- Email: [email protected]
Time-evolution of the entropy of fluctuations in some biological systems as investigated by NMR
Volume 62, number 2
TIME-EVOLUTION AS INVESTIGATED
CHEMICAL PHYSICS LETTERS
OF THE ENTROPY BY
OF FLUCTUATIONS
1 April 1979
IN SOME BIOLOGKAL
SYSTEMS
NMR
RLENK* Dgpartement de Physique de la Mat&e CondensSe de I’lhiversitt! de GenPve. CH-1211 Genera 4. Switzerland
Received 9 November 1978
A simple expression for the entropy of fluctuations has been developed, using the tunnclling-effect model. This gives the possibility to estimate the changes and evolution of entropy in non-crystalline and biological samples by NhlR investigations. On the other hand, the oscillatory character of the time-evolution of some properties, experimenttiy found in the investigted samples of plants, is interpreted in terms of the generalized master equation \\ithan exponential memory function_
in which pi is the population
of the ith level and k is
Tile investigations of entropy represent 3 significant aspect of the characterization of non-crystalline
the Boltzmann constant.
or partially oriented organic samples and biological systems. The latter have many physico-chemical properties, similar to polymer-elastomers or liquid crystals_ The internal energy, Ui, in these samples is determined by the action of the fast microbrownian fluctuations, and it is related with the mean correlation frequency fc, by the relation
In the following we need to establish a relationship between rhe correlation frequency fc and the entropy of fluctuations, S,. This is generally a difficult task, in this approach we wiil develop a partial solution of this problem, using a simple model of tunneiling in a two-level quantum system- Using relation (3), the statistical entropy in this two-level system is given by
uiafF
S=-k
*
(1)
[(l -_p2)!n(l
-&+P-,
InP21-
(4)
where n = l--2 _ Let us recall that the correlation frequency is related with the correlation time, ~c, by the formula
In this relation p2 is the population of the upper quantum level, which is given by the Bo!tzmann distribution
f, = lhc
~2 = fw(
(2)
and that the correlation time rc is given in terms of the time integral over the corresponding correlation function of the random motion. The most significant contnbution to the entropy in non-crystalline or biologica! systems is also given by the motional phenomena_ We will discuss this contribution, using the statistical formula S=-kEpiinpi, i
* Now also at Laboratotie de Physiologic V&%le. de Genke, Geneva, Swiitzerland.
(3) Universit6
- L&),
65)
where E, is the act3ation energy of the tunnel effect and p = 1 fkT. Furthermore, the frequency of correlation = frequency of tunnelling, is given by fc(T)
=f,
exp( - PEaI -
(6)
in which f,
is the tunnelhng frequency at f= m_ Combining eqs. (5) and (6) yields the following formula for the upper-level population
P2
=f,lfL. -
(7)
399
Volume 62. number 2
CHEMICAL PHYSICS LETTERS
1 Apnl 1979
Inserting (7) into (4), one obtains, after some simplifications Sf =
WC/f,) Mt~/f,) -
It is clear that for the corresponding
(8) domain
of ap-
plications, the entropy of fluctuations increases monotonicaIiy with the increasing correlation frequency. Very roughIy one can put
Sraf,
-
(9)
From the experimental point of view, nuciear magnetic resonance (NMR) represents an adequate method for the determination of the correIation frequency [ 1) 3] and consequently, it can contribute to some problems of the quantum thermodynamics of noncrysMine and bioiogicai systems. It can be shown [I ,2] that the correlation frequency in the usual “high temperature” approximation is rebated to the spin refaxation rates in a very simple
manner I/& lx w-1 t
(IOa)
IJf, a ID-7 a A 7
(IOb)
where T, is the-spin-lattice
rehxxrtion time, Tz is the reia.xation time and A is the NMR spectral
spin-spin iine-width. As ir;hlR is able to determine the correIation frequency, NblR investigations can supply information concerning the entropy changes and evolution in noncrystalline umples and biological systems. As a contribution to this probiem, we have performed some NbfR investigations of spinach leaves [2] _ Generaii~ , the leaves represent a particuiar interest in pIant physiolcgy, because of their significant role in bioiogicaI development- The moiectdar dynamics of the Ieaves by NMR has been investigated in their naturaI state (information of the rnoIecuIar dynamics is obtained through the protons in molecules of water) and iyophyiized samples. swdien by C6F6 (information is obtained through the protons of biomoiecuks and the l9 F nuclei of C6 F6)_ The NMR results are presented in figs_ I, 2 and 3Fig. 1 sholxs a comparison of NM R spectra of two different physiological states. Both spectra are given by a poorly resolved multipfet of biomolecuies, centered at about + 0.7 ppm and a peak of the residua1 water- They are quite identical, only the spectral line400
Fig- 1. ‘H NMR spectra of the lyophjlized spinach leaves. in t\io different physioIo&al states: V - vegetative state; F floral state. widths are different. The NblR spectrum with narrow components corresponds to the vegetative (V) state, whiIe the spectrum of the floral state (F) is characterized by the broader NhlR components. Note that the
vegetative state is an “early”
state and the floral state
represents a “more developed” state in the biological evolution. Consequently, the situation in fig. 1, interpreted by eqs_ (8) and (lob). signifies that the entropy of ffuctuationsdecreases during the biological deveiopment On the other hand, the NhlR spectrum of living leaves is determined by the resonance of protons in water_ Note that the Hz0 molecules create a poiymerlike aggregate, which is oriented and strongIy coupled with the remainder of the biological system. Consequently, the II,0 moiecuies reflect the dynamic state of the iiving system_ The Nh4R spectrum of the water in the living leaves consists of a singIe line, with A = 24 ppmDuring the development of the plant, the spectra1 line-width is not constant, but it ispetfodicaliy dependent on time of evolution te : A =f(t,)_ The period of these bioIogica1 osciktions is about 22-24 h (see fig. 2) and this phenomenon can be related to the circadian rhythms in energy metabolism [4]-
Volume 62, number 2
io
2
CHEMICAL
PHYSICS LETTERS
(hrs)
t
Fir;. 2. Oscili~ory behmiour of the time-evolution of the correlation frequency,& (arbitrary units), in the living spinach lea\ es.
We would like to interpret these oscillatory effects by the existence of strong couplings in the investigated systems. This situation is mathematically described by a generalized master equation in statisticai thermodynamics [5], which for a property A(t) is given as follows
afi(t) = -qmd(tj
(11)
_
This equation describes a non-markovian, irreversible process, because it contains the memory function K(t), which involves the history of the investigated system at earlier times (see ref_ [z], ch. 7). The memory function can have many particutar mathematical forms. J-et US examine the case of an exponential memory, such as K(t) = exp( - cut) ,
(12)
where o is the “memory rate”. Casingthe Laplace transform of eqs. (11) and (12) and inverting the result, one obtains the following timeevolution of the property A(fj
A(f) = A(0) exp( - i at) [cos bt 7 (&3-b) sin bt] . (13) in which b = (1 - $&)I/~
1 April 1979
_
Fig. 3. Graphical representation of the time-solution of a property, gobemed by the generalized master equation -.\ith the e&ponentiaI memory funcrion. given by eq. (13). for different values of the memory rate: (8) o = 0 (undamped oscillations); (b) o = 0.5 (damped oscillations): (c) (I = 1.9 (monotonic decay).
Relation (13) demonstrates the oscillatory modulation of the time-evolution of a property in the strongly coupled system, for some particular values of the memory rate, (Y, as shown in fig. 3. I thank Professor H_ Greppin, Jkboratoire de Physiologic WgCtale, Geneva, for supplying the samples of spinach leaves.
References [l] I. Sclomon, Phys Rev. 99 (1955) 559. [I?1 R. Lenk, Broarkn motion and spin relaxation-(Elsevier, Amslerdxn, 1977). [3] R. Lenk, .\I. Bonzon. P_ Descouts and II. Greppin. British Radio Spectroscopy Group, Meeting York, 3~1) 1977_ [4] B-G. Cumminz and E. Wzgux, Ann. Re\. Plant Physiol. 19 (1968) XI_ [5] R Zsanzig. Ph)sica 30 (1964) 1109, and references therein.
401
TIME-EVOLUTION AS INVESTIGATED
CHEMICAL PHYSICS LETTERS
OF THE ENTROPY BY
OF FLUCTUATIONS
1 April 1979
IN SOME BIOLOGKAL
SYSTEMS
NMR
RLENK* Dgpartement de Physique de la Mat&e CondensSe de I’lhiversitt! de GenPve. CH-1211 Genera 4. Switzerland
Received 9 November 1978
A simple expression for the entropy of fluctuations has been developed, using the tunnclling-effect model. This gives the possibility to estimate the changes and evolution of entropy in non-crystalline and biological samples by NhlR investigations. On the other hand, the oscillatory character of the time-evolution of some properties, experimenttiy found in the investigted samples of plants, is interpreted in terms of the generalized master equation \\ithan exponential memory function_
in which pi is the population
of the ith level and k is
Tile investigations of entropy represent 3 significant aspect of the characterization of non-crystalline
the Boltzmann constant.
or partially oriented organic samples and biological systems. The latter have many physico-chemical properties, similar to polymer-elastomers or liquid crystals_ The internal energy, Ui, in these samples is determined by the action of the fast microbrownian fluctuations, and it is related with the mean correlation frequency fc, by the relation
In the following we need to establish a relationship between rhe correlation frequency fc and the entropy of fluctuations, S,. This is generally a difficult task, in this approach we wiil develop a partial solution of this problem, using a simple model of tunneiling in a two-level quantum system- Using relation (3), the statistical entropy in this two-level system is given by
uiafF
S=-k
*
(1)
[(l -_p2)!n(l
-&+P-,
InP21-
(4)
where n = l--2 _ Let us recall that the correlation frequency is related with the correlation time, ~c, by the formula
In this relation p2 is the population of the upper quantum level, which is given by the Bo!tzmann distribution
f, = lhc
~2 = fw(
(2)
and that the correlation time rc is given in terms of the time integral over the corresponding correlation function of the random motion. The most significant contnbution to the entropy in non-crystalline or biologica! systems is also given by the motional phenomena_ We will discuss this contribution, using the statistical formula S=-kEpiinpi, i
* Now also at Laboratotie de Physiologic V&%le. de Genke, Geneva, Swiitzerland.
(3) Universit6
- L&),
65)
where E, is the act3ation energy of the tunnel effect and p = 1 fkT. Furthermore, the frequency of correlation = frequency of tunnelling, is given by fc(T)
=f,
exp( - PEaI -
(6)
in which f,
is the tunnelhng frequency at f= m_ Combining eqs. (5) and (6) yields the following formula for the upper-level population
P2
=f,lfL. -
(7)
399
Volume 62. number 2
CHEMICAL PHYSICS LETTERS
1 Apnl 1979
Inserting (7) into (4), one obtains, after some simplifications Sf =
WC/f,) Mt~/f,) -
It is clear that for the corresponding
(8) domain
of ap-
plications, the entropy of fluctuations increases monotonicaIiy with the increasing correlation frequency. Very roughIy one can put
Sraf,
-
(9)
From the experimental point of view, nuciear magnetic resonance (NMR) represents an adequate method for the determination of the correIation frequency [ 1) 3] and consequently, it can contribute to some problems of the quantum thermodynamics of noncrysMine and bioiogicai systems. It can be shown [I ,2] that the correlation frequency in the usual “high temperature” approximation is rebated to the spin refaxation rates in a very simple
manner I/& lx w-1 t
(IOa)
IJf, a ID-7 a A 7
(IOb)
where T, is the-spin-lattice
rehxxrtion time, Tz is the reia.xation time and A is the NMR spectral
spin-spin iine-width. As ir;hlR is able to determine the correIation frequency, NblR investigations can supply information concerning the entropy changes and evolution in noncrystalline umples and biological systems. As a contribution to this probiem, we have performed some NbfR investigations of spinach leaves [2] _ Generaii~ , the leaves represent a particuiar interest in pIant physiolcgy, because of their significant role in bioiogicaI development- The moiectdar dynamics of the Ieaves by NMR has been investigated in their naturaI state (information of the rnoIecuIar dynamics is obtained through the protons in molecules of water) and iyophyiized samples. swdien by C6F6 (information is obtained through the protons of biomoiecuks and the l9 F nuclei of C6 F6)_ The NMR results are presented in figs_ I, 2 and 3Fig. 1 sholxs a comparison of NM R spectra of two different physiological states. Both spectra are given by a poorly resolved multipfet of biomolecuies, centered at about + 0.7 ppm and a peak of the residua1 water- They are quite identical, only the spectral line400
Fig- 1. ‘H NMR spectra of the lyophjlized spinach leaves. in t\io different physioIo&al states: V - vegetative state; F floral state. widths are different. The NblR spectrum with narrow components corresponds to the vegetative (V) state, whiIe the spectrum of the floral state (F) is characterized by the broader NhlR components. Note that the
vegetative state is an “early”
state and the floral state
represents a “more developed” state in the biological evolution. Consequently, the situation in fig. 1, interpreted by eqs_ (8) and (lob). signifies that the entropy of ffuctuationsdecreases during the biological deveiopment On the other hand, the NhlR spectrum of living leaves is determined by the resonance of protons in water_ Note that the Hz0 molecules create a poiymerlike aggregate, which is oriented and strongIy coupled with the remainder of the biological system. Consequently, the II,0 moiecuies reflect the dynamic state of the iiving system_ The Nh4R spectrum of the water in the living leaves consists of a singIe line, with A = 24 ppmDuring the development of the plant, the spectra1 line-width is not constant, but it ispetfodicaliy dependent on time of evolution te : A =f(t,)_ The period of these bioIogica1 osciktions is about 22-24 h (see fig. 2) and this phenomenon can be related to the circadian rhythms in energy metabolism [4]-
Volume 62, number 2
io
2
CHEMICAL
PHYSICS LETTERS
(hrs)
t
Fir;. 2. Oscili~ory behmiour of the time-evolution of the correlation frequency,& (arbitrary units), in the living spinach lea\ es.
We would like to interpret these oscillatory effects by the existence of strong couplings in the investigated systems. This situation is mathematically described by a generalized master equation in statisticai thermodynamics [5], which for a property A(t) is given as follows
afi(t) = -qmd(tj
(11)
_
This equation describes a non-markovian, irreversible process, because it contains the memory function K(t), which involves the history of the investigated system at earlier times (see ref_ [z], ch. 7). The memory function can have many particutar mathematical forms. J-et US examine the case of an exponential memory, such as K(t) = exp( - cut) ,
(12)
where o is the “memory rate”. Casingthe Laplace transform of eqs. (11) and (12) and inverting the result, one obtains the following timeevolution of the property A(fj
A(f) = A(0) exp( - i at) [cos bt 7 (&3-b) sin bt] . (13) in which b = (1 - $&)I/~
1 April 1979
_
Fig. 3. Graphical representation of the time-solution of a property, gobemed by the generalized master equation -.\ith the e&ponentiaI memory funcrion. given by eq. (13). for different values of the memory rate: (8) o = 0 (undamped oscillations); (b) o = 0.5 (damped oscillations): (c) (I = 1.9 (monotonic decay).
Relation (13) demonstrates the oscillatory modulation of the time-evolution of a property in the strongly coupled system, for some particular values of the memory rate, (Y, as shown in fig. 3. I thank Professor H_ Greppin, Jkboratoire de Physiologic WgCtale, Geneva, for supplying the samples of spinach leaves.
References [l] I. Sclomon, Phys Rev. 99 (1955) 559. [I?1 R. Lenk, Broarkn motion and spin relaxation-(Elsevier, Amslerdxn, 1977). [3] R. Lenk, .\I. Bonzon. P_ Descouts and II. Greppin. British Radio Spectroscopy Group, Meeting York, 3~1) 1977_ [4] B-G. Cumminz and E. Wzgux, Ann. Re\. Plant Physiol. 19 (1968) XI_ [5] R Zsanzig. Ph)sica 30 (1964) 1109, and references therein.
401