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Spontaneous Signal Generation in Living Cells
Bulletin of Mathematical Biology (2001) 63, 643–654 doi:10.1006/bulm.2001.0236 Available online at http://www.idealibrary.com on
Spontaneous Signal Generation in Living Cells FUMIO OOSAWA Aichi Institute of Technology, Yagusa, Toyota 470-0392, Japan Living cells often generate signals spontaneously in the absence of external stimuli. Those signals play an important role in their tactic behaviors. This paper presents a theoretical treatment on the mechanism of spontaneous signal generation. The mechanism consists of two steps: (1) production of the basic fluctuation of the intracellular electric potential due to the open–closed fluctuation of the gates of ion channels and (2) generation of a spike-like fluctuation of potential depending on the positive shift of the basic fluctuation. The first step is described by an equation of the Langevin type, where the random force is proportional to the circulating ion current across the membrane; the average of the square of the random force is proportional to the rate of free-energy consumption by the current. The second step is described by a rate equation of transition of field-sensitive channel gates which contains the fluctuating electric field in the exponential term. There, the fluctuation has a nonlinear effect. Such a two-step process may work in various kinds of living cells. The presence of circulating ion current in the resting state is a most important key. Some cells may be quiet and some cells may be active to generate spontaneous signals. c 2001 Society for Mathematical Biology
1.
I NTRODUCTION
Living cells respond to various kinds of stimuli from the environment and control their behaviors. Sometimes, even without external stimuli, they actively change the behaviors. Take, for example, paramecium cells swimming in water. They swim in a straight line and occasionally change their swimming direction, and then swim straight again (Jennings, 1906; Naitoh and Eckert, 1972; Nakaoka and Oosawa, 1977; Oosawa and Nakaoka, 1977). The directional change occurs spontaneously. In a homogeneous environment, the time interval of successive directional changes has a nearly exponential distribution. It is very likely that stochastic processes are involved to induce directional changes. Paramecium cells swim by beating a large number of cilia on the cell surface. The change of swimming direction is caused by a transient reversal of the direction of the beating cilia on a limited area of the cell surface. As in the case of muscle 0092-8240/01/040643 + 12
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contraction, an electric signal in the paramecium cell initiates the reversal of ciliary beating (Naitoh and Eckert, 1972; Machmer, 1988). A glass capillary microelectrode was inserted into a paramecium cell fixed on a plate and the electric potential inside the cell was measured. The potential was usually about −20 to −30 mV on average. It was not kept constant, but showed a random fluctuation of the amplitude of one to a few mV, including a sharp spikelike fluctuation of a larger amplitude (Moolenaar et al., 1976; Majima, 1979, 1980). This spike-like fluctuation was a signal for partial reversal of ciliary beating on the cell surface. Not all kinds of living cells produce such a large fluctuation of the potential. Some cells are quiet, while others are noisy or active. The purpose of this paper is to present a theoretical treatment on the mechanism of spontaneous signal generation in living cells.
2.
M EMBRANE P OTENTIAL F LUCTUATION AND THE I ON C URRENT
Living cells usually produce an inside-negative electric potential through combination of ion pumps and ion channels in the cell membrane. Ion pumps cause a concentration difference of the various ions between the inside and outside of the cell. Those ions pass through ion channels in the membrane. Let us consider a cell membrane of area S and capacitance C, in which there are two kinds of ion channels for cation A+ and cation B+ . The concentration of cation A+ is denoted by [A+ ]i inside the cell and [A+ ]o outside the cell; and that of cation B+ by [B+ ]i inside and [B+ ]o outside. The electric potential inside the cell is denoted by V . (The potential outside the cell is zero.) The free-energy differences FA and FB of cations A+ and B+ between the inside and outside of the cell is given by FA = eV − kT log([A+ ]o /[A+ ]i ) = e(V − V A )
(1.1)
FB = eV − kT log([B+ ]o /[B+ ]i ) = e(V − V B )
(1.2)
where e is the electric charge of cations, which were assumed to be monovalent; k is the Boltzmann constant and T is the absolute temperature; and VA = (kT /e) log([A+ ]o /[A+ ]i )
(2.1)
VB = (kT /e) log([B+ ]o /[B+ ]i ).
(2.2)
More exactly, ion concentrations in the above equations should be replaced with ion activities. [If ions are divalent, e in (1.1), (1.2) and (2.1), (2.2) should be replaced with 2e.]
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The currents IA and IB of cations A+ and B+ through the channels in the membrane are approximately proportional to the free-energy difference; IA = −G A (V − V A )
(3.1)
IB = −G B (V − V B )
(3.2)
where G A and G B are the conductances due to the ion channels. (I is positive for the inward current.) If [A+ ]i is larger than [A+ ]o and [B+ ]i is smaller than [B+ ]o , VA is negative and VB is positive. The total number of channels for cation A+ is NA and that for B+ is NB , and they assume two states, open and closed. The numbers of open channels of A+ and B+ are denoted as n A and n B , respectively. The conductances G A and G B are expressed as G A = n A gA
(4.1)
G B = n B gB
(4.2)
respectively; where gA and gB are conductances of single channels. In a stationary state, the total current across the membrane must be zero. The outward current of cation A+ and the inward current of B+ cancel each other: IA + IB = 0.
(5)
This equation with the above equations for I gives the electric potential V as V = (G A VA + G B VB )/(G A + G B )
(6)
IA = −IB = (G A G B /(G A + G B ))(VA − VB ).
(7)
with Actually, ion channels undergo thermal fluctuation between two states. The number of open channels fluctuates with time. Therefore, the potential V and the current I in the above equations must be the average values. Let us denote the deviation of the potential from its average V as v and the deviation of the number of open channels from its average n A and n B as δn A and δn B , respectively; then the conductances G A and G B are expressed as G A + δG A = (n A + δn A )gA , and G B +δG B = (n B +δn B )gB . Because of the open–closed fluctuation of the channels, the total current across the membrane is not always kept zero. The excess current is used to charge up or down the membrane capacitance. Therefore, instead of equation (5), the equation C(dv/dt) = IA + IB
(8)
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must be applied. Using the average value of the potential given by equation (6), the following equation is obtained: IA + IB = −(G A + δG A )(V + v − VA ) − (G B + δG B )(V + v − VB ) = −(G A + G B )v − δG A (V + v − VA ) − δG B (V + v − VB ).
(9)
In the first order of fluctuation, − C(dv/dt) = +(G A + G B )v + δG A (V − VA ) + δG B (V − VB )
(10)
or C(dv(t)/dt) = −G o v(t) − Io (δn A (t)/n A − δn B (t)/n B )
(11)
where G o (= G A + G B ) is the average value of the total conductance and Io (= |IA | = |IB |) is the average value of the current of cation A+ or cation B+ , or the circulating current across the membrane. The behavior of the fluctuating potential produced by open–closed fluctuation of the channels is described by this equation, where C, G o and Io are all proportional to the area S of the membrane. Thus, we reached an equation of the Langevin type. It indicates that the random force to produce the potential fluctuation is proportional to the magnitude of the circulating current. In the usual Langevin equation, which describes movements of fine particles in water, the random force arises from a purely thermal fluctuation in the equilibrium. The square average of the random force is proportional to the average thermal energy kT . On the other hand, in the present case, the square average of the random force is proportional to the rate of free-energy consumption by the circulating current. (This point will be discussed again in the later section.) Previously, the potential fluctuation was discussed based on the open–closed fluctuation of the ion channels, but it has never been compared with the fluctuation described by the usual Langevin-type equation. In most of the cell membrane, the relaxation time of the potential fluctuation defined by C/G o is very much shorter than the characteristic time of the open–closed fluctuation of the ion channels. Then, the potential v can follow the fluctuation of the conductance. The average value of the square of the potential fluctuation v is given by hv 2 i = (Io /G o )2 (hδn 2A i/n 2A + hδn 2B i/n 2B ) (12) where hδn 2 i is the average of the square of the number fluctuations of the open channels. Thus, the average amplitude of the fluctuating potential is proportional to the magnitude of the circulating current. This is also rewritten as hv 2 i = (G A G B /(G A + G B ))2 (VA − VB )2 (hδn 2A i/n 2A + hδn 2B i/n 2B ).
(13)
If the open–closed fluctuations of the ion channels are independent of one another, the ratios hδn 2A i/n 2A and hδn 2B i/n 2B are proportional to 1/n A and 1/n B .
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The circulating current I is related to the concentration ratios of cations A+ and B inside and outside the cell. If the concentration difference between the inside and outside of the cell decreases, the circulating current decreases. Finally, when the circulating current vanishes, the potential fluctuation vanishes. More exactly, only the thermal Nyquist noise of the membrane conductance remains. In other words, the large potential fluctuation is produced, consuming the free energy stored in the concentration differences. If the membrane has channels for only one kind of ion, the current does not flow through the channels in the stationary state. The open–closed fluctuation of these channels does not produce the large potential fluctuation even when the cation concentrations differ between the inside and the outside of the cell. +
3.
S IGNAL G ENERATION IN A F LUCTUATING F IELD
Now, let us suppose the third channel for cation C+ in the membrane. The gates of these channels take two states, open and closed. The free energies in these states are assumed to depend on the electric field at the gates; the gate has an electric dipole, the direction of which changes with the open–closed transition. Then, the energy barrier for the transition between two states also depends on the electric field. The rate constants of the open–closed transition and the reverse transition, k+ and k− , are assumed to be expressed in the following form: k+ = k+o exp(−β E)
(14.1)
k− = k−o exp(+β E)
(14.2)
where E is the electric field, given by V /d and d is the thickness of the membrane (Oosawa, 1975). A positive value of β means that the open state becomes more favorable with a positive shift of the potential. Let us denote the probability of the open state as P. The change of P with time is described by the equation d P(t)/dt = −k+ P(t) + k− (1 − P(t)).
(15)
If V or E fluctuates, the values of rate constant k fluctuate. This equation is rewritten as d P(t)/dt = −k+ (t)P(t) + k− (t)(1 − P(t)) (16) with ∗ k+ (t) = k+ exp(−βε(t))
(17.1)
∗ k− (t) = k− exp(+βε(t))
(17.2)
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where the average field is contained in constant k ∗ and ε(t) in the exponential term is the fluctuating component of the field; ε = v(t)/d. ∗ ∗ If k+ is much larger than k− , the probability P is nearly zero when the field has no fluctuating component. When the field begins to fluctuate, the rate of transition to the open state increases rapidly. In contrast with an equation of the Langevin type, the above equation contains a fluctuating quantity in the exponential term. The fluctuation has a nonlinear effect. According to equations (17.1), (17.2), the average of the values of k with fluctuation is always larger than the values of k without fluctuation and the fluctuating field always increases the average of P towards 1/2 (Oosawa, 1975, 1990; Oosawa et al., 1985). Here, it is necessary to consider the influence of opening of these channels to the potential V . The current of cation C+ flowing through the channels is given by IC = G C (V − VC )
(18)
VC = (kT /e) log([C+ ]o /[C+ ]i ).
(19)
with The conductance G C in this case is written as NC P(t) gC , where NC is the number of the third channels and gC is their conductance. To equation (8) in the previous section, the above current must be added. Then, instead of equation (11), we have −C dv/dt = G o v + Io (δn A /n A − δn B /n B ) +NC P(t)gC (V − VC ).
(20)
Let us suppose that upon the positive shift of the fluctuating potential v, the probability P(t) increases. If (V − VC ) is negative, cations C+ flow into the cell and a more positive shift of v is induced. Then, the probability P further increases according to equation (16). More channels are opened and the shift of v is amplified. A sharp spike-like increase of the potential is initiated. According to equation (20), the average value of v becomes positive. Thus, the process of spontaneous signal generation in the cell is described by combination of equations (20) and (16). The process consists of two steps. The first step is to produce a basic fluctuation of the potential in the cell; it requires the circulating current across the membrane and the thermal fluctuation of the gates of ion channels. The second step is to amplify the potential fluctuation and generate spikes at random intervals; it requires field-sensitive ion channels. The first step prepares the fluctuating field for the channels in the second step.
4.
S PATIAL C ORRELATION OF F LUCTUATION
Various kinds of ion channels are not uniformly distributed in the cell membrane and each channel undergoes the open–closed fluctuation at random. Then, the fluc-
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tuating potential is a function not only of time but also spatial coordinates; that is, v = v(x, t), where spatial coordinates are represented by x. In the previous equations, N and n represent number densities and I current densities in the membrane. The contribution of the transverse current in the cell should be added to equation (11) or (20). Thus, instead of equation (19), −C(∂v(x, t)/∂t) = + ρ(∂ 2 v(x, t)/∂ x 2 ) + G o (x)v(x, t) + Io (x)(δn A (x, t)/n A (x) − δn B (x, t)/n B (x)) + NC (x)P(x, t)gC (V − VC )
(21)
where ρ is the conductance of the cytoplasm; if this conductance is large enough, the fluctuating potential is quickly made uniform inside the cell. The probability P and the fluctuating field ε in equation (16) are functions of time and spatial coordinates. Therefore, instead of equation (16) ∗ ∗ d P(x, t)/dt = −k+ exp(βε(x, t))P(x, t) + k− exp(βε(x, t))(1 − P(x, t)). (22) By this equation, the spatial correlation of the probability P at two different coordinates x1 and x2 is related to the spatial and temporal correlation of the fluctuating field approximately in the following way: ∗ ∗ 2 ∗ ∗ 3 hP(x1 , t)P(x2 , t)i − hP(x1 , t)2 i = (k+ k− ) /(k+ + k− ) Z ∗ ∗ exp(−(k+ + k− )τ )4 sinh((β 2 /2)h(ε(x1 , τ )ε(x2 , 0)i)dτ .
(23)
The open–closed fluctuations of two field-sensitive channels are not independent of one another because of spatial correlation of the fluctuating field around them (Oosawa, 1975, 1990).
5.
D ISCUSSIONS
(i) What kinds of cells can generate spontaneous signals? For paramecium cells, the spontaneous change of swimming direction is absolutely necessary to look for the best place to live in their native environment which is spatially and temporarily changing. For this purpose, they evolved a nice device to generate spontaneous signals. The most important key for the generation of spontaneous signals is a circulating current across the membrane. In their case, the outward current is carried mainly by potassium ions and the inward current is carried by calcium ions and other ions. Consider the case of a nerve axon. The axon has two kinds of channels for potassium ions (K+ ) and sodium ions (Na+ ) (Hodgkin and Huxley, 1952). In the
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resting state, only K+ channels are opened and Na+ channels are closed. Therefore, there is no circulating current. The amplitude of the fluctuating potential is about one-hundredth of that of paramecium cells (Fishman et al., 1975). The Na+ and K+ channels of the nerve axon are field sensitive, but the basic potential fluctuation is too small to induce spikes spontaneously. The physiological function of the nerve axon is to propagate given signals, so that generation of spontaneous signals must be suppressed. In nerve synapses, spontaneous quantum release of transmitters was demonstrated. Perhaps, some of the synapses, not all, generate spontaneous signals and it may be related to the presence of the circulating current in the synaptic membrane. In addition, in an ensemble of nerve cells, some cells may be quiet and some others may be noisy or spontaneously active. (ii) Comparison with experimental data The above theoretical treatment showed that spontaneous signals are generated by a two-step process, a basic potential fluctuation and spike-like fluctuation depending on the basic fluctuation. This is actually the case for paramecium cells. The intracellular electric potential of a paramecium cell fixed on a plate was measured by inserting a glass microelectrode. The potential fluctuation appeared to consist of two components (Majima, 1980). The basic fluctuation has an average amplitude of one to a few mV. It has a Lorentzian spectrum; the corner frequency is related to the rate of open–closed fluctuation of the channel gates. The concentration of potassium ions (K+ ) in the cell is much larger than outside, so that in the stationary state, they flow from the inside to the outside. On the other hand, the concentration of calcium ions (Ca2+ ) is much larger outside than inside and they flow into the cell. Some other cations also flow inwards. Previously, it was shown that the average amplitude of basic potential fluctuation increased with the increase of the motive force of the potassium current, V − VK (Moolenaar et al., 1976). Therefore, the potential fluctuation was interpreted to be mainly due to the open–closed fluctuation of channels for potassium ions. However, the analysis has indicated that not only channels for potassium ions but also those for calcium ions make contributions to the potential fluctuation [Majima (1980); more details are given in Majima (1979)]. The average amplitude of the basic fluctuation can be explained assuming that the total number of channels is of the order of one thousand. Positive spikes are generated at random intervals. The probability of positive spike generation was found to increase with the positive shift of the potential in basic fluctuation. This increase appeared to be nearly exponential (Toyotama, 1981). Actually, opening of calcium channels located in the membrane of cilia is field sensitive, and by the mechanism discussed above, a certain number of these channels are opened cooperatively. (Paramecium cells from which cilia were removed show only the basic potential fluctuation and do not generate positive spikes.) Calcium ions which enter through the opened channels induce the rever-
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sal of ciliary beating, although the molecular mechanism of reversal is not known. Thus, the general feature of membrane potential fluctuation in paramecium cells is described by equations (20) and (16). Paramecium cells have an additional mechanism to close the field-sensitive calcium channels. When these channels in the membrane of cilia are opened, some of the calcium ions flowing into the cell bind to the exit and close the channel (Brehm and Eckert, 1978). Too much amplification of the spike is inhibited and the flow of calcium ions continues only in a limited time. This mechanism is not included in the above theoretical treatment. However, it must be remarked that in the case of spontaneous spikes, their height is so small that the downward shift of the basic fluctuation has an effect similar to the inactivation of opened channels. Therefore, the continuation time of the spike is made rather short. According to the mechanism described by equations (20) and (16), the distribution of duration time and interval of positive spikes depends on the amplitude and time correlation of basic fluctuation and the sensitivity of field-sensitive channels. Computer simulation is effective to analyse these problems. The results of simulation will be reported in another paper. The reversal of ciliary beating occurs in a limited area of the cell surface. How is the area of reversal controlled in living cells? This problem is related to the spatial correlation of the potential fluctuation. At present, the measurements of the electric potential inside the cell have no spatial and temporal resolution sufficient to specify the area of spike generation. Usually, the potential is assumed to be uniform in the cell because of high ionic conductivity. However, the cytoplasm contains complex structures such as highly charged filaments. The relaxation time of the potential fluctuation around such filaments is very long, so that transient heterogeneity of the potential is not always negligible (Oosawa, 1970). The theoretical analysis according to equation (21) is a future problem. Paramecium cells can also generate negative spikes spontaneously. They are signals for transient increase of the swimming velocity. Generation of such spikes can be discussed based on a similar mechanism. (iii) Large potential fluctuation associated with the free-energy flow Ion pumps in the membrane work to make the concentration difference of various ions between the inside and outside of the cell. The free energy stored in the form of the concentration difference is dissipated by the circulating ion current across the membrane. The open–closed fluctuation of the gate channels is purely thermal. However, as shown by equation (11), the force to produce large membrane potential fluctuation is proportional to the circulating current. This means that the potential fluctuation requires the supply of free-energy stored in the concentration difference. The rate of free-energy consumption is given by (IA /e)FA + (IB /e)FB = (IA − IB )(VB − VA )/2 = Io2 /(2G A G B /(G A + G B )).
(24)
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Therefore, the average amplitude of the random force in equation (11) of the Langevin type is proportional to the square root of the rate of free-energy consumption or the free-energy flux. The square average of the fluctuating potential becomes proportional to the free-energy flux. The energy stored in the membrane is given by (1/2)C(V + v)2 . The average of the fluctuating component is (1/2)Cv 2 . Calculations based on the above equation show that this quantity is approximately equal to the amount of free energy flowing per channel within the characteristic time of the membrane C/G o . To maintain the concentration difference in spite of the circulating current, ion pumps must work, usually using the chemical free energy of the hydrolysis of ATP. In other words, a part of this free energy is utilized to make the large potential fluctuation. The open–closed fluctuation of the gates of field-sensitive channels described by equation (16) is also associated with the free-energy consumption. The fluctuation of P(t) in equation (16) was proved to have asymmetry with respect to time reversal (Yonezawa and Saito, 1999). This means that a certain amount of free energy is consumed during the gate fluctuation. The free energy flows from the fluctuating field to the channel gates. In other words, to maintain the fluctuating field in spite of the reaction from gates, the free energy must be supplied. This reaction was not taken into consideration in the present treatment. (iv) Concluding remarks Living organisms do not always show deterministic behaviors. Under some conditions, they show probabilistic behaviors. Such behaviors are often related to spontaneous signals generated in those organisms. Regulation of spontaneous signals depending on the environmental condition is a key process for their tactic behaviors (Oosawa and Nakaoka, 1977). In the case of multicellular organisms, it is interesting to investigate which cells are the source of spontaneous signals and ask if those cells have the circulating current in the resting state to produce large potential fluctuation and generate spontaneous signals. In the present work, a Langevin-type equation with the random force associated with the free-energy flux was derived. A similar equation may be useful to describe the process of activation of random movements by the free-energy consuming reaction in various biological systems. For example, thermal movements of F-actin filaments are greatly activated by interaction with myosin molecules hydrolysing ATP molecules (Vale and Oosawa, 1990; Oosawa, 2000). In this case, the square average of the random force is possibly proportional to the free-energy flux given by the ATP hydrolysis. Various types of nonlinear stochastic differential or integral equations have been proposed. Among them, equation (16) has a reasonable physical basis. The mathematical analyses of this type of equations have been performed in a few works (Hida and Potthoff, 1990; Kuo and Potthoff, 1990; Yonezawa and Saito, 1999). It should also be noted that the recently developed idea of stochastic resonance
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may be related to the previously proposed idea contained in equation (16) (Douglass et al., 1993; Collins et al., 1995).
ACKNOWLEDGEMENTS This work was initiated by stimulating discussions with M Yamanoi (Meijo Univ., Nagoya). Further extension of the work is in progress in collaboration with him. Results of computer simulation performed in his laboratory will be reported in the near future.
R EFERENCES Brehm, P. and R. Eckert (1978). Calcium entry leads to inactivation of calcium channel in paramecium. Science 202, 1203–1206. Collins, J., C. Chow and T. Imhoff (1995). Stochastic resonance without tuning. Nature 376, 236–238. Douglass, J., L. Wilkens, E. Pantazelou and F. Moss (1993). Noise enhancement of information transfer in crayfish mechanoreceptors by stochastic resonance. Nature 365, 337–340. Fishman, H., L. Moore and D. Poussart (1975). Potassium-ion conductance noise in squid axon membrane. J. Membr. Biol. 24, 305–328. Hida, T. and J. Potthoff (1990). White noise analysis—An overview, in White Noise Analyses, Hida et al. (Eds), World Scientific, pp. 140–165. Hodgkin, A. and A. Huxley (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544. Jennings, H. (1906). Behavior of Lower Organisms, Indiana University Press. Kuo, H. and J. Potthoff (1990). Anticipating stochastic integrals and stochastic differential equations, in White Noise Analyses, T. Hida et al. (Eds), World Scientific, pp. 256–273. Machmer, H. (1988). Electrophysiology in Paramecium, H. Gortz (Ed.), Springer, pp. 185–215. Majima, J. (1979). Membrane potential fluctuation in paramecium, PhD thesis, Department of Biophysical Engineering, Osaka University (in English). Majima, J. (1980). Membrane potential fluctuation in paramecium. Biophys. Chem. 11, 101–108. Moolenaar, W., J. de Goede and A. Verveen (1976). Membrane noise in paramecium. Nature 260, 344–345. Naitoh, Y. and R. Eckert (1972). Electrophysiology of the ciliate protozoa, in Experiments in Physiology and Biochemistry, G. Kerkut (Ed.), Academic Press, pp. 17–31. Nakaoka, Y. and F. Oosawa (1977). Temperature sensitive behaviors of paramecium caudatum. J. Protozool. 24, 575–580. Oosawa, F. (1970). Counter ion fluctuation and dielectric dispersion in linear polyelectrolytes. Biopolymers 9, 677–688.
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Oosawa, F. (1975). Effect of field fluctuation on a macromolecular system. J. Theor. Biol. 52, 175–186. Oosawa, F. (1990). Hierarchy of noise production in living cells, in White Noise Analysis, T. Hida et al. (Eds), World Scientific, pp. 315–329. Oosawa, F. (2000). The loose coupling mechanism in molecular machines of living cells. Genes to Cells 5, 9–15. Oosawa, F. and Y. Nakaoka (1977). Behaviors of microorganisms as particles with internal state variables. J. Theor. Biol. 66, 747–761. Oosawa, F., M. Tsuchiya and T. Kubori (1985). Fluctuation in living cells, effect of field fluctuation and asymmetry of fluctuation. Lectures Notes in Biomathematics 70, 158–170. Toyotama, H. (1981). Thermo-receptor potential in paramecium, PhD thesis, Department of Biophysical Engineering, Osaka University. Vale, D. and F. Oosawa (1990). Protein motors and Maxwell’s demon; does mechanochemical transduction involve a thermal ratchet? Adv. Biophys. 26, 97–131. Yonezawa, H. and K. Saito (1999). The asymmetry in time for some biological phenomena. Res. Rep. Meijo University 39, 53–59. Received 10 December 2000 and accepted 21 February 2001
Spontaneous Signal Generation in Living Cells FUMIO OOSAWA Aichi Institute of Technology, Yagusa, Toyota 470-0392, Japan Living cells often generate signals spontaneously in the absence of external stimuli. Those signals play an important role in their tactic behaviors. This paper presents a theoretical treatment on the mechanism of spontaneous signal generation. The mechanism consists of two steps: (1) production of the basic fluctuation of the intracellular electric potential due to the open–closed fluctuation of the gates of ion channels and (2) generation of a spike-like fluctuation of potential depending on the positive shift of the basic fluctuation. The first step is described by an equation of the Langevin type, where the random force is proportional to the circulating ion current across the membrane; the average of the square of the random force is proportional to the rate of free-energy consumption by the current. The second step is described by a rate equation of transition of field-sensitive channel gates which contains the fluctuating electric field in the exponential term. There, the fluctuation has a nonlinear effect. Such a two-step process may work in various kinds of living cells. The presence of circulating ion current in the resting state is a most important key. Some cells may be quiet and some cells may be active to generate spontaneous signals. c 2001 Society for Mathematical Biology
1.
I NTRODUCTION
Living cells respond to various kinds of stimuli from the environment and control their behaviors. Sometimes, even without external stimuli, they actively change the behaviors. Take, for example, paramecium cells swimming in water. They swim in a straight line and occasionally change their swimming direction, and then swim straight again (Jennings, 1906; Naitoh and Eckert, 1972; Nakaoka and Oosawa, 1977; Oosawa and Nakaoka, 1977). The directional change occurs spontaneously. In a homogeneous environment, the time interval of successive directional changes has a nearly exponential distribution. It is very likely that stochastic processes are involved to induce directional changes. Paramecium cells swim by beating a large number of cilia on the cell surface. The change of swimming direction is caused by a transient reversal of the direction of the beating cilia on a limited area of the cell surface. As in the case of muscle 0092-8240/01/040643 + 12
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contraction, an electric signal in the paramecium cell initiates the reversal of ciliary beating (Naitoh and Eckert, 1972; Machmer, 1988). A glass capillary microelectrode was inserted into a paramecium cell fixed on a plate and the electric potential inside the cell was measured. The potential was usually about −20 to −30 mV on average. It was not kept constant, but showed a random fluctuation of the amplitude of one to a few mV, including a sharp spikelike fluctuation of a larger amplitude (Moolenaar et al., 1976; Majima, 1979, 1980). This spike-like fluctuation was a signal for partial reversal of ciliary beating on the cell surface. Not all kinds of living cells produce such a large fluctuation of the potential. Some cells are quiet, while others are noisy or active. The purpose of this paper is to present a theoretical treatment on the mechanism of spontaneous signal generation in living cells.
2.
M EMBRANE P OTENTIAL F LUCTUATION AND THE I ON C URRENT
Living cells usually produce an inside-negative electric potential through combination of ion pumps and ion channels in the cell membrane. Ion pumps cause a concentration difference of the various ions between the inside and outside of the cell. Those ions pass through ion channels in the membrane. Let us consider a cell membrane of area S and capacitance C, in which there are two kinds of ion channels for cation A+ and cation B+ . The concentration of cation A+ is denoted by [A+ ]i inside the cell and [A+ ]o outside the cell; and that of cation B+ by [B+ ]i inside and [B+ ]o outside. The electric potential inside the cell is denoted by V . (The potential outside the cell is zero.) The free-energy differences FA and FB of cations A+ and B+ between the inside and outside of the cell is given by FA = eV − kT log([A+ ]o /[A+ ]i ) = e(V − V A )
(1.1)
FB = eV − kT log([B+ ]o /[B+ ]i ) = e(V − V B )
(1.2)
where e is the electric charge of cations, which were assumed to be monovalent; k is the Boltzmann constant and T is the absolute temperature; and VA = (kT /e) log([A+ ]o /[A+ ]i )
(2.1)
VB = (kT /e) log([B+ ]o /[B+ ]i ).
(2.2)
More exactly, ion concentrations in the above equations should be replaced with ion activities. [If ions are divalent, e in (1.1), (1.2) and (2.1), (2.2) should be replaced with 2e.]
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The currents IA and IB of cations A+ and B+ through the channels in the membrane are approximately proportional to the free-energy difference; IA = −G A (V − V A )
(3.1)
IB = −G B (V − V B )
(3.2)
where G A and G B are the conductances due to the ion channels. (I is positive for the inward current.) If [A+ ]i is larger than [A+ ]o and [B+ ]i is smaller than [B+ ]o , VA is negative and VB is positive. The total number of channels for cation A+ is NA and that for B+ is NB , and they assume two states, open and closed. The numbers of open channels of A+ and B+ are denoted as n A and n B , respectively. The conductances G A and G B are expressed as G A = n A gA
(4.1)
G B = n B gB
(4.2)
respectively; where gA and gB are conductances of single channels. In a stationary state, the total current across the membrane must be zero. The outward current of cation A+ and the inward current of B+ cancel each other: IA + IB = 0.
(5)
This equation with the above equations for I gives the electric potential V as V = (G A VA + G B VB )/(G A + G B )
(6)
IA = −IB = (G A G B /(G A + G B ))(VA − VB ).
(7)
with Actually, ion channels undergo thermal fluctuation between two states. The number of open channels fluctuates with time. Therefore, the potential V and the current I in the above equations must be the average values. Let us denote the deviation of the potential from its average V as v and the deviation of the number of open channels from its average n A and n B as δn A and δn B , respectively; then the conductances G A and G B are expressed as G A + δG A = (n A + δn A )gA , and G B +δG B = (n B +δn B )gB . Because of the open–closed fluctuation of the channels, the total current across the membrane is not always kept zero. The excess current is used to charge up or down the membrane capacitance. Therefore, instead of equation (5), the equation C(dv/dt) = IA + IB
(8)
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F. Oosawa
must be applied. Using the average value of the potential given by equation (6), the following equation is obtained: IA + IB = −(G A + δG A )(V + v − VA ) − (G B + δG B )(V + v − VB ) = −(G A + G B )v − δG A (V + v − VA ) − δG B (V + v − VB ).
(9)
In the first order of fluctuation, − C(dv/dt) = +(G A + G B )v + δG A (V − VA ) + δG B (V − VB )
(10)
or C(dv(t)/dt) = −G o v(t) − Io (δn A (t)/n A − δn B (t)/n B )
(11)
where G o (= G A + G B ) is the average value of the total conductance and Io (= |IA | = |IB |) is the average value of the current of cation A+ or cation B+ , or the circulating current across the membrane. The behavior of the fluctuating potential produced by open–closed fluctuation of the channels is described by this equation, where C, G o and Io are all proportional to the area S of the membrane. Thus, we reached an equation of the Langevin type. It indicates that the random force to produce the potential fluctuation is proportional to the magnitude of the circulating current. In the usual Langevin equation, which describes movements of fine particles in water, the random force arises from a purely thermal fluctuation in the equilibrium. The square average of the random force is proportional to the average thermal energy kT . On the other hand, in the present case, the square average of the random force is proportional to the rate of free-energy consumption by the circulating current. (This point will be discussed again in the later section.) Previously, the potential fluctuation was discussed based on the open–closed fluctuation of the ion channels, but it has never been compared with the fluctuation described by the usual Langevin-type equation. In most of the cell membrane, the relaxation time of the potential fluctuation defined by C/G o is very much shorter than the characteristic time of the open–closed fluctuation of the ion channels. Then, the potential v can follow the fluctuation of the conductance. The average value of the square of the potential fluctuation v is given by hv 2 i = (Io /G o )2 (hδn 2A i/n 2A + hδn 2B i/n 2B ) (12) where hδn 2 i is the average of the square of the number fluctuations of the open channels. Thus, the average amplitude of the fluctuating potential is proportional to the magnitude of the circulating current. This is also rewritten as hv 2 i = (G A G B /(G A + G B ))2 (VA − VB )2 (hδn 2A i/n 2A + hδn 2B i/n 2B ).
(13)
If the open–closed fluctuations of the ion channels are independent of one another, the ratios hδn 2A i/n 2A and hδn 2B i/n 2B are proportional to 1/n A and 1/n B .
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The circulating current I is related to the concentration ratios of cations A+ and B inside and outside the cell. If the concentration difference between the inside and outside of the cell decreases, the circulating current decreases. Finally, when the circulating current vanishes, the potential fluctuation vanishes. More exactly, only the thermal Nyquist noise of the membrane conductance remains. In other words, the large potential fluctuation is produced, consuming the free energy stored in the concentration differences. If the membrane has channels for only one kind of ion, the current does not flow through the channels in the stationary state. The open–closed fluctuation of these channels does not produce the large potential fluctuation even when the cation concentrations differ between the inside and the outside of the cell. +
3.
S IGNAL G ENERATION IN A F LUCTUATING F IELD
Now, let us suppose the third channel for cation C+ in the membrane. The gates of these channels take two states, open and closed. The free energies in these states are assumed to depend on the electric field at the gates; the gate has an electric dipole, the direction of which changes with the open–closed transition. Then, the energy barrier for the transition between two states also depends on the electric field. The rate constants of the open–closed transition and the reverse transition, k+ and k− , are assumed to be expressed in the following form: k+ = k+o exp(−β E)
(14.1)
k− = k−o exp(+β E)
(14.2)
where E is the electric field, given by V /d and d is the thickness of the membrane (Oosawa, 1975). A positive value of β means that the open state becomes more favorable with a positive shift of the potential. Let us denote the probability of the open state as P. The change of P with time is described by the equation d P(t)/dt = −k+ P(t) + k− (1 − P(t)).
(15)
If V or E fluctuates, the values of rate constant k fluctuate. This equation is rewritten as d P(t)/dt = −k+ (t)P(t) + k− (t)(1 − P(t)) (16) with ∗ k+ (t) = k+ exp(−βε(t))
(17.1)
∗ k− (t) = k− exp(+βε(t))
(17.2)
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F. Oosawa
where the average field is contained in constant k ∗ and ε(t) in the exponential term is the fluctuating component of the field; ε = v(t)/d. ∗ ∗ If k+ is much larger than k− , the probability P is nearly zero when the field has no fluctuating component. When the field begins to fluctuate, the rate of transition to the open state increases rapidly. In contrast with an equation of the Langevin type, the above equation contains a fluctuating quantity in the exponential term. The fluctuation has a nonlinear effect. According to equations (17.1), (17.2), the average of the values of k with fluctuation is always larger than the values of k without fluctuation and the fluctuating field always increases the average of P towards 1/2 (Oosawa, 1975, 1990; Oosawa et al., 1985). Here, it is necessary to consider the influence of opening of these channels to the potential V . The current of cation C+ flowing through the channels is given by IC = G C (V − VC )
(18)
VC = (kT /e) log([C+ ]o /[C+ ]i ).
(19)
with The conductance G C in this case is written as NC P(t) gC , where NC is the number of the third channels and gC is their conductance. To equation (8) in the previous section, the above current must be added. Then, instead of equation (11), we have −C dv/dt = G o v + Io (δn A /n A − δn B /n B ) +NC P(t)gC (V − VC ).
(20)
Let us suppose that upon the positive shift of the fluctuating potential v, the probability P(t) increases. If (V − VC ) is negative, cations C+ flow into the cell and a more positive shift of v is induced. Then, the probability P further increases according to equation (16). More channels are opened and the shift of v is amplified. A sharp spike-like increase of the potential is initiated. According to equation (20), the average value of v becomes positive. Thus, the process of spontaneous signal generation in the cell is described by combination of equations (20) and (16). The process consists of two steps. The first step is to produce a basic fluctuation of the potential in the cell; it requires the circulating current across the membrane and the thermal fluctuation of the gates of ion channels. The second step is to amplify the potential fluctuation and generate spikes at random intervals; it requires field-sensitive ion channels. The first step prepares the fluctuating field for the channels in the second step.
4.
S PATIAL C ORRELATION OF F LUCTUATION
Various kinds of ion channels are not uniformly distributed in the cell membrane and each channel undergoes the open–closed fluctuation at random. Then, the fluc-
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tuating potential is a function not only of time but also spatial coordinates; that is, v = v(x, t), where spatial coordinates are represented by x. In the previous equations, N and n represent number densities and I current densities in the membrane. The contribution of the transverse current in the cell should be added to equation (11) or (20). Thus, instead of equation (19), −C(∂v(x, t)/∂t) = + ρ(∂ 2 v(x, t)/∂ x 2 ) + G o (x)v(x, t) + Io (x)(δn A (x, t)/n A (x) − δn B (x, t)/n B (x)) + NC (x)P(x, t)gC (V − VC )
(21)
where ρ is the conductance of the cytoplasm; if this conductance is large enough, the fluctuating potential is quickly made uniform inside the cell. The probability P and the fluctuating field ε in equation (16) are functions of time and spatial coordinates. Therefore, instead of equation (16) ∗ ∗ d P(x, t)/dt = −k+ exp(βε(x, t))P(x, t) + k− exp(βε(x, t))(1 − P(x, t)). (22) By this equation, the spatial correlation of the probability P at two different coordinates x1 and x2 is related to the spatial and temporal correlation of the fluctuating field approximately in the following way: ∗ ∗ 2 ∗ ∗ 3 hP(x1 , t)P(x2 , t)i − hP(x1 , t)2 i = (k+ k− ) /(k+ + k− ) Z ∗ ∗ exp(−(k+ + k− )τ )4 sinh((β 2 /2)h(ε(x1 , τ )ε(x2 , 0)i)dτ .
(23)
The open–closed fluctuations of two field-sensitive channels are not independent of one another because of spatial correlation of the fluctuating field around them (Oosawa, 1975, 1990).
5.
D ISCUSSIONS
(i) What kinds of cells can generate spontaneous signals? For paramecium cells, the spontaneous change of swimming direction is absolutely necessary to look for the best place to live in their native environment which is spatially and temporarily changing. For this purpose, they evolved a nice device to generate spontaneous signals. The most important key for the generation of spontaneous signals is a circulating current across the membrane. In their case, the outward current is carried mainly by potassium ions and the inward current is carried by calcium ions and other ions. Consider the case of a nerve axon. The axon has two kinds of channels for potassium ions (K+ ) and sodium ions (Na+ ) (Hodgkin and Huxley, 1952). In the
650
F. Oosawa
resting state, only K+ channels are opened and Na+ channels are closed. Therefore, there is no circulating current. The amplitude of the fluctuating potential is about one-hundredth of that of paramecium cells (Fishman et al., 1975). The Na+ and K+ channels of the nerve axon are field sensitive, but the basic potential fluctuation is too small to induce spikes spontaneously. The physiological function of the nerve axon is to propagate given signals, so that generation of spontaneous signals must be suppressed. In nerve synapses, spontaneous quantum release of transmitters was demonstrated. Perhaps, some of the synapses, not all, generate spontaneous signals and it may be related to the presence of the circulating current in the synaptic membrane. In addition, in an ensemble of nerve cells, some cells may be quiet and some others may be noisy or spontaneously active. (ii) Comparison with experimental data The above theoretical treatment showed that spontaneous signals are generated by a two-step process, a basic potential fluctuation and spike-like fluctuation depending on the basic fluctuation. This is actually the case for paramecium cells. The intracellular electric potential of a paramecium cell fixed on a plate was measured by inserting a glass microelectrode. The potential fluctuation appeared to consist of two components (Majima, 1980). The basic fluctuation has an average amplitude of one to a few mV. It has a Lorentzian spectrum; the corner frequency is related to the rate of open–closed fluctuation of the channel gates. The concentration of potassium ions (K+ ) in the cell is much larger than outside, so that in the stationary state, they flow from the inside to the outside. On the other hand, the concentration of calcium ions (Ca2+ ) is much larger outside than inside and they flow into the cell. Some other cations also flow inwards. Previously, it was shown that the average amplitude of basic potential fluctuation increased with the increase of the motive force of the potassium current, V − VK (Moolenaar et al., 1976). Therefore, the potential fluctuation was interpreted to be mainly due to the open–closed fluctuation of channels for potassium ions. However, the analysis has indicated that not only channels for potassium ions but also those for calcium ions make contributions to the potential fluctuation [Majima (1980); more details are given in Majima (1979)]. The average amplitude of the basic fluctuation can be explained assuming that the total number of channels is of the order of one thousand. Positive spikes are generated at random intervals. The probability of positive spike generation was found to increase with the positive shift of the potential in basic fluctuation. This increase appeared to be nearly exponential (Toyotama, 1981). Actually, opening of calcium channels located in the membrane of cilia is field sensitive, and by the mechanism discussed above, a certain number of these channels are opened cooperatively. (Paramecium cells from which cilia were removed show only the basic potential fluctuation and do not generate positive spikes.) Calcium ions which enter through the opened channels induce the rever-
Spontaneous Signal Generation in Living Cells
651
sal of ciliary beating, although the molecular mechanism of reversal is not known. Thus, the general feature of membrane potential fluctuation in paramecium cells is described by equations (20) and (16). Paramecium cells have an additional mechanism to close the field-sensitive calcium channels. When these channels in the membrane of cilia are opened, some of the calcium ions flowing into the cell bind to the exit and close the channel (Brehm and Eckert, 1978). Too much amplification of the spike is inhibited and the flow of calcium ions continues only in a limited time. This mechanism is not included in the above theoretical treatment. However, it must be remarked that in the case of spontaneous spikes, their height is so small that the downward shift of the basic fluctuation has an effect similar to the inactivation of opened channels. Therefore, the continuation time of the spike is made rather short. According to the mechanism described by equations (20) and (16), the distribution of duration time and interval of positive spikes depends on the amplitude and time correlation of basic fluctuation and the sensitivity of field-sensitive channels. Computer simulation is effective to analyse these problems. The results of simulation will be reported in another paper. The reversal of ciliary beating occurs in a limited area of the cell surface. How is the area of reversal controlled in living cells? This problem is related to the spatial correlation of the potential fluctuation. At present, the measurements of the electric potential inside the cell have no spatial and temporal resolution sufficient to specify the area of spike generation. Usually, the potential is assumed to be uniform in the cell because of high ionic conductivity. However, the cytoplasm contains complex structures such as highly charged filaments. The relaxation time of the potential fluctuation around such filaments is very long, so that transient heterogeneity of the potential is not always negligible (Oosawa, 1970). The theoretical analysis according to equation (21) is a future problem. Paramecium cells can also generate negative spikes spontaneously. They are signals for transient increase of the swimming velocity. Generation of such spikes can be discussed based on a similar mechanism. (iii) Large potential fluctuation associated with the free-energy flow Ion pumps in the membrane work to make the concentration difference of various ions between the inside and outside of the cell. The free energy stored in the form of the concentration difference is dissipated by the circulating ion current across the membrane. The open–closed fluctuation of the gate channels is purely thermal. However, as shown by equation (11), the force to produce large membrane potential fluctuation is proportional to the circulating current. This means that the potential fluctuation requires the supply of free-energy stored in the concentration difference. The rate of free-energy consumption is given by (IA /e)FA + (IB /e)FB = (IA − IB )(VB − VA )/2 = Io2 /(2G A G B /(G A + G B )).
(24)
652
F. Oosawa
Therefore, the average amplitude of the random force in equation (11) of the Langevin type is proportional to the square root of the rate of free-energy consumption or the free-energy flux. The square average of the fluctuating potential becomes proportional to the free-energy flux. The energy stored in the membrane is given by (1/2)C(V + v)2 . The average of the fluctuating component is (1/2)Cv 2 . Calculations based on the above equation show that this quantity is approximately equal to the amount of free energy flowing per channel within the characteristic time of the membrane C/G o . To maintain the concentration difference in spite of the circulating current, ion pumps must work, usually using the chemical free energy of the hydrolysis of ATP. In other words, a part of this free energy is utilized to make the large potential fluctuation. The open–closed fluctuation of the gates of field-sensitive channels described by equation (16) is also associated with the free-energy consumption. The fluctuation of P(t) in equation (16) was proved to have asymmetry with respect to time reversal (Yonezawa and Saito, 1999). This means that a certain amount of free energy is consumed during the gate fluctuation. The free energy flows from the fluctuating field to the channel gates. In other words, to maintain the fluctuating field in spite of the reaction from gates, the free energy must be supplied. This reaction was not taken into consideration in the present treatment. (iv) Concluding remarks Living organisms do not always show deterministic behaviors. Under some conditions, they show probabilistic behaviors. Such behaviors are often related to spontaneous signals generated in those organisms. Regulation of spontaneous signals depending on the environmental condition is a key process for their tactic behaviors (Oosawa and Nakaoka, 1977). In the case of multicellular organisms, it is interesting to investigate which cells are the source of spontaneous signals and ask if those cells have the circulating current in the resting state to produce large potential fluctuation and generate spontaneous signals. In the present work, a Langevin-type equation with the random force associated with the free-energy flux was derived. A similar equation may be useful to describe the process of activation of random movements by the free-energy consuming reaction in various biological systems. For example, thermal movements of F-actin filaments are greatly activated by interaction with myosin molecules hydrolysing ATP molecules (Vale and Oosawa, 1990; Oosawa, 2000). In this case, the square average of the random force is possibly proportional to the free-energy flux given by the ATP hydrolysis. Various types of nonlinear stochastic differential or integral equations have been proposed. Among them, equation (16) has a reasonable physical basis. The mathematical analyses of this type of equations have been performed in a few works (Hida and Potthoff, 1990; Kuo and Potthoff, 1990; Yonezawa and Saito, 1999). It should also be noted that the recently developed idea of stochastic resonance
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may be related to the previously proposed idea contained in equation (16) (Douglass et al., 1993; Collins et al., 1995).
ACKNOWLEDGEMENTS This work was initiated by stimulating discussions with M Yamanoi (Meijo Univ., Nagoya). Further extension of the work is in progress in collaboration with him. Results of computer simulation performed in his laboratory will be reported in the near future.
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Oosawa, F. (1975). Effect of field fluctuation on a macromolecular system. J. Theor. Biol. 52, 175–186. Oosawa, F. (1990). Hierarchy of noise production in living cells, in White Noise Analysis, T. Hida et al. (Eds), World Scientific, pp. 315–329. Oosawa, F. (2000). The loose coupling mechanism in molecular machines of living cells. Genes to Cells 5, 9–15. Oosawa, F. and Y. Nakaoka (1977). Behaviors of microorganisms as particles with internal state variables. J. Theor. Biol. 66, 747–761. Oosawa, F., M. Tsuchiya and T. Kubori (1985). Fluctuation in living cells, effect of field fluctuation and asymmetry of fluctuation. Lectures Notes in Biomathematics 70, 158–170. Toyotama, H. (1981). Thermo-receptor potential in paramecium, PhD thesis, Department of Biophysical Engineering, Osaka University. Vale, D. and F. Oosawa (1990). Protein motors and Maxwell’s demon; does mechanochemical transduction involve a thermal ratchet? Adv. Biophys. 26, 97–131. Yonezawa, H. and K. Saito (1999). The asymmetry in time for some biological phenomena. Res. Rep. Meijo University 39, 53–59. Received 10 December 2000 and accepted 21 February 2001