibrium shall be thermodynamically a continuous one. Obviously, this reflects the constraints of evolution of biological systems under most appropriate conditions. Consequently, ~No can be factorized into a term h(T), vEl depending on the heat bath only, and a term f2(e), corresponding to system 2: (12)
By combin~ng (6) for hE3 = 0 (~~~ = In(~) a~.), and (7) for bE2 = 0 (:~~ = ~;), we then obtam (13)
Equ. (13) follows from the facts that, for e = T, we have DEI = 0 according to equ. (11), hence requiring In(pI~2) = 1 at the same time for bE3 = 0 according to (6) and (13). This provides ~No =~, which is satisfied by (13). Consequently, for In (Pl/ vEl • p2) = 1, it follows: ke = f from (9), which fulfills the condition ilNo = No for . ilEI kT . . f'les t hese boundary cond'mons. 0E2= 0 at t he same time. Equ. ( 13) satls Hence, we have to conclude that (13) is the only possible relation for aN., provided I
we exclude discontinuities between the closed and ideal open system. iJE Let us now introduce a chemical potential p. characterizing the deviation of system 2 from thermal equilibrium. We use the well-known definition E£ = exp \~l PI CkT
(14)
J
Equalizing (14) and (10) by insertion of (11) and (13), we then obtain the final relation (15), which has been derived repeatedly by different other ways (Popp, 1979; Popp et aI., 1979): I' =
E
(1 -
~) -
kT.
(15)
Temperature Response of Biological Systems Equ. (15) is at a first view not a new description of non-equilibrium systems. Rather, it is a self-evident relation between chemical potential p.(f) and «excitation» temperature e(l:) for all systems, which are not governed by thermal equilibrium. Z. Pjlanzenphysiol. Bd. 114. S. 1-13. 1984.
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FRITZ-ALBERT POPP, KE-SHUE LI
and WALTER NAGL
Considering, for instance, the uptake (or release) of a photon from (or to) the heat bath as a virtual measuring process, an irreversibility loss of at least AS. _AEmin_k
T
mill -
-
has to be taken into account, as Zucker (1974) pointed out. Measuring the probability distribution, f(8) = exp(-E/k8), by release of a photon of energy E with the thermal «uncertainty» (noise) Af(T) = exp(- k~)' we receive an actual information ASA =
-k ·In
Gf)
= -€
(4 -i).
We can use there the same value E for the energy of photons and the corresponding excited energy level according to equ. (6). For this gain of information, the energy J.I- is used, reducing the entropy by AS =
_Po.
T
According to the Brillouins's principle (Zucker, 1974), we then obtain
For equality we get consequently
-~ =
-€
(4 - i) + k,
which is obviously identical with (15). However, equ. (15) provides a much better basis for understanding the temperature response of biological systems than the Arrhenius plot, mainly, because the mechanism corresponds to the real situation of a non-equilibrium system. Considering the physical background it may be easy to find practical predictions of the relation between J.I- and 8. 8 is a measurable quantity, and at the same time a term that is responsible for biochemical (and biological) regulation. Actually, reasonable connections between 8(E) and biological functions, such as growth control, biochemical reactivity etc., have been described (Popp, 1979; Nagl and Popp, 1983; Popp and Nagl, 1983). Fig. 2 displays the measured energy dependence of 8(E) for cucumber seedlings (Popp et al., 1981; Popp, 1983). Since 8 increases with increasing E, p 2/pl according to (14) becomes practically independent on E. This can be seen by insertion of (15) into (14) after using 8 - E (for a more detailed discussion, see, for instance, Popp, 1979). The independence of (14) on E may be a fundamental biological property, providing permanent phase transitions at the one hand, and a stabilization of activation energies due to changes of temperature on the other hand, as has been repeatedly pointed out (Popp, 1979, 1983). The general validity was actually confirmed by Z. Pjlanzenphysiol. Bd. 114.
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Thermodynamic approach to the temperature response of biological systems
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900
800
Fig. 2: Measured excitation temperatures 8(A) of untreated cucumber seedlings (b) and those which have been treated by acetone for poissoning (a). Although the intensity of the ultraweak photon emission of the dying cells is much higher (which corresponds to the higher excitation temperatures at all A), the agreement of theoretical predictions, namely that 8(A) aI/A, and the experimental results is quite well. The excitation temperature is the temperature, that a black body would have if it exhibited the same radiation intensity which was measured.
_ 700 :.: Q
CD
600 500 400 300
400
500
600
700
lI[nmj
measurements of Slawinski et al. (1981). Of course, when one compares the biological «rate» equation (2) with (14), the main difference can be seen in the fact that (14) is stabilized with respect to changes of E by alterations of temperature, while (2) is obviously not. Thus, by upholding the principal validity of a rate equation for description of biological temperature response, we use (14) instead of (2): (I6)
After insertion of (15) into (16), the QIO-value takes, after straight-forward calculations the form (17 a)
(I7b)
Equ. (17a) is subject ofthe measurable quantities 8 and :~. To obtain an idea of (17a) and (17b), let us consider the simple case of
ajl _ aT
const.
=
a
'
which may serve as a first approximation. By differentiation of (15) we obtain the following differential equation with the solution (19). eTa8 E --=-+{k+a) e2 aT 8
Z. Pjlanzenphysiol. Bd. 114.
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FRITZ-ALBEIlT POPP, KE-SHUE LI
e=
and WALTEIl NAGL
T
(19)
a - Tlt(k+a)'
where a is a system-specific constant 0 ~ a ~ 1. After insertion of (19) into (17a), we then have
ATIO( t)
QlO=l+---a T kT
(20)
(k;)
Since E ~ kT for most physiological reactions, the term .:l~1O a may well be of the order of 1, hence reflecting the real situation (Precht et aI., 1973). At the same time, the well-known decrease of QlO with increasing T is truthfully given by (20). In fact, after differentiation of (20) with respect to T, we obtain (21)
A more complicated case arises for the time-dependence of biological responses after alterations of T. However, it can also be described in terms of (14). For simplicity, we use here only e, because the calculation in terms of (14) would be only a straight-forward one. Repeated differentiation of (15) with respect to the time t yields
e - 8 (1 + 2~e) + ~~ ~) + e[G: -~)
(*
eT. tTe . J=0. ( +ke) t tTT 1
(22)
+--p.--p.
Equ. (22) represents a modified relaxation oscillation which may be discussed in the following way. During a time t (O~t~to) the temperature T shall increase constantly from a value T 1 to a value T 2. For it we provide a relation (23)
Of course, equ. (23) is reasonable for our model under study. The energy by heating up system 2 (<
e (1 - f3tTe2) _ 8 [1:T (1 + 2ke) _j3etT 1:JT _ t 2
-8
22 :: + e
[[
Z. Pjlanzenphysiol. Ed. 114. s. 1-13. 1984.
(*r -~] (1
+
k~)J
=
o.
(24)
Thermodynamic approach to the temperature response of biological systems
e
9
e,
Since e does not change considerably compared to and it treated as a constant value for evaluation of (24). It can then be solved exactly:
.
e(t) =
A -B + (1 + B) exp ( -At)
+ Ct
lS
(25)
. (1 + 2ke _ ~e:\
A=!
where
e
T
~e2
eT)
1-eT
Equ. (25) could now be integrated to give e(t). However, this is not necessary for our discussion. > We see that > 0 for at least the time interval 0 < t :% to, provided In this case e(t), which represents here x(t) of equ. (2), increases with increasing temperature. After t has been switched off at the time to, we get then the wellknown «overshoot» of the systems's response which is frequently observed (Precht et aI., 1973). Of course, we are concerned now with the equation
;e
e
*.
(26)
which has the solution 9(t) =
8(to) 1 _ (t_to)2j3e~(to) eT-j3e 2
(27)
According to the theory equ. (27) the following temperature response will take place. At a definite time t>to, the denominator [of (27)] approaches the value zero, providing a strong increase of e. This leads automatically to a second resonance term for
e< o. Again the overshoot reaction occurs now for decreasing e, hence providing e ~ .f!t-, which
which now in turn causes a sudden decrease of e in view of
{J
now leads to a new increase. Since the time has continued, the next maximum appears at a lower value of e. Thus, oscillations of e have to be expected in this range. Finally, a time t is reached, where becomes according to (27) inevitably less
e
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FRITZ-ALBERT POPP, KE-SHUE Lr and WALTER NAGL
10
'"
T = T2 - - - - - - - - -
time,t
Fig. 3: Typical course of temporal biological temperature response. This course is predicted by formula (24) for the excitation temperature 8(t}. Thereby, the external temperature T is r{sed from T 1 to T 2 within the time interval to. The course shows the typical overshoot and the frequently observed oscillations. The maximum value of 8(t} provides a measure of the energetic coupling within the system, including the hypothetical activation energy. The number of oscillations is related to the velocity of energy transport. 200
~'.'.
150 100 ~ 50 \/I
fr 0
~
.... ·~~1
~
T2
~ T2=35°C .~
a.
150
• T2=40°C
~.
100 50 \
0
2 ' -..."T:30 C
T,-+I'
o
~40°C
..........
T 2=35 C
I<---- T1 - - - I •
;:;:; 200
~
T2
5
10
tlh
15
20
Fig. 4: Temperature dependence of photon emission from cucumber seedlings. The PE at T 1 = 19°C has been normalized at the value SO cps for all samples after 2 h of observation. The decay during the first hours corresponds to «delayed luminescence» and has nothing to do with the temperature effects under study. Then, after different time intervals (upper curves 6 h, lower curves 3 h), the temperature has been rised to various temperatures T 2 within one hour. The measured PE is a function of T 2, the time lag before heating and therefore of the pretreatment of the seedlings. Despite of this pretreatment, the characteristics of Fig.3 are well reproduced by these measurements. In any case, the dependence is similar to that of respiration and other biological functions, indicating that PE is based on physiological processes. It can therefore be taken as an example of temperature response.
then zero. Consequently, e (and therefore x) drops down to lower values. Because 1 I- 0 counts for t - 00, the response dies away. Fig.3 displays the typical course of 8(t) according to equ. (24).
e
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Thermodynamic approach to the temperature response of biological systems
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This theory is well reflected by experimental results. Fig. 4 shows, as an example, the measurements of thermoluminescence of plant seedlings (cucumber, Cucumis sativus, cv. Chinesische Schlangen). The results are consistent with both the theoretical predictions and the results on the temporal course of temperature response as obtained in many other systems (Precht et aI., 1973).
Appendix The aim of this appendix is the discussion of the most fundamental properties of an «idea}" open system which can be deduced from the wellknown grand canonical ensemble under definite constraints. It describes the temperature response of biological systems. The number of systems in the t:!1semble is_m; each system has the same volume V; the total number of particles is fixed at mN, where N is the average number of particles in a system. The total number of particles of a system shall not exceed the number No, which is the ma~i mum particle number for an average energy E per system. The total energy E is fixed at mE. The energy levels for a system are a function of V resp. No. For simplicity, we assume that each particle number N, which in a particular system is variable, corresponds to a definite energy EN. In a given distribution, there are mN1 systems of the ensemble in the quantum state EN1 , mN2 in the state EN2 , etc. The probability pN of finding a system in the state EN is
(1) The number of ways of having a given distribution is
m' . t=II-·
(2)
N mN!
We seek for the most probable distribution, that is consistent with the following three constraints: a) the number of systems in the ensemble is fixed (3) b) the total energy (4) and c) the total number of particles (5) are fixed.
(3)
1:mN = m, 1:mNEN = mE, 1:mN· N = mN.
(4) (5)
The conditions for the most probable distribution then are -lnmN + ct
-
{3EN + NinA = 0,
(6)
where ct, {3, In A are the Lagrange multipliers. Let us now introduce the special constraint
(7)
{3E N = N . InA. (7) is identical to
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FRITZ-ALBERT POPP, KE-SHUE
LI and WALTER NAGL
aEN =
_
("~NS)EN (~)N aEN
aN
(8)
where S(N, EN, ... ) is the entropy of the system under consideration. To take an example: for a thermal equilibrium system equ. (8) means that
(9) or,
(10) The system has to exhibit a chemical potential which reduces the number of particles with increasing energy. As is well-known from thermodynamics, equ. (9) is not possible for a closed system. Actually, we are concerned with an open system, where the particles are condensed into the same state by taking up energy. Such a behaviour can be observed for dissipative structures, where the absorbance of energy under special conditions gives rise to an increasing collective behaviour of originally individual particles. The freezing of degrees of freedom enforces us to reduce the number of particles, which are subjected to the thermodynamic countability. In a general sense, equ. (8) may be interpreted as the defInition of a quasi particle number at phase transitions far away from thermal equilibrium. Let us briefly recall some properties of the system (6) under condition (7) and (8), respectively. From equ. (6) wo obtain after substituting equ. (7)
(11)
mN = exp(a). since PN = mN and EpN = 1, we have m
mN m
1 No
PN=-=-,or
exp(a)
=
(12)
m
mN =-, No
By using the grand partition function
Z (A, (J, v) under the constraint (7), there follows:
=
E AN exp (-{J EN) N
Z - NO(A,{J, v).
(13)
Since In No = In N+{J EN, and (il~ntlN)A,v is no longer a function of {J according to (10), (ilI~rO)A'V cannot be a function of {J, too. We get, therefore, the very important result
Z. Pj/anzenphysiol. Bd. 114. s. 1-13. 1984.
Thermodynamic approach to the temperature response of biological systems
13
This means that the spectral energy fluctuation of the system under study is (theoretically) zero. Ideal open systems exhibit a completely noiseless behaviour. Acknowledgement We are thankful to Professor Dr. W. Haupt, Erlangen, for critical reading of the manuscript and helpful suggestions.
References BURKE, J. F. and D. ISH-HoROWICZ: Expression of Drosophila heat shock genes is regulated in rat-1 cells. Nucleic Acids Res. 10, 3821-3830 (1982). DAVEY, C. B., R.J. MiLLER, and L. A. NELSON: Temperature-dependent anomalies in growth of microorganisms. J. Bacterio!' 91, 1827-1830 (1966). HOFFMANN, B. and J. L. INGRAHAM: A cold-sensitive mutant of Salmonella typhimurium which requires tryptophan for growth at 20 degrees. Biochim. Biophys. Acta 201, 300-308 (1970). JAYNES, E. T.: Information theory and statistical mechanism. Phys. Rev. 106, 620-630 (1957). KEY, J. L., C. Y. LIN, and Y. M. CHEN: Heat shock proteins of higher plants. Proc. Nat!' Acad. Sci. USA 78, 3526-3530 (1981). NAGL, W. and F. A. Popp: A physical (electromagnetic) model of differentiation. 1. Basic considerations. Cytobios 31, 45-62 (1983). OLSEN, R. H. and E. S. METCALF: Conversion of mesophilic to psychophilic bacteria. Science 162,288-289 (1968). Popp, F. A.: Photon storage in biological systems. In: F. A. Popp, G. BECKER, H. L. KONIG, and W. PESCHKE (Eds.): Electromagnetic Bio-Information, 123-149. Urban & Schwarzenberg, Munchen, Wien, Baltimore, 1979. - Neue Horizonte in der Medizin. K. F. Haug-Verlag, Heidelberg, 1983. Popp, F. A. and W. NAGL: A physical (electromagnetic) model of differentiation. 2. Applications and examples. Cytobios 31, 71-83 (1983). Popp, F. A., H. KLIMA, and H. G. SCHMIDT: Aspects of growth regulation in biological systems. Bio-Photon-Physics 3, 23-29 (1979). Popp, F. A., B. RUTH, W. BAHR, J. BoHM, P. GRASS, G. GROLIG, M. RATTEMEYER, H. G. SCHMIDT, and P. WULLE: Emission of visible and ultraviolet radiation by active biological systems. Collect. Phenom. 3, 187~214 (1981). PRECHT, H., J. CHRISTOPHERSEN, H. HENSEL, and W. LAACHER (Eds.): Temperature and Life. Springer, Berlin, Heidelberg, New York, 1973. PRIGOGINE, L, G. NICOLlS, and A. BABLOYANTZ: Thermodynamics of evolution. Physics Today 11,23-28 (1972). RYAN, F. J. and K. KIRITANI: Effect of temperature on natural J. Gen. Microbio!. 20, 644-653 (1959). Mutation in E. coli. SLAWINSKI, J., I. MAJCHOWICZ, and L. CIESLA: Spectral distribution of luminescence from ger· minating plants. J. Lumines. 24/25, 791-794 (1981). ZUCKER, F. J.: Information, Entropie, Komplementaritat und Zeit. In: WEIZSXCKER, E. v. (Ed.): Offene Systeme I, 67. Klett, Stuttgart, 1974.
Z. Pjlanzenphysiol. Ed. 114. S. 1-13. 1984.
1) Laboratory of Biophotons, LudwigstraBe 31, 6520 Worms, F.R.G. 2) Institute of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing, People's Republic of China.
3) Department of Biology, The University, P. O. Box 3049,6750 Kaiserslautern, F.R.G. Received August 31,1983 . Accepted October 6,1983
Summary The temperature response of biological systems (QlO) is described in terms of the physical model of dissipative structures of thermodynamically open systems, which are far away from thermal equilibrium. The theory is shown to be consistent with experimental results on physiological temperature responses of cucumber seedlings.
Key words: Cucurbita pepo, biophotons, luminescence, temperature response, thermodynamics.
Introduction The response of biological systems to alterations of the temperature of their environment was carefully investigated (Precht et al., 1973). For its characterization, commonly the parameters QlO and )( are used. The QlO value is the factor, by which the biological effect under study (in terms of a measurable quantity )() increases for a rise of 10 degrees of the temperature T: (1)
)( corresponds to a «rate constant» of the biological phenomenon, which is rather abstract for practical purposes. It is assigned to a hypothetical Arrhenius equation (2)
where R is the gas constant. )(0 describes a hypothetical response at T- 00. Obviously, E describes the temperature characteristics of the biological process in terms of an activation energy of an equilibrium system. The parameters QlO and E have their origins in the projection of the biological ef*) To whom reprint requests should be addressed.
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FRITZ-ALBERT POPP, KE-SHUE
LI and WALTER NAGL
fect to biochemical reactions at about thermal equilibrium, which are referred to as the proper regulators of biological temperature response. Despite of the advantage of such a simple approach, a general criticism of its validity is justified for several reasons. For instance: 1. Even in the case of simplest systems (microorganisms), there appear nonlinear phenomena in the Arrhenius plot, which point to a regulation of temperature that originates not alone from enzymatic activities, but from much more complex overall-interactions within the system (Davey et al., 1966). 2. Interference of the temperature response of biological systems with DNA activities are well established, for instance in the case of mutations (Hoffmann and Ingraham, 1970), activation of heatshock genes (Key et al., 1981; Burke and Ish-Horowicz, 1982), growth control (Ryan and Kiritani, 1959), and adaptation to new temperature environments (Olsen and Metcalf, 1968). 3. Since Prigogine introduced the theory of dissipative structures (Prigogine et al., 1972), it appears to be very unlikely that the living state is governed by thermal equilibrium. The latter assumption, however, underlies the concept of usual treatment of biological response to temperature. The model of dissipative structures points to a general mechanism serving as the common regulator of temperature response, irrespectively of the particular enzymatic pattern or other individual configurations of genetic expression. This new concept is based on thermodynamic laws far away from thermal equilibrium, considered to govern the overall-regulation of biological systems. The temperature response of a definite biological system has to be expressed in such a model in terms of special modifications of the general equation of temperature responses. Hence, we are primarily concerned with the problem of finding a common nominator of temperature response which can serve as the regulator of physiological processes. Its adaption to the specific constraints and effects within the systems under study is then the topic of further experimental research. Let us, therefore, exceptionally start with a purely theoretical investigation of the temperature response of open systems in the range of phase transitions. In fact, such phenomena have often been suggested to underly the temperature response of biological systems (Precht et al., 1973). The second part of this study deals with the application to biological systems, in particular to the thermoluminescence of cucumber seedlings, which was experimentally investigated by our group (Popp et al., 1981). The results can serve as a paradigm by which the theoretical predictions can be examined.
Properties 0/ an Ideal Open System Coupled to a Heat Bath The coupling of (1) an ideal open system 1, representing the active regulator of physiological functions to (2), a two level system 2 with energy difference 1', corresponding to the spectral thermodynamic properties of the passive parts of a cell popZ. Pjlanzenphysiol. Bd. 114. S. 1-13. 1984.
Thermodynamic approach to the temperature response of biological systems
3
- - 1. Environment ,r--+
Fig. 1: Model of photon interaction of a cell population and its environment. I represents the external world as a heat bath of temperature T. The cell population is not in thermal equilibrium. Therefore, it is represented by a transition zone II of «excitation» temperature e(>.), and an active photon store III (e.g. DNA) with excitation temperature 6(>'). In contrast to T, e and 6 depend on the interaction energy E, corresponding to the wavelength >..
ulation (including nutrition), and to (3) a heat bath of temperature T, representing the chaotic part of the external world (Popp, 1979), can deal as a useful simple thermodynamic model of a biological system (Fig. I}. The «ideal» open system is defined as an ensemble with a variable (quasi-}particle number No reflecting the degrees of freedom. No shall be a function of the energy content EI (and the volume). In contrast to a closed system, which is under the constraints of energy conservation, an ideal open system is completely open, this means that it exists without any constraints. The thermodynamics of such a system can be obtained from the grand canonical ensemble under special conditions (Appendix). To get an idea of an ideal open system, we may imagine, for instance, the open exciplex system of DNA bases (Li et al., 1983), which couple and decouple due to their energy content, thus changing their particle number, and at the same time the degrees of freedom. The entropy 51 of an ideal open system (Appendix) becomes a function of No(EI}: S,
=
-k . In No,
(3)
where k represents the Boltzmann constant. The system 2, which is coupled to systems 1 and 3, takes the entropy 52
=
k . (p1lnpl + P2Inp2).
(4)
Ph P2 are the occupation probabilities of the lower and upper energy levels, respectively. 5ince PI +P2 = 1, we should be aware that 52 is a function of PI only (or P2 = 1 - PI). The entropy 53 of the external heat bath is written in the well-known form (5)
where E3 and F3 represent the internal and free energies, respectively. Coupling these three sub-systems to one isolated system, we obtain, by combining the energy conservation law (~~OEi = O) and the condition of maximum entropy Z. Pjlanzenphysiol. Bd. 114. s. 1-13. 1984.
4
FRITZ-ALBERT
Popp,
KE-SHUE
LI and W ALTEIl NAGL
(which, according to Jaynes, 1957, can be extented to any system, namely ElliSj = 0), the formula (6): (6)
liE2 has been substituted by - (liEI +liE3). liS2 = -k ·In(pdp2)· lipi (according to (4) and PI +P2 = 1) is the change of entropy in system 2. For liE2 = 0 (liEI = -OE 3 ), the system 1 becomes unstable, since in this case we have a coupling only between the ideal open system and the heat bath. This can be deduced from (6): for which we obtain:
No > o. (aNo) aE, kT
(7)
=
As is shown in appendix I, system 1 remains stable only for
<0 (aNO) aE, ' thereby diminishing its degrees of freedom by uptake of energy. Dissipative structures satisfy just this condition. We can conclude that system 2 is necessary for stabiliza~ion of system 1 by means of nutrition (oE2 < 0). Of course, even in the case of liE3 = 0, the ideal open system can be stabilized for
in
fEl) ~(aNo) No ilE,
\P2
=
:0:;;
o.
(8)
This means that the buffer system 2 has to be overpopulated (P2 > pI), or at least to be at threshold (P2 = pI). If system 2 does not transport energy to system 1 or take up energy from system 1, it would decay into the Boltzmann distribution (9), involving destabilization of system 1, too. This results also for liEI = 0 from (6):
;=exp(~;)
(9)
The general solution of (6) takes then the form
=-exp --[-€ (1- + -1(aNo))] p, 1 + aE kT N() aE, n2
(10)
I
aEz
For a more profound understanding, we may consider system 2 as a heat pump which removes heat (liE3) from a lower-temperature reservoir at T, delivering it to a higher one with 8( > T), while absorbing work (liE,), and vice versa. Z Pjlanzenphysiol. Ed. 114.
s. 1-13. 1984.
Thermodynamic approach to the temperature response of biological systems
5
As is well-known, the coefficient of performance is OE3=~ oEI 8-T
(11)
In order to evaluate (10) in terms of e, we have to introduce the real physical mechanism of the ideal open system, represented by the term e~;). It describes the energy which is necessary for an alteration of the degrees of freedom. We confine ourselves to a system, which can be derived from a grand canonical ensemble (Appendix). This means that the transition from the equilibrium state into the actual state far away from thermal equilibrium shall be thermodynamically a continuous one. Obviously, this reflects the constraints of evolution of biological systems under most appropriate conditions. Consequently, ~No can be factorized into a term h(T), vEl depending on the heat bath only, and a term f2(e), corresponding to system 2: (12)
By combin~ng (6) for hE3 = 0 (~~~ = In(~) a~.), and (7) for bE2 = 0 (:~~ = ~;), we then obtam (13)
Equ. (13) follows from the facts that, for e = T, we have DEI = 0 according to equ. (11), hence requiring In(pI~2) = 1 at the same time for bE3 = 0 according to (6) and (13). This provides ~No =~, which is satisfied by (13). Consequently, for In (Pl/ vEl • p2) = 1, it follows: ke = f from (9), which fulfills the condition ilNo = No for . ilEI kT . . f'les t hese boundary cond'mons. 0E2= 0 at t he same time. Equ. ( 13) satls Hence, we have to conclude that (13) is the only possible relation for aN., provided I
we exclude discontinuities between the closed and ideal open system. iJE Let us now introduce a chemical potential p. characterizing the deviation of system 2 from thermal equilibrium. We use the well-known definition E£ = exp \~l PI CkT
(14)
J
Equalizing (14) and (10) by insertion of (11) and (13), we then obtain the final relation (15), which has been derived repeatedly by different other ways (Popp, 1979; Popp et aI., 1979): I' =
E
(1 -
~) -
kT.
(15)
Temperature Response of Biological Systems Equ. (15) is at a first view not a new description of non-equilibrium systems. Rather, it is a self-evident relation between chemical potential p.(f) and «excitation» temperature e(l:) for all systems, which are not governed by thermal equilibrium. Z. Pjlanzenphysiol. Bd. 114. S. 1-13. 1984.
6
FRITZ-ALBERT POPP, KE-SHUE LI
and WALTER NAGL
Considering, for instance, the uptake (or release) of a photon from (or to) the heat bath as a virtual measuring process, an irreversibility loss of at least AS. _AEmin_k
T
mill -
-
has to be taken into account, as Zucker (1974) pointed out. Measuring the probability distribution, f(8) = exp(-E/k8), by release of a photon of energy E with the thermal «uncertainty» (noise) Af(T) = exp(- k~)' we receive an actual information ASA =
-k ·In
Gf)
= -€
(4 -i).
We can use there the same value E for the energy of photons and the corresponding excited energy level according to equ. (6). For this gain of information, the energy J.I- is used, reducing the entropy by AS =
_Po.
T
According to the Brillouins's principle (Zucker, 1974), we then obtain
For equality we get consequently
-~ =
-€
(4 - i) + k,
which is obviously identical with (15). However, equ. (15) provides a much better basis for understanding the temperature response of biological systems than the Arrhenius plot, mainly, because the mechanism corresponds to the real situation of a non-equilibrium system. Considering the physical background it may be easy to find practical predictions of the relation between J.I- and 8. 8 is a measurable quantity, and at the same time a term that is responsible for biochemical (and biological) regulation. Actually, reasonable connections between 8(E) and biological functions, such as growth control, biochemical reactivity etc., have been described (Popp, 1979; Nagl and Popp, 1983; Popp and Nagl, 1983). Fig. 2 displays the measured energy dependence of 8(E) for cucumber seedlings (Popp et al., 1981; Popp, 1983). Since 8 increases with increasing E, p 2/pl according to (14) becomes practically independent on E. This can be seen by insertion of (15) into (14) after using 8 - E (for a more detailed discussion, see, for instance, Popp, 1979). The independence of (14) on E may be a fundamental biological property, providing permanent phase transitions at the one hand, and a stabilization of activation energies due to changes of temperature on the other hand, as has been repeatedly pointed out (Popp, 1979, 1983). The general validity was actually confirmed by Z. Pjlanzenphysiol. Bd. 114.
s.
1-13. 1984.
Thermodynamic approach to the temperature response of biological systems
7
900
800
Fig. 2: Measured excitation temperatures 8(A) of untreated cucumber seedlings (b) and those which have been treated by acetone for poissoning (a). Although the intensity of the ultraweak photon emission of the dying cells is much higher (which corresponds to the higher excitation temperatures at all A), the agreement of theoretical predictions, namely that 8(A) aI/A, and the experimental results is quite well. The excitation temperature is the temperature, that a black body would have if it exhibited the same radiation intensity which was measured.
_ 700 :.: Q
CD
600 500 400 300
400
500
600
700
lI[nmj
measurements of Slawinski et al. (1981). Of course, when one compares the biological «rate» equation (2) with (14), the main difference can be seen in the fact that (14) is stabilized with respect to changes of E by alterations of temperature, while (2) is obviously not. Thus, by upholding the principal validity of a rate equation for description of biological temperature response, we use (14) instead of (2): (I6)
After insertion of (15) into (16), the QIO-value takes, after straight-forward calculations the form (17 a)
(I7b)
Equ. (17a) is subject ofthe measurable quantities 8 and :~. To obtain an idea of (17a) and (17b), let us consider the simple case of
ajl _ aT
const.
=
a
'
which may serve as a first approximation. By differentiation of (15) we obtain the following differential equation with the solution (19). eTa8 E --=-+{k+a) e2 aT 8
Z. Pjlanzenphysiol. Bd. 114.
(18)
s. 1-13. 1984.
8
FRITZ-ALBEIlT POPP, KE-SHUE LI
e=
and WALTEIl NAGL
T
(19)
a - Tlt(k+a)'
where a is a system-specific constant 0 ~ a ~ 1. After insertion of (19) into (17a), we then have
ATIO( t)
QlO=l+---a T kT
(20)
(k;)
Since E ~ kT for most physiological reactions, the term .:l~1O a may well be of the order of 1, hence reflecting the real situation (Precht et aI., 1973). At the same time, the well-known decrease of QlO with increasing T is truthfully given by (20). In fact, after differentiation of (20) with respect to T, we obtain (21)
A more complicated case arises for the time-dependence of biological responses after alterations of T. However, it can also be described in terms of (14). For simplicity, we use here only e, because the calculation in terms of (14) would be only a straight-forward one. Repeated differentiation of (15) with respect to the time t yields
e - 8 (1 + 2~e) + ~~ ~) + e[G: -~)
(*
eT. tTe . J=0. ( +ke) t tTT 1
(22)
+--p.--p.
Equ. (22) represents a modified relaxation oscillation which may be discussed in the following way. During a time t (O~t~to) the temperature T shall increase constantly from a value T 1 to a value T 2. For it we provide a relation (23)
Of course, equ. (23) is reasonable for our model under study. The energy by heating up system 2 (<
e (1 - f3tTe2) _ 8 [1:T (1 + 2ke) _j3etT 1:JT _ t 2
-8
22 :: + e
[[
Z. Pjlanzenphysiol. Ed. 114. s. 1-13. 1984.
(*r -~] (1
+
k~)J
=
o.
(24)
Thermodynamic approach to the temperature response of biological systems
e
9
e,
Since e does not change considerably compared to and it treated as a constant value for evaluation of (24). It can then be solved exactly:
.
e(t) =
A -B + (1 + B) exp ( -At)
+ Ct
lS
(25)
. (1 + 2ke _ ~e:\
A=!
where
e
T
~e2
eT)
1-eT
Equ. (25) could now be integrated to give e(t). However, this is not necessary for our discussion. > We see that > 0 for at least the time interval 0 < t :% to, provided In this case e(t), which represents here x(t) of equ. (2), increases with increasing temperature. After t has been switched off at the time to, we get then the wellknown «overshoot» of the systems's response which is frequently observed (Precht et aI., 1973). Of course, we are concerned now with the equation
;e
e
*.
(26)
which has the solution 9(t) =
8(to) 1 _ (t_to)2j3e~(to) eT-j3e 2
(27)
According to the theory equ. (27) the following temperature response will take place. At a definite time t>to, the denominator [of (27)] approaches the value zero, providing a strong increase of e. This leads automatically to a second resonance term for
e< o. Again the overshoot reaction occurs now for decreasing e, hence providing e ~ .f!t-, which
which now in turn causes a sudden decrease of e in view of
{J
now leads to a new increase. Since the time has continued, the next maximum appears at a lower value of e. Thus, oscillations of e have to be expected in this range. Finally, a time t is reached, where becomes according to (27) inevitably less
e
Z. Pjlanzenphysiol. Ed. 114. s. 1-13. 1984.
FRITZ-ALBERT POPP, KE-SHUE Lr and WALTER NAGL
10
'"
T = T2 - - - - - - - - -
time,t
Fig. 3: Typical course of temporal biological temperature response. This course is predicted by formula (24) for the excitation temperature 8(t}. Thereby, the external temperature T is r{sed from T 1 to T 2 within the time interval to. The course shows the typical overshoot and the frequently observed oscillations. The maximum value of 8(t} provides a measure of the energetic coupling within the system, including the hypothetical activation energy. The number of oscillations is related to the velocity of energy transport. 200
~'.'.
150 100 ~ 50 \/I
fr 0
~
.... ·~~1
~
T2
~ T2=35°C .~
a.
150
• T2=40°C
~.
100 50 \
0
2 ' -..."T:30 C
T,-+I'
o
~40°C
..........
T 2=35 C
I<---- T1 - - - I •
;:;:; 200
~
T2
5
10
tlh
15
20
Fig. 4: Temperature dependence of photon emission from cucumber seedlings. The PE at T 1 = 19°C has been normalized at the value SO cps for all samples after 2 h of observation. The decay during the first hours corresponds to «delayed luminescence» and has nothing to do with the temperature effects under study. Then, after different time intervals (upper curves 6 h, lower curves 3 h), the temperature has been rised to various temperatures T 2 within one hour. The measured PE is a function of T 2, the time lag before heating and therefore of the pretreatment of the seedlings. Despite of this pretreatment, the characteristics of Fig.3 are well reproduced by these measurements. In any case, the dependence is similar to that of respiration and other biological functions, indicating that PE is based on physiological processes. It can therefore be taken as an example of temperature response.
then zero. Consequently, e (and therefore x) drops down to lower values. Because 1 I- 0 counts for t - 00, the response dies away. Fig.3 displays the typical course of 8(t) according to equ. (24).
e
Z. Pjlanzenphysiol. Bd. 114. S. 1-13. 1984.
Thermodynamic approach to the temperature response of biological systems
11
This theory is well reflected by experimental results. Fig. 4 shows, as an example, the measurements of thermoluminescence of plant seedlings (cucumber, Cucumis sativus, cv. Chinesische Schlangen). The results are consistent with both the theoretical predictions and the results on the temporal course of temperature response as obtained in many other systems (Precht et aI., 1973).
Appendix The aim of this appendix is the discussion of the most fundamental properties of an «idea}" open system which can be deduced from the wellknown grand canonical ensemble under definite constraints. It describes the temperature response of biological systems. The number of systems in the t:!1semble is_m; each system has the same volume V; the total number of particles is fixed at mN, where N is the average number of particles in a system. The total number of particles of a system shall not exceed the number No, which is the ma~i mum particle number for an average energy E per system. The total energy E is fixed at mE. The energy levels for a system are a function of V resp. No. For simplicity, we assume that each particle number N, which in a particular system is variable, corresponds to a definite energy EN. In a given distribution, there are mN1 systems of the ensemble in the quantum state EN1 , mN2 in the state EN2 , etc. The probability pN of finding a system in the state EN is
(1) The number of ways of having a given distribution is
m' . t=II-·
(2)
N mN!
We seek for the most probable distribution, that is consistent with the following three constraints: a) the number of systems in the ensemble is fixed (3) b) the total energy (4) and c) the total number of particles (5) are fixed.
(3)
1:mN = m, 1:mNEN = mE, 1:mN· N = mN.
(4) (5)
The conditions for the most probable distribution then are -lnmN + ct
-
{3EN + NinA = 0,
(6)
where ct, {3, In A are the Lagrange multipliers. Let us now introduce the special constraint
(7)
{3E N = N . InA. (7) is identical to
Z. Pjlanzenphysiol. Bd. 114.
s.
1-13. 1984.
12
FRITZ-ALBERT POPP, KE-SHUE
LI and WALTER NAGL
aEN =
_
("~NS)EN (~)N aEN
aN
(8)
where S(N, EN, ... ) is the entropy of the system under consideration. To take an example: for a thermal equilibrium system equ. (8) means that
(9) or,
(10) The system has to exhibit a chemical potential which reduces the number of particles with increasing energy. As is well-known from thermodynamics, equ. (9) is not possible for a closed system. Actually, we are concerned with an open system, where the particles are condensed into the same state by taking up energy. Such a behaviour can be observed for dissipative structures, where the absorbance of energy under special conditions gives rise to an increasing collective behaviour of originally individual particles. The freezing of degrees of freedom enforces us to reduce the number of particles, which are subjected to the thermodynamic countability. In a general sense, equ. (8) may be interpreted as the defInition of a quasi particle number at phase transitions far away from thermal equilibrium. Let us briefly recall some properties of the system (6) under condition (7) and (8), respectively. From equ. (6) wo obtain after substituting equ. (7)
(11)
mN = exp(a). since PN = mN and EpN = 1, we have m
mN m
1 No
PN=-=-,or
exp(a)
=
(12)
m
mN =-, No
By using the grand partition function
Z (A, (J, v) under the constraint (7), there follows:
=
E AN exp (-{J EN) N
Z - NO(A,{J, v).
(13)
Since In No = In N+{J EN, and (il~ntlN)A,v is no longer a function of {J according to (10), (ilI~rO)A'V cannot be a function of {J, too. We get, therefore, the very important result
Z. Pj/anzenphysiol. Bd. 114. s. 1-13. 1984.
Thermodynamic approach to the temperature response of biological systems
13
This means that the spectral energy fluctuation of the system under study is (theoretically) zero. Ideal open systems exhibit a completely noiseless behaviour. Acknowledgement We are thankful to Professor Dr. W. Haupt, Erlangen, for critical reading of the manuscript and helpful suggestions.
References BURKE, J. F. and D. ISH-HoROWICZ: Expression of Drosophila heat shock genes is regulated in rat-1 cells. Nucleic Acids Res. 10, 3821-3830 (1982). DAVEY, C. B., R.J. MiLLER, and L. A. NELSON: Temperature-dependent anomalies in growth of microorganisms. J. Bacterio!' 91, 1827-1830 (1966). HOFFMANN, B. and J. L. INGRAHAM: A cold-sensitive mutant of Salmonella typhimurium which requires tryptophan for growth at 20 degrees. Biochim. Biophys. Acta 201, 300-308 (1970). JAYNES, E. T.: Information theory and statistical mechanism. Phys. Rev. 106, 620-630 (1957). KEY, J. L., C. Y. LIN, and Y. M. CHEN: Heat shock proteins of higher plants. Proc. Nat!' Acad. Sci. USA 78, 3526-3530 (1981). NAGL, W. and F. A. Popp: A physical (electromagnetic) model of differentiation. 1. Basic considerations. Cytobios 31, 45-62 (1983). OLSEN, R. H. and E. S. METCALF: Conversion of mesophilic to psychophilic bacteria. Science 162,288-289 (1968). Popp, F. A.: Photon storage in biological systems. In: F. A. Popp, G. BECKER, H. L. KONIG, and W. PESCHKE (Eds.): Electromagnetic Bio-Information, 123-149. Urban & Schwarzenberg, Munchen, Wien, Baltimore, 1979. - Neue Horizonte in der Medizin. K. F. Haug-Verlag, Heidelberg, 1983. Popp, F. A. and W. NAGL: A physical (electromagnetic) model of differentiation. 2. Applications and examples. Cytobios 31, 71-83 (1983). Popp, F. A., H. KLIMA, and H. G. SCHMIDT: Aspects of growth regulation in biological systems. Bio-Photon-Physics 3, 23-29 (1979). Popp, F. A., B. RUTH, W. BAHR, J. BoHM, P. GRASS, G. GROLIG, M. RATTEMEYER, H. G. SCHMIDT, and P. WULLE: Emission of visible and ultraviolet radiation by active biological systems. Collect. Phenom. 3, 187~214 (1981). PRECHT, H., J. CHRISTOPHERSEN, H. HENSEL, and W. LAACHER (Eds.): Temperature and Life. Springer, Berlin, Heidelberg, New York, 1973. PRIGOGINE, L, G. NICOLlS, and A. BABLOYANTZ: Thermodynamics of evolution. Physics Today 11,23-28 (1972). RYAN, F. J. and K. KIRITANI: Effect of temperature on natural J. Gen. Microbio!. 20, 644-653 (1959). Mutation in E. coli. SLAWINSKI, J., I. MAJCHOWICZ, and L. CIESLA: Spectral distribution of luminescence from ger· minating plants. J. Lumines. 24/25, 791-794 (1981). ZUCKER, F. J.: Information, Entropie, Komplementaritat und Zeit. In: WEIZSXCKER, E. v. (Ed.): Offene Systeme I, 67. Klett, Stuttgart, 1974.
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