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exist the resting state Ri, in which a cell can remain for a long time, and the resting state R2, in which a cell can be blocked by the phase G2. Switching from the resting state to proliferation and vise versa is under die influence of different factors that can sometimes act spontaneously. Mechanism of this switching is the main object of modeling.

According to Y. Romanovsky, N. Stepanova, D. Chernavsky [162], we accept the hypothesis that the regulator of the cell cycle is situated in the cell membrane. The problem is to find a model for the origin of cell membrane signals that influence transition from one phase of the cell cycle to another.

There are two types of processes in a cell membrane: chemical and physical. Let us consider these two processes.

Relating to the chemical processes is:

1. Oxidation of lipids with participation of free radicals. There exists an intermediate state in which lipids themselves become radicals, and this process is auto-catalytic. Characteristic concentrations of radicals are 1CT6 mole/liter and the half-renovation time is 1 sec.

2. Removing from membrane lipids that are oxidized too much so that they stop becoming radicals.

3. Introducing new lipids into a membrane with the help of special protein-carriers. Characteristic half-renovation time of lipids is 10 hours.

4. Weak radicals and anti-oxidizers regulate the process of lipid. Characteristic half-renovation time of anti-oxidizers in a membrane is 1 minute.

Thus, MM of tihe process of oxidation of lipids in a membrane has the form (dimensionless) [162]

where S, R, and RA are concentrations of lipids, radicals of super-oxide-type, and anti-oxidizers.

Characteristic values of parameters in order are xs = 104 sec, xR = 1 sec, x& = 1G2 sec, the others have order 1.

The fastest process cannot be a stationary one because the null cline dRidt = 0 contains unstable parts.

The intermediate in the speed process is always stable and we can put dRNdt = 0, i.e., RA = y/(R +62). The reduced system takes the form

The null-cline of this system is the attract that under y < K has N-shaped character. The phase portrait of the system shows the following possibilities:

XsdS/dt = v - aSR - DSy XgdR/dt = k + aSR- yRf(R +82)-i?2- Sjt.

1. Stationary stable behavior under small R and large S. This behavior coordinates with die cell cycle resting state Rx.

2. Stable state under small S and large R. This case coordinates with the resting state R2.

3. Auto-oscillating behavior under the condition that the point of intersection of the clines dS/dt = dRJdt = 0 is situated on the dropping line of die attractor. The fast phases of the attractor are accompanied with harsh changes of S and R. We can consider jumping to large S as a signal for the start of the DNK synthesis and jumping to large R as a signal for the start of mitosis.

4. Trigger-type behavior under the condition that die cline dS/dt - 0 intersects the attractor three times. Here we have two stable states, R\ and R2, and transfer from each to another is possible under a finite perturbation. In addition, transfer from R2 to Ri is accompanied by cell division. Switching the resting state to proliferation can be the result of hanging of parameters. So, increase of y and v and decrease of k promote transfer to the resting state Ry.

However, this model is insufficiently crude (i.e., under small perturbations of parameters all its various behaviors can overlap), and it does not include the state of cell membrane. This model better suits the properties of the cancer cell cycle than of the normal cell cycle.

The minimal model regarding physical processes in a cell membrane has the form where %a is the time for phase equilibrium in a membrane and F(a,S) is a hysteresis-type function characterizing viscosity of the compound lipid layer of a membrane. Viscosity of a membrane in the "fluid" and "solid" state of membrane can differ.

We cannot reduce the model (23) to the case of two equations. But the whole process can be represented on the phase plane (S, R) projecting on it the motion of depicted point on two planes a = a* and a = «2, which correspond to die certain values S = S, and S = S2.

As a result, the model (23) has the same four main properties as die model (22). In addition, the model (23) is sufficiently crude.

There are some other MM of the cell cycle (see below and [1,51,146]).

Very complicated models of the system of partial differential equations are constructed and investigated by B. Sendov [168], assuming the hypothesis that the regulator of the cell cycle is located in the cell nucleus.

According to [51], mitosis and cell division are controlled by a protein called MPF (M-phase promoting factor). MPF, which consists of two subunits (cdc2 and cyclin), can be active or inactive. Active MPF is known to promote its own activation. An analysis of the mechanism of interactions between cdc2 and cyclin leads to a pair of ODEs similar to a well-known MM of the oregonator tgdSIdt = v~ aSR - DS, XgdRidt =k+ aSR - yR/(R +82)-R2-6 A

where x = active MPF, z = total cyclin, and where /and q are parameters. Originally, the model (24) arose to describe the well-known Belousov-Zhabotinskiy (BZ) reaction (A, Zhabotinskiy [215]). One of the important properties of this reaction is oscillation when 0.5 < / < 2,414. Similarly to the BZ reaction, the MPF control circuit can be spontaneously oscillatory (as in early embryonic division) or excitable (as in somatic cell division). As expected, waves of nuclear division can propagate through the somatic tissue. These results were developed further in [146], where MM and phase-plane portraits of the Gl, G2, and M-phase checkpoints of the cell cycle have been described.

The hyper-cycle MM is developed by Eigen and Schuster (see, for example, G. Rowe [163]). The hyper-cycle is suggested by the cyclic property of the DNA-protein synthesis reactions in modem organisms. DNA contains the information for constructing proteins, which are in turn necessary to construct more DNA, and so on.

We take the simple step of closing the loop by requiring the last products as a

We suppose that we have n different species of information molecules' (such as RNA or DNA sequences), each of which contains the instructions for manufacturing one protein E,-. This protein is assumed to be a 'replicase,* whose function is to recognize and translate the next information molecule I,+1 into its replicase Eí+1.

A general hyper-cyclic system will have n species in the loop, with p species participating in each reaction. The rate term in such a system will have the general form where p - 1 indices r denote all the other species participating in the reaction that The respective system of ODE can be written in the form where S, arises from flows into or out of the system. We assume that where S is some overall flux and C is the total concentration of all species:

If we impose the constraint of constant concentration, C = Co, then the system ODEs where the sum is taken over all sets of indices [r, s,t) such that the resulting product of factors is one of the rate terms of

The general stationary point analysis for autonomous systems in the case of the equal rate constants results in the following (G. Rowe [163]).

When n = 2, 3, or 4, there is one stationary point with all species present, and this point is asymptotically stable. For n > 4, a stable limit cycle appears in which all species oscillate in quantity, but still participate all the time.

The general conclusion is that the hyper-cycle equations provide a chemical system in which all the information present in a set of molecules can be maintained over long periods.

"One can envisage hyper-cycles encompassing more and more information by inserting extra links in the chain as more information-carrying molecules evolve" [163], p. 95.

For Eigen and Schuster's application of the hyper-cycle MM to the origin of life, refer to [163].


Let I(t) be the positive direction for the membrane current outwards from the axon. This current is due to individual ions, which pass through the membrane, and to the contribution from the time variation in the transmembrane potential (the membrane capacitance contribution). Thus we have (J. Murray [143])

where C is die capacitance and ht is the current contribution from the ion movement across the membrane. Based on experimental observation Hodgkin and Huxley (1952) took where V is the potential and /Nai /K, and /L are respectively die sodium, potassium ami 'leakage' currents, /L is a contribution from all the other ions that contribute to the current.

The gs are constant conductances with, for example, gNam3/i sodium conductance, and Vns, Vk, and VL, which are constant equilibrium potentials.

The m, n% and h, bounded by 0 and 1, are variables that are determined by the differential equations

where the a and (3 are given functions of V (again empirically determined by fitting the results to the data); aJV) and aJY) are qualitatively like (1+ tanh V)/2 while a*(V) is qualitatively like (1 - tanh V)/2, which is a 'turn-off switch, when Vis moderately large.

If an applied current Ia(t) is imposed, then the governing equation using (29) becomes

The system (33) with (32) constitutes the 4-variable Hodgkin and Huxley MM, which was solved numerically.

If Jo = 0, the rest state of the model (32) and (33) is linearly stable but excitable.

That is, if the perturbation from the steady state is sufficiently large, there is a large excursion of the variables in their phase space before returning to the steady state.

If Ia ^ 0, there is a range of values where regular repetitive firing occurs. The mechanism displays limited cycle characteristics. Both types of phenomena have been observed experimentally.

Because of the complexity of the equation system, various simpler mathematical models, which capture the key features of the full system, have been proposed ([134, 143, and 146]). At the same time, various generalizations and specifications have been worked out ([134, 146, and 163]. Furthermore, different neuro-simulators [134] and numerical methods for neuronal modeling (M. Mascagni, A. Sherman [130]) were developed.

It should be noted that A. Sherman [170] considers the models of electrical activity with the typical Z-shaped phase portraits with application to the pancreatic |i-cell. The cell energetic metabolism is determined by the electrical activity and production of insulin. The time scales for m, n> and h in (32) are not the same order. The time scale for m is much faster than for the others. So we can put dmfdt - 0.

If we set h = constant, the system still retains many features that are experimentally observed.

The resulting 2-variable model in V and n can be qualitatively approximated by the dimensionless system where 0 < a < 1 and b and y are positive constants. Here v is like the membrane potential V, and w plays the role in all three variables m, n and h.

With 7a = 0 or with a constant, the system (34) is simply a two-variable phase plane system, the phase portrait for which can be easily constructed. There can be, for example, one or three steady states.

The excitability characteristic, a key feature in the Hodgkin-Huxley system, is now evident. That is, a perturbation, for example, from 0 to a point on the v-axis with v > a, undergoes a large phase trajectory excursion before returning to 0.

dv/dt =J{v) - w + la , dw/dt = bv ~jw, J{v) = y(a-v)(v~l),

Spatial Systems naturally arise from the general conservation law of mathematical physics (see J. Murray [143]; A. Samarskii, A. Mikhailov [164]; V, Vladimirov [195]).

Let S be an arbitrary surface enclosing a volume V. The general conservation law states that the rate of change of the amount of material in V is equal to the rate of flow of material across S into Vplus the rate of material created in V, i.e., where the vector J = QXf J,, J2) is the velocity or flux of material and / is the source of material that may be a function of density c, x, and t. Applying the divergence theorem to the surface integral, one can transfer the relation (35) into the form iv [dcfdt + VJ-j{c, x, t)]dv = 0, VJ = dJJdx + dJ/dy + dJJdz. (36)

Since the volume Vis arbitrary, (36) implies die so-called conservation equation for c:

In a cellular context, c is the cell density n, with logistic population growth/- rn(N-n), where r is the initial proliferative rate and N is the maximum cell density in the absence of any other effects. In addition, the flux J can be at the expense of not only the classical, Fickian diffusion, but also of more general diffusion, including long range contribution as well as convection, haptotaxis, galvanotaxis, and chemotaxis (see the last equation in (62) below). Still more complicated models can be found in [3land 129]. Introducing, for example, die pressure field p(x, t)f we obtain the so-called Navier-Stokes equations; in particular, the Euler's equations (see C. Doering, J, Gibbon [40]).

The kinetic equation with regard to interactions of components and diffusion has the where kinetic variables x,- = x((t> r) depend on both the time t and coordinate r, Dy are coefficients of diffusion, and functions Ff determine the total rates of variations xf under their interactions. In biochemistry, variables x, are concentrations of reacting substances; in biology, they are enzyme-substrate complex, biomass or the numbers of organisms of the given species in the unit of volume, etc. The particular cases of the system (38) are widely studied Fisher-Kolmogorov equations. The system (1) is the point system

Let us consider in more detail the case Du - Dh = 0, and dDfir = 0. Then the linearized system for (38) in the vicinity of die stationary values x* of homogenous state of die initial system, i.e., the root of die system equations (6), has the form dx'fât = £ [a^c'j + AôV/ar2] (J = 1,n%au = &Ffèxj)\ x'i = x'{-i = 1,2»n, The solutions of (30) are linear combinations of the waves x'i = Aßxp (pfci + ikr), i = 1,2 n,i = (-l)

where k is the wave number, determining the wave length Xk = 2n/ky and pk = 8è + are the solutions of the so-called equation of variance

If the point x* is unstable, then there is at least one k for which 5* > 0. Unsteadiness here Is divided into two forms. If the equation (41) for the wavelength A* has an even number of roots with 5* > 0, then unsteadiness is called vibrating. In the case of odd roots we have unsteadiness by Turing, reducing to the so-called stationary dissipative or damping structures (DS).

For n = 2, = 0, and dDfir = 0, the system (38) can be rewritten in the form dxßt = P(x,y) + DjPxidf, dyßt = Q(x,y) + D^yßS.

A = k (Dx + By) - aii - am B = ana22 - «2i«i2 + k4DxDy- &(anDy + anDx. (44)

The particular case of the system (42) is the Van-der-Pol self-vibrating system: dxBt = y + Dxd2x/S?, dy/dt = - (six + 28# - 28^ + Dy tfyfor*. (45)

In this case, pk = - A72 ± ((A'/2)2 - B')1/2. (46)

Another class of MM for partial systems arises in quantum mechanics and its applications in biology (J. Adam [3]; A, Davydov [38]).

For examples, we have Schrddinger time-independent equation in the form vV(x) + [£-V(x)]\|f(x) = 0,

where |\{/(x)| is a measure of the probability that the particle of energy E > 0 will be found at location x, given that the particular potential V(x) is present, or the particular case of Schrddinger nonlinear equation in the form

where h = 1.054-10"27 erg; E, V, and t)|2 have the same sense as above; D is the strain potential of the molecular chain with distance R between adjacent molecules interactions, which are determined by the potential 2J due to the electric dipole moment d directed under the angle 9 with respect to the chain line; G is the parameter of non-linearity determining the part Az of the chain that is energizing circuit; and Az = nRJlAG.

In the case (39), x = x9 V2 = tfVafr2, and V(x) = V > E > 0, 0 < x < a, V(x) = 0, otherwise, denoting k = (E)m and K = (V~E)m, the solution has the following form y(x) = Aelkx+ Be** (x < 0), Ce^ + DeF* (0<x< a), AS(E)em™y(x>a%

where the constants A, B, C, and D can be easily found from the condition of continuity at x = 0 and x = a; the transmission amplitude

S(E) = 2MkKi{2lkKch(Ka) + (i^-^2)sh2(&)]}, i = (-1)

The quantity AS(E) * 0, meaning that the wave or quantum particle has a non-zero probability of being found to the right of the step. This phenomenon is called tunneling.

A related quantity jS(£)| can be regarded as the energy transmission factor: \S(E)\2 = 4£(V-£)/[4e(V-£) + VW(&)].

In the case of (49), the exact solution with the condition t)\d% = 1 has the following form t) = (p/2)i/2exp{ Ï[hv/(2J)(^)-Etm] p. = G/(4J), (53)

where vR = V is the rate of perturbation propagation along the chain V < VA and VA is the rate of sound propagation along the chain. In addition, is the probability of perturbation of separate molecules along the chain. The value Çq is the argument of the maximal probability and deformation at the instant t = 0.

The perturbations described by the functions (53) and (54) are called solitons. They are capable to carry over perturbation energy very effectively. In particular, they arose in

The important tool of mathematical modeling is a connection between the system of ODE with a random term to the end of it and with the respective system of PDE. À typical form of the equations describing biochemical reactions is where C, i, A, and r represent the concentrations, time, stoichiometric matrix, and the rate law, respectively. Molecular fluctuations can be incorporated explicitly, for example, by including a white noise term to the end of (55):

Then the evolution of the probability density function p(Cj) is described by the Fokker-Planck or Kolmogorov's forward equation dp(C,t)/dt = - V[Ar{Qp(C,t)] + irïLij d2Gip(CMdCtdCJ), (57)

where the matrix <5y is the eovariance of the noise process x(t).

In the case of cells, the quantity p(€,t)dC is die probability of finding a cell with a concentration of a certain chemical between C and C + 3C at a time t.

MM (57) is usually analyzed using Monte-Carlo methods, i.e., solving numerically the equation (56) many times and then using statistics to estimate the probability density

All these relations (36)--(57) are widespread under investigation of various biological events, in particular, events in cells. In particular, works by C. Rao, D. Wolf, A, Arkin [158] apply (55)—(57) for modeling of control, exploitation and tolerance of


MM of tumor angiogenesis are described in detail by M. Chaplain [33]. Tumor angioginesis factor (TAF) having concentration c(x, t) is secreted by die solid tumor and diffuses into the surrounding tissue. Upon reaching endothelial cells in, for example, the

Limbal vessels, the TAF stimulate the release of enzymes by the endothelial cells, which degrade their basement membrane. The initial response of the endothelial cells is to migrate towards the source of TAF. Capillary sprouts are formed and cells begin to

MM for TAF and endothelial cells having concentration «(x, t) are

where the first, second, and third term into the first equation mean accordingly diffusion, production, and decay of moiety and the first two, next, and last term in the second equation mean accordingly cell migration, mitotic generation, and cell loss.

where c0(x) is a prescribed function chosen to describe qualitatively the profile of TAF in die external tissue when it reaches die limbal vessels, c* is a constant value c on the boundary of die tumor, L is a distance to the limbus, and no is an initial endothelial cell

Given the complex nature of tumors' micro-ecology, authors of [146] have developed a series of models, which address the interactive nature of the sub-populations

Assume, for example, a tumor is composed of two different kinds of cells, X and Y, that represent accordingly the proliferating tumor cells and the hypoxic/anoxic tumor population. Then the micro-ecology in which it is growing can be modeled by where r's are the Malthusian growth parameters of the individual population, the iTs are the carrying capacities for the individual populations, die c's are the inter-specific competition rates (and could represent negative growth factor signaling), and m is the

Anyone also can find in M. Mackey [123] a more sophisticated approach and certain


According to J. Murray [143], the time-scale of embryonic motions during development is long (hours) and the spatial scale is small (less than a millimeter or two). We can therefore ignore inertial effect in the mechanical equation for the cell-extracellular matrix (ECM) interaction.

The mechanical cell-matrix equation is then where F is the external force acting on the matrix and 0 is the stress tensor (applied force is balanced by elastic force).

Here are the various models of 0, F, and the matrix material p = p(r,t):

^viscous — IWf + MaVl. 0** = E/(l+ v)[e + v/(l - 2v)fll], e = V*(Vu + VuT),

F = -su, dp/dt + VipUt) = S(n,p,u), dn/dt = -Vindufdt] + V[D,Vn - D2V(V2«)] - V-iifoVp - a2V*p] + rn(N - n), (62)

where the subscript t denotes partial differentiation, I is the unit tensor, jii and are shear and bulk viscosities of ECM, e is the strain tensor, $ is the dilation, $ = V u, E and v are die Young's modulus and Poisson ratio respectively, x is a measure of the traction force generated by a cell, % is a measure of how the force is reduced because of neighboring cells of number n, y is the measure of the non-local long range cell-ECM interaction, s is an elastic parameter characterizing the substrate attachment, and 5(«,p,u) is the rate of secretion of the matrix by the cells. Du D2, au flj, r* and N are positive parameters.

The last equality in (62) is die cell conservation equation, where the first term corresponds to die convention, the second to diffusion, the third to haptotaxis, and the fourth to die example of mitosis.

The additional galvanotactic and chemotaetic contributions to the flux can be written respectively as

ff = <*ECM + Gcelb <*ECM = ^viscous + ^elastic»

where 9 is the electrical potential and c is the concentration of a cheraotactic chemical. The parameters g and % are positive.

J. Murray [143] contains detailed analysis of the complicated systems (62) and (63) with applications to the various processes of pattern formation and to certain medical problems.

4. Possibility for Description of Biological Events on Molecular Level

If external conditions are constant, then after a certain time, called the time of relaxation, there is thermodynamic equilibrium of the system with the external surrounding.

The state of such system in quantum statistical physics is determined by the statistical density operator (A. Davydov [38])

where k is Boltunann constant, k = 1.41Q"23 joules per degree; T is the absolute temperature; H is the Hamiltonian operator of energy; NA is the operator of a number of particles; jx is the chemical potential; P is the pressure; V is the volume of the system; G is Gibbous thermodynamic potential or Gibbous free energy.

From the condition of normalization Spp = 1, one can find implicit expression for Gibbous free energy where the symbol Sp[A] means the speak of the operator A, i.e., the sum of its diagonal matrix elements by all quantum states of die system. The magnitude is called the sum of states of the system. The mean values of energy E, number of particles N9 and volume V are determined by the following formulae

E = Sp[p#] = - 92[3/36(G/B)], N = Sp[pNA] = - dGfd^ V = Sp[pV] = dG/dP. (67)

Thus, to determine all the mean values it is sufficiently to know the sum of states Z. Entropy of the system is determined as p = exp [(G-H + \\NA-PF)/Q], 6 = kT,

On to strength of (64), (65), (67), and (68), we have

For the quasi-stationary processes (the processes running their course sufficiently slow so that the system considered is always very closed to the state of equilibrium), from (69) it follows that the variation of Gibbous free energy under constant F, and P

where AE is the variation of internal energy of the system, TAS = Q is a quantity of warmth incoming from the external surrounding, \iAN is the work connecting to variation of number of particles in the system, and PAV is the work under increasing volume of the system.

Besides, since all extensive variables (the variables depending on size of the system) are usually relative to one mole, AG, AE, AS, and AV characterize the respective variations of one mole of the system, and AN characterizes the variation of number of moles in the system.

If such variations take place in 25° C and in normal atmospheric pressure (P = 1), then they are standard and are designated by the same symbols, but with the subscript or superscript of 0.

In biological systems, ions eZ (Z = ± 1, ± 2, ..,) transpose frequently unlike neutral particles.

Under the transfer of electric charge Aq from the system with the electric potential Uiat to the external surrounding with the potential Uexi the system results in the work

Since the charge of one mole of ions is equal to eZN0 (NQ = 6x1023), transfer of AN moles requires

where F is Faraday constant.

Therefore, (71) can be rewritten in the form

Combining this work with the work - pAtf for transfer of neutral particles, we have the general work for transfer of ions

The magnitude |xA is called electrochemical potential. Thus, |iA characterizes variation of free Gibbous energy per one mole of additional charged substance. Therefore, the complete variation of free energy of charged particles

Let us consider the system of ions in the solution under constant pressure and temperature, divided by semi-permeable partition on two subsystems: 1 and 2. According to (75), under the variation of number of ions in each subsystem, the variation of free Gibbous energy is determined by the expression where \iAu and |xA2(- are electrochemical potentials of ions of types i in the subsystems 1 The value of free Gibbous energy under equilibrium is minimal. Therefore,

If there are no chemical reactions in die system, then the numbers of ions of each type are the same, mid the equality (77) takes place under

Thus, the condition of equilibrium of two subsystems is the equality of the chemical potentials of ions of each type in both subsystems.

Free Gibbous energy G, internal energy E, and entropy S are the functions of steady states. Usually, their variations do not depend on the way by which the process is running from one steady state to another.

Biological systems not in equilibrium arise under their effect by various external perturbations. Unsteady states later converge to steady ones.

In addition, the total entropy of die system and the external surrounding usually increases, attaining maximal value under equilibrium. At die same time, free Gibbous

Thus, thermodynamics and statistical physics, determining the directions of process passing, do not give us the rate of passing or the respective molecular mechanism. The latter can be achieved on the basis of the dynamic theory.

However, due to a very large number of degrees of freedom, dynamical mathematical description in biology seems to be impossible. Fortunately, this pessimism

Many important biological processes do not have a large number of selected, usually collective, degrees of freedom. These processes have a long time of relaxation because of weak connections with the other degrees of freedom. Finally, they have a very short

The variation of the collective degrees of freedom is related essentially to unsteady processes that can be described by the dynamical laws of quantum or classical physics.

One of the problems that molecular biophysics faces is determination of the main properties of such strong- and weak-relaxations and their respective mechanisms.

4.1. MM OF TRAFFIC THROUGH CELL MEMBRANES 4.1.1. Passive transport

Let c(x,t) be molar concentration of the same kind of neutral molecules on the two sides (1 and 2) of homogeneous membrane situated along the y-axis. The rate of change of the amount of material in that region is equal to the rate of flow J across the boundary plus any that is created within the boundary. If no material is created, then dc/dt = -dJ/3x. The gradient dc/dx causes diffusion (passive transport).

Molar density of flow J of molecules (i.e., the molar number molecules passing through the unit of membrane length per 1 sec) is determined by Fickian law, J = -DdcBhe, where D is the material diffusivity (cm/sec). Therefore, we have dc/dt = d(Ddc/dx)/dx.

If D is constant and c(x,Q) = ß5(x), where 5(x) is the Dime delta function, then the solution of (79) is (see [143])

Note that sometimes the membrane is characterized by the factor of permeability P = DA, where /, instead of D, is die length of the membrane.

In die case of ions of charge eZh where e is the charge of a proton and Zk = ±1, ±2, .,., and where electric field inside the membrane has the potential U, the motion of ions through the membrane is expressed by the density of partial current

where F = eNA = 96,500 Kcal/mole (Faraday constant) and v* is mobility of ions through the membrane.

Based on the relation by Einstein Dk - vkRT, where T is the absolute temperature measured in Kelvin and where R is the universal gaseous constant, (81) can be rewritten in the form

The expression in the brackets of (82) is called electrochemical gradient.

According to (82), the mobility of ions of ¿-type does not depend on other ions (principle of independence of flows). This is usually true for small concentrations of ions. However, this principle is not true at least for potassium (P) ions in the case of the cell membrane [38].

Let n be the total number of ions of different type. Then the equations (82) for k= 1, 2, n, are the system of n equations relative n+1 functions c^x) and U(x). Adding to this system die condition of electric neutrality inside the membrane

and introducing the boundary conditions, i.e., die values ck(0)9 U(0), c*(l)f and £/(!), we obtain the determined system for the functions c*(x) and U(x).

Along with passive transport, Hving cells also have active transport, in which molecules and ions are pumped from low concentrations to high concentrations against their

To transfer one mole of substance from the side of the membrane with low concentration c\ to the side with high concentration c2 it is necessary to use the work that is equal to the change of Gibbous free energy

AC = RT In (c-Jci) - 0.863 Ig (cjcx) [kcal/mole]. (84)

This work has to be accompanied by another process creating and releasing the

To transfer ions, it is necessary to have the respective difference of electrical

AU = - RT/F In [<£ kP^+m + £ kP& £ ¿P*^ +1 kP*c **)], (85)

where c\ and c ¡g are the molar concentrations of positive and negative ions inside (j is in) and outside of the membrane (j is out). The values of AU for living cells in their rest state can be situated in the range from 20 to 200 mV,

For the nerve axon of the giant squid, the values of the molar concentrations of three base ions P+, Na+, and CI" are equal respectively to 340, 49, and 114 (inside) and 10.4, 463, and 592 (outside). In addition, in the rest state of the nerve axon, permeability factors can be found as the relation PP: PNa: Pcl = \ : 0.04 : 0.45. Using (85), one can

AU = -RT/F In [(340 + 0.45592)/(10.4 + 0.45-114)]« - 59.7 mV, (86)

which is in good agreement with the experimental value of - 60 mV.

Transport systems of membranes creating gradients of concentrations of transferring

There are Na^K^ATPase pumps, Ca2+ pumps, proton pumps, etc. Na^K+ATPases provide a small ratio concentration of Na+ to K+ inside cells in spite of the fact that such a ratio outside cells is usually large. Na+/K+ATPases and Ca2+ pumps work at the expense of consumption of energy. ATP molecules are hydrolyzed to ADP and to the inorganic phosphate (P*). These pumps can work backwards, using gradients of ion concentrations for the synthesis of ATP from the molecules ADP and P,.

It should be noted that we considered here only the simplest well-known MM of traffic substances through membranes. Significantly complete and complicated MM of transport of matter through biological membranes, including transport by vesicle formation, are given in [1; E. Fromter [50]; 134; H. Othmer, et al. [146]].


The problem of structure (C. Delisi [39]) is to find the minimum free energy structure. An accurate prediction depends on an accurate assignment of free energies to various types of structures (Fig. 4). Q in Fig. 4 is the progress variable, representing the fraction of native contacts present at a given stage of folding and E is the average free energy.

Beginning of helix formation

Beginning of helix formation

Native structure

Figure 4. A three-dimensional representation ofafoldmg process

The problem was decomposed by noting that the free energy of any structure can be, to a good first approximation, thought of as the sum of free energies of more basic structures, which include helical runs of AU and GC, internal loops, bulges, and hairpin

Suppose a particular structure has a total of nGC and nAV base pairs, «,■ bulges of length i, fij internal loops of length js and nk hairpins of length k (where i, j and k run from 1 through some finite limit, which depends on the chain's sequence and length). Let Gau, Ggc, Gi(Z), and GHp(0 (/ = 1,2,...) represent the various base pair and loop free energies. Then GT, the total free energy of the secondary structure, is

Gt = nocGcc + Rau^au + S. »¡GW*) +S; nfitffl + £k nkGm(k). (87)

The free energies for the various types of base pairing as well as for very short loops of various types were obtained experimentally. The free energy for loop closure is proportional to the natural algorithm of the equilibrium constant for bond formation, and where for a randomly coiling chain {S(R)A(Bsfy,y)]N is the complete (spatial and angular) probability density function for the location of the second bond separated from the first by N units of the chain and where R* and (9*,cp*,\|/*) are spatial coordinate and angular orientation of the second bond within Sr and <5k5v respectively of the first, Kq is the geometric requirements on the bases. If the probability density is calculated as a function of the length of the intervening sequence, the angular function quickly becomes uniform and the distance distribution becomes nearly Gaussian:

Among the grand challenges is the problem of protein folding [39]. Computational approaches start from a fundamental assumption that a system at equilibrium will be at its minimum free energy. Two main obstacles lie in the well-known way of progress. The first requires a useful potential function; the second requires rapidly executable algorithms that can find a structure close to die global free energy minimum. One of possible ways to find such a structure discussed in s. 3.2, ch, 16. For certain additional results in this direction see also L. Adam et al. [4]; [129]; T. Schlick [165].


where k is reaction rate, Ea is activation energy, T is absolute temperature, kB is the Boltzmann constant, and A is a constant subject to further identification. Conventional transition-state theory gives k = ksT/hexp (-AG°/kB7),

where AG° is Gibbs free energy and h is the Plank constant. Comparing (90) and (91) and using AG0 = MP - RT, we find tot the pre-exponential factor A in (90) corresponds to kBTOexp [(AS0 + R)/R] and Ea = AH0 - RT, The Brownian motion driven by thermal forces of the particle is described by the Langevin equation méxldi1 + ydx/dt + dV/dx = Fexl(/),

where m is the particle mass, x is a coordinate, V is the potential energy surface of the particle, y is friction (viscosity) coefficient, and F^ is the external thermal excitation from the bath (solvent). It was shown in Kramer's theory, by solving the diffusion equation that the rate of escape in the high friction limit is k = lfyexp (-AG°/kBI),

4.3.2. Application to the protein folding without barriers

The escape rates ku and k^ for two-state folding are given (O. Bieri, T. Kiefhaber [25] by ku = v„exp (-AG°/kB7), vrexp (-AG°/kB7)»

where Cu and Cf are constants depending on the potential energy surface, rjsolvent is the solvent viscosity, and <j is internal viscosity arising from internal friction.

Since the equilibrium of a reaction is always determined by the escape rates for folding and unfolding, the free energy for folding

Folding starts from a solvated polypeptide chain in water with little internal friction compared to solvent friction. We can neglect the internal friction for the refolding reaction. For the unfolding reaction, starting from the native protein, the internal viscosity will be significant due to the restricted motion of the polypeptide chain.

This and similar considerations show that in the case of myoglobin the difference in the pre-exponential factors for unfolding and refolding will contribute maximally lkcal/mol to the equilibrium stability of the native protein.

In general, vu and V/depend on the solvent conditions such as denaturant concentration, temperature, and pressure.


DNA, organized into two pairs (double helix, see Fig. 5) of 24 distinct types of chromosomes, is a nucleic acid polymer, made up of small molecules called nucleotides that can be distinguished by the four bases: adenine (A), cytosine (C), guanine (G), and thymine (T).

The nucleotides A and T as well as C and G form hydrogen bonds and so A, T and C, G are called the complementary base pairs.

Linear (for small motions) and nonlinear (for arbitrary motions) physics of DNA are described in a work by L. Yakushevich [211]. It turns out that one of the typical solutions of the nonlinear models is the so-called solitom, which require the least energy for their motions.

The rod-like ideal model takes into account three types of internal motions: stretching, twisting and bending.

So, the Hamiltonian of the system where the first three terms describe stretching, twisting, and bending, respectively, ami the rest of the terms describe interactions among these three types of motions.

The expected general form of the non-linear equations corresponding to Hamiltonian H is

Figure 5, Sketch of a double helix

Figure 5, Sketch of a double helix

sputt = Yuu + nonlinear term + + £/g_b, itptf = Ccp^ + nonlinear term + + <£t_b,

where (p and u are angular and longitudinal displacements of the nth disk; F is Young's modulus; C is the torsion rigidity of the rod; S is the area of the rod section; p is a density; / is the moment of inertia of the disk; and U+l9 £/s_b> ^t-b* *b-t are the terms describing interactions between interna! motions in DNA.

The problem of derivation of all the terms in (97) has not yet been solved. A work by L. Yakushevich [211] contains a survey of many existing models and of their original results. The ideal models did not take into account such factors as the environment, non-homogeneity, and the helicity. Effects of all those factors are considered in the book. In particular, when accounting for the helix, it implies the consideration of the well-

having soliton-like solutions; kinks and anti-kinks:

where v is the soliton's velocity and Zq is a constant.

Appendix of this book contains the quantitative values of structural and dynamical parameters of DNA such as the radius of DNA helix, die distance between base pairs, the mass of a base pair, the moment of inertia of a base pair, die velocity of sounds in DNA, the velocity of torsion waves, the force constants for torsion, the rolls, the tilts, the rises, the bends, and the hydrogen bond stretching motions.

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