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Soliton Kinetic Equations with Non-Kolmogorovian Structure : A New Tool for Biological Modeling? Aerts, Diederik; Czachor, Marek; Gabora, Liane; Polk, Philippe 2006

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Soliton Kinetic Equations with Non-KolmogorovianStructure: A New Tool for Biological Modeling?Diederik Aerts∗, Marek Czachor†,∗, Liane Gabora∗∗ and Philippe Polk‡∗Centrum Leo Apostel (CLEA) and Foundations of the Exact Sciences (FUND)Vrije Universiteit Brussel, 1050 Brussels, Belgium†Katedra Fizyki Teoretycznej i Metod Matematycznych, Politechnika Gdan´ska, 80-952 Gdan´sk, Poland∗∗Department of Psychology, University of British Columbia, Canada‡Department of Biology, Vrije Universiteit Brussel, 1050 Brussels, BelgiumAbstract. Non-commutative diagrams, where X → Y → Z is allowed and X → Z → Y is not, may equally well apply toMalusian experiments with photons traversing polarizers, and to sequences of elementary chemical reactions. This is whynon-commutative probabilistic, logical, and dynamical structures necessarily occur in chemical or biological dynamics. Wediscuss several explicit examples of such systems and propose an exactly solvable nonlinear toy model of a “brain–heart”system. The model involves non-Kolmogorovian probability calculus and soliton kinetic equations integrable by Darbouxtransformations.Keywords: solitons, Darboux transformations, kinetic equations, non-Kolmogorovian probability, von Neumann equationPACS: 82.39.-k, 03.65.Ca, 05.45.YvINTRODUCTIONThis paper can stand on its own as showing how the dynamics of chemical reactions entail a quantum structure evenif their microscopic (and obvious) quantum nature is neglected. However, it is most fully appreciated when viewed aspart of a large and systematic effort to identify the structure of — quantum, classical or quantum-like — entities andprocesses in different domains of science. The guiding insight behind our approach is that whether a particular typeof entity or process is quantum, classical or quantum-like depends on the nature (quantum, classical or quantum-like)of the structure of the model that describes well the entity or process1, and not on the scientific domain to whichthis entity or process pertains. As such it is for example possible to encounter quantum structure in the macroscopicdomain and classical structure in the micro-domain2. In this paper we shall show that simple catalytic schemes leadto probabilities having a non-Kolmogorovian structure3. In chemistry, when one switches to a dynamical level, oneusually arrives at nonlinear kinetic equations. If, however, the governing probabilities are non-Kolmogorovian, aswe will show in this paper, one could look for links to nonlinear evolutions of non-Kolmogorovian states or, say,to density matrices if the model is Hilbertian4. And indeed, it turns out that nonlinear von Neumann equations may1 More specifically, we call an entity a “quantum entity" if its behavior is well described by the standard formalism of quantum mecanics, i.e.the formalism as put forward by John von Neumann in [1]. An entity is a classical entity if its behavior is well described by one of the classicalmechanics theories. Remark that the theories used in modelling within complexity and chaos approaches are classical mechanical theories. An entityentailing quantum mechanical aspects which cannot be completely modeled within standard quantum mechanics, we call “quantum-like".2 In [2, 3] a macroscopic entity consisting of a configuration of vessels of water is proposed where Bell inequalities are violated. It has beenshown that the violation of Bell inequalities proves the presence of quantum structure, more specifically the presence of a non classical probabilitystructure [4]. Guided by the same idea classical models for quantum spin were elaborated [5, 6], and the structure encountered as such was used tounderstand better the structural difference between quantum and classical [7]. In section 2 of this paper we consider a chemical reactor and provethat its macroscopic dynamics entails quantum structure. When a classical structure appears in the micro-world one terms it a superselection rule,and is modeled within the standard quantum mechanics formalism in an ad hoc way, more specifically by leaving out the superpositions that wouldintroduce nonclassical structure [8].3 One way of showing the presence of quantum structure is by looking at the structure of the probability model that governs the change of the entity.Classical entities and processes entail a probability model that satisfies the axioms of Kolmogorov, and hence is called Kolmogorovian, while theprobability model of quantum entities and processes does not satisfy all the axioms of Kolmogorov, and hence is called non-Kolmogorovian.4 If a certain structure is non-classical, for example a probability model which is non-Kolmogorovian, a logical first attempt is to try to fit this non-classical structure into the structure of standard quantum mechanics [1]. Mathematically this comes to looking for a dynamics such as Schrödingerequation on a set of states (of the entity under consideration), where each state is represented by a vector in a complex Hilbert space. A slightly moreSoliton Kinetic Equations with Non-Kolmogorovian Structure: A New Tool for Biological Modeling?October 13, 20051be regarded as a “Lax representation” of a classs of integrable nonlinear kinetic equations[16]. This is one of thereasons why one may work with von Neumann-type equations even though the system in question is not “quantum"in the microscopic sense. At the level of Lax representations one can introduce various soliton techniques and findexact solutions. The equations we work with are new in the sense of being simultanously kinetic and solitonic. Theresult we obtain in this paper fortifies the scheme we worked out in [17], where we introduce a general frameworkfor the description of change-of-state in biological as well as non-biological entities. The framework is referred to asCAP, for context-driven actualization of potential. Processes of change differ with respect to the degree to which theyare sensitive to, internalize, and depend upon a particular context, and whether the change of state is deterministicor nondeterministic. The CAP framework has been fruitful for illustrating in broad terms how unusual Darwinianevolution is, and clarifying in what sense change-of-state of living organisms (and also cultural entities) is (and isnot) Darwinian. Part of the original goal of the overarching effort was to find and zero in on exactly those areas ofbiology which have not been successfully analyzed in Darwinian terms and see if formalisms developed for describingchange-of-state in other disciplines are instead applicable here. It turns out that, irrespective of the discipline, whenchange-of-state of an entity depends on how that entity interacts with a context, the resulting probabilities can benon-Kolmogorovian, and the appropriate formalisms for describing this process are either the quantum formalismsor mathematical generalizations of them. This means that our critique within CAP on the Neo-Darwinian Synthesis,where the process of evolution is narrowed down to the interplay of variation and selection, is structural. A processof interplay of variation and selection as imagined within the Neo-Darwinian Synthesis will lead to an underlyingprobability model which is Kolmogorovian, and in this sense cannot take into account the influence of context asconceived within CAP. In practice it means that the narrowed down evolution process, as within the Neo-DarwinianSynthesis, where only variation and selection play, neglects what happens in the realm of the potential states of theconsidered entities. It only considers actualized entities, and their actualized interactions, and believes that all ofevolution will be steered by these actualized entities, and their actualized interactions. Within CAP also potential statesof entities, and potential interactions between these entities will play a role in the process of evolution. In this respectCAP is a generalization of the Neo-Darwinian process, and in this respect furthers the work of scientists who saw theimportance of self-organization in biology. Kauffman [18] showed using random Boolean networks, self-organizedautocatalytic sets, and related models, that much or perhaps even most change in biological organisms cannot beexplained with recourse to Darwin. Others followed suit, and it has meanwhile become evident that neo-Darwinism,powerful though it is, can only to a limited degree account for biological change (see also [19, 20]). The concept ofnatural selection offers little in the way of explanation for biological forms and phenotypes arise in the first place.(One can ask, for example, if natural selection is such a powerful tool for describing biological change, what can ittell me about the fitness of the offspring I would have with one healthy mate as opposed to another.) Moreover, non-Darwinian processes — such as autopoiesis [21], emergence [22, 23, 24], symbiosis [25], punctuated equilibrium [26]and epigenetic mechanisms [27] — play a vital role. Moreover, the generation of variation is not completely random;convergent pressures are already at work prior to the physical realization of organisms. First, mating is often assortative— mates are chosen on the basis of traits they possess or lack, rather than at random — and relatives are avoided asmates. Second, since Cairns’ initial report [28], there is increasing evidence of directed mutation, where the frequencyof beneficial mutations is much higher than chance, particularly in environments to which an organism is not welladapted. Furthermore, the concept of fitness, a cornerstone of the neo-Darwinian enterprise, is problematic [29]. Insum, there is more going on in evolution than random variation and natural selection. The CAP framework concentrateson how quantum-like structure enters the realm of biological change. In the present paper we show that this happensalready at the level of the dynamics of simple chemical reactions. As one of the applications we describe a situationwhere we have a composite system whose subsystem is driven by a “pacemaker”. Although the driving oscillation isharmonic, the subsystem exhibits characteristic bursting behavior typical of biological oscillators, occcuring due to anonlinear pacemaker-subsystem coupling. We stress the analogy between the subsystems in question and subsystems(“organs”) of a biological dynamical system (to focus attention we refer to the subsystems as the“brain” and the“heart” — a sinusoidal rhythm of the “brain” is translated into bursts of the “heart”, and the carrier of information isnonlinear kinetics of a chemical type). Another application is a simple two-qubit system whose evolution possessesthree phases separated by “birth” and “death” (before “birth” and after “death” the system consists of two subsystemsgeneral mathematical representation of standard quantum mechanics is the one where the states of the entity are represented by density matricesof a complex Hilbert space, and the dynamics is described by von Neumann equations. This is the representation of standard quantum mechanicsthat we consider in this paper. Since we need to model non-linear behavior, we however have to consider the non-linear von Neumann equations asstudied in [14, 15]Soliton Kinetic Equations with Non-Kolmogorovian Structure: A New Tool for Biological Modeling?October 13, 20052that are effectively noninteracting and evolve independently, and their entropies do not change; the intermediate phaseinvolves exchanges of energy and modifications of entropies of the subsystems).We begin with a discussion of various non-Kolmogorovian aspects of chemical kinetics. Next, we briefly discuss thelinks between kinetic equations and their Lax-von Neumann forms. The systems we begin with are those exhibitingrhythmic properties. We introduce their formal description in terms of composite systems and derive an effectiveequation describing the “heart”. This equation can be solved by soliton techniques. We derive the solution and plot thebursts of an oscillating quantity that has a physical meaning of a center of mass of the “heart” (average position of theoscillator). The next section contains a brief discussion of the two-qubit organism, along the lines introduced in[14, 15].Finally, we end the discussion with explicit examples of kinetic equations associated with von Neumann dymanics.The simplicity and efficiency of the von Neumann formalism as compared with the kinetic equations becomes thenespecially clear.NON-KOLMOGOROVIAN ASPECTS OF CHEMICAL KINETICSConsider the following catalytic network [16](a) B k→ A+B, (b) A+X k′→ X +(products), (c) A+B k→ A+(products), (d) Y k′′→ B. (1)Assume that the reaction (b) is of 0-th order in A. This may occur if, for example, [A]> [X ] for all t, and X catalyzesa decay of A 5. Let the reaction (c) be of 0-th order in B (say, if [B] > [A] for all t) and A catalyzes a decay of B. Thekinetic equations read[ ˙A] = k[B]− k′[X ], [ ˙B] = k′′[Y ]− k[A], [ ˙X ] = [ ˙Y ] = 0. (2)If we define two new (in general non-positive) variables x = [A]− (k′′/k)[Y ], y = [B]− (k′/k)[X ], we can rewrite thesystem as a harmonic oscillator i ˙ξ = kξ where ξ = x+ iy. In order to find its “Lax representation”, let us introducethe two matricesH =(k 00 0), ρ =(a ξ¯ξ b), (3)a≥ 0, b≥ 0, a+b> 0. The equations i ˙ξ = kξ , a˙ = ˙b = 0 are equivalent to the von Neumann equation iρ˙ = [H,ρ]. Theequation has the “Lax form” since the right-hand side is given by a commutator. If ρ > 0 then for any projector A wehave [A] = TrρA ≥ 0. Let now A and B project, respectively, on(11)and(1−i). Then x = [A]− (a+b)/2 andy = [B]− (a+b)/2, i.e. (k′/k)[X ] = (k′′/k)[Y ] = (a+b)/2. We thus conclude that the kinetic scheme (a)-(d) impliesthe dynamics that, for certain initial conditions and appropriately adjusted kinetic constants, is von Neumannian. Thekinetic variables satisfy [A(t)] = Trρ(t)A and [B(t)] = Trρ(t)B and the projectors A and B do not commute.There is also a simpler argument why chemical kinetics is non-Kolmogorovian. There are numerous examples ofreactions X →Y that cannot happen directly, but involve an intermediate product Z. The decomposition into elementaryreactions is thus X → Z → Y , and not X → Y . Formally, the situation is analogous to the Malus law with three or twopolarizers.Finally, consider the chemical reactor shown in Fig. 1 [16, 2]. We fill the bottle with a liquid and performmeasurements of its property by means of a beaker. With each of the spigots we associate a random variable, A,A′, B, or B′, which yields +1 if a given property (e.g. color) is found, and −1 otherwise.The reactor consists of three chambers, the lowest two are connected by means of a fan whose rotation leads tofluid mixing in the two bottom chambers. Measurements of A or B do not change states of the fluid in the lowestchambers. However, a measurement of either A′ or B′ mixes the fluids in the bottom chambers since the fan will rotatedue to the flow of the fluid. We assume that the conditions of the experiment allow for collecting some amount of fluidthrough spigots A′ or B′ without noticing the effect of the mixing in the earlier of the two measurements, but the secondmesurement will reveal that the fluids got mixed. Fig. 1 symbolically illustrates the structure of the measurement.5 In chemical notation [A] denotes a nonnegative dynamical variable associated with A, usually referred to as a “concentration”.Soliton Kinetic Equations with Non-Kolmogorovian Structure: A New Tool for Biological Modeling?October 13, 20053A BA' B'(a) (b)(c) (d)FIGURE 1. (a) The reactor before measurements. The symbols of the four spigots correspond to the four random variables. (b)Reactor after measurement of A (A = +1). (c) Reactor after measurement of A′ — flow of the liquid produced mixing of the twolowest layers, but the beaker contains the fluid characteristic of the upper layer (A′ =+1). (d) Reactor after measurements of first A′and then B′ — the left beaker contains the fluid from the upper layer (A′ =+1), and the right one contains the mixture (B′ =−1).Soliton Kinetic Equations with Non-Kolmogorovian Structure: A New Tool for Biological Modeling?October 13, 20054Having one beaker we have to decide on the order of measurements. Clearly, the measurements A and B will alwaysproduce +1 (the fluid is red, say), independently of the order in which the four measurements are performed. The sameresult will be found during the first measurement of either A′ or B′. However, once A′ has been measured, the stateof the fluid is changed and a reaction changes the property in question (say, the fluid in the bottom chambers is nowgreen). Therefore, the result of a subsequent measurement of B′ will produce −1. An analogous situation is found ifone first measures B′.The results of possible runs of the experiment can be collected in two quadruples that depend on the order ofmeasurements of A′ and B′: (A,B,A′,B′) = (+1,+1,+1,−1), (A,B,B′,A′) = (+1,+1,+1,−1). These quadruples canbe used to define the random variable C = AB+AB′+A′B−A′B′ = 1+B′+A′(1−B′) = 2. The average of C equals〈C〉= 2. This is consistent with the Bell inequality|〈AB〉+ 〈A′B〉+ 〈AB′〉−〈A′B′〉| ≤ 2. (4)But what happens if in a single measurement we measure only one of the four product random variables occuringin (4)? Mesurements of AB, AB′, and A′B (performed independently) always give +1, but the measurement of A′B′produces −1. The averages therefore read, 〈AB〉= 〈AB′〉= 〈A′B〉=+1, 〈A′B′〉=−1, and〈AB〉+ 〈A′B〉+ 〈AB′〉−〈A′B′〉= 4. (5)If all the four random variables are measured on the same reactor than either AB′ or A′B equals −1 because either A′or B′ equals −1, and we always find A = B = +1. If we perform experiments independently than the minus sign canappear only if A′ and B′ are measured on the same reactor. All these properties imply that the probability model of thereactor is non-Kolmogorovian. It is interesting that the model is also non-Hilbertian [16].SOLITON KINETIC EQUATIONSThe linear von Neumann equation occured in the previous section as a “Lax form” of a system of kinetic equationsassociated with certain catalytic reactions. Linearity was implied by the fact that reactions were of zero order. However,the generic case one finds in biochemical systems does not involve zero-order reactions and, hence, is nonlinear. Asone of the simple generalizations of the von Neumann equation one may consider its nonlinear versioniρ˙ = [H, f (ρ)], (6)introduced in [30] in the context of nonlinear quantum mechanics. There are reasons to assume that f (ρ) is an operatorfunction satisfying the “no-feedback, no-nonlinearity” property, formally meaning that f (ρ) = ρ if ρ2 = ρ [15, 31].The latter condition is fulfilled, up to rescaling of time, by any polynomial; also any function satisfying f (0) = 0,f (1) = 1 will do the job since by spectral theorem it will map a projecor into itself.It is important that real and imaginary parts of all matrix elements of ρ can be directly associated with probabilities.The construction is simple. Consider an operator H that possesses a discrete part of spectrum, and let H| j〉 = E j| j〉,〈 j|k〉= δ jk, and ρ jk = 〈 j|ρ|k〉= x jk+ iy jk. Now, let | jk〉= (| j〉+ |k〉)/√2, | jk′〉= (| j〉− i|k〉)/√2, Pjk = | jk〉〈 jk|, P′jk =| jk′〉〈 jk′|, Pj = | j〉〈 j|. Then x jk = p jk − 12 p j − 12 pk, y jk = p′jk − 12 p j − 12 pk, where pk = TrPkρ = ρkk, p jk = TrPjkρ ,p′jk = TrP′jkρ . The latter formulas imply that 0 ≤ pk, p jk(t), p′jk(t) ≤ 1, for any t, and thus are some probabilities.The probabilities pk are time independent. If f (ρ) is a polynomial then (6) is equivalent to a set of nonlinear kineticequations with polynomial nonlinearities. The system is conservative if t is real. One can add dissipation by replacingt with a complex parameter. In particular, an imaginary time turns our equation into a kind of heat equation.Still, we have more than just a set of nonlinear kinetic equations. The probabilities are rooted in a nonclassical(Hilbert-space) model of probability so that we have much more control over the structure of the probabilitiesin question. In particular, we can derive certain uncertainty relations between different probabilities. This is notsurprising if one thinks of the chemical reactor violating the Bell inequality where manipulations with the spigots ledto noncommuting random variables. For any three operators satisfying [A,B] = iC one can prove ∆A∆B≥ 12 |〈C〉|where∆A =√〈A2〉−〈A〉2, 〈A〉 = Tr(Aρ), etc. If A = P = P2 is a one-dimensional projector then 〈P〉 = p is a probabilityand one finds ∆P =√p(1− p). We say that two propositions P1 and P2 are complementary if ∆P1∆P2 ≥ ε > 0.In order to show that the propositions, say Pj and Pjk, are complementary, we compute[Pj,Pjk] = 12(| j〉〈k|− |k〉〈 j|)= iC (7)Soliton Kinetic Equations with Non-Kolmogorovian Structure: A New Tool for Biological Modeling?October 13, 20055and 〈C〉= yk j. Finally, √p j(1− p j)√p jk(1− p jk)≥ 12 |yk j|. (8)The variable yk j measures complementarity of Pj and Pjk. The complementarity varies in time, analogously to themorphogenesis of complementarity discussed in [15]. The fact that complementarity should be time dependent wasalso clear from the reactor example.Another important property of the dynamics is that (6) is integrable in the sense of soliton theory for any f . Asoliton technique of solving (6), based on Darboux transformations, was introduced in [14], and further developed in[32, 33, 34]. Therefore, as opposed to standard kinetic equations that typically have to be solved numerically, we canwork with exact analytic solutions.BIOLOGICAL OSCILLATORS, COUPLED SYSTEMS, AND RHYTHMICPHENOMENAIf one looks for the most striking manifestations of biochemical nonlinear evolutions one arrives at phenomena thatare rhythmic. Periodic switching, with periods ranging from fraction of a second to years, is encountered at all levelsof biological organization, and its successful modelling in terms of nonlinear kinetic equations is one of the greatachievements of computational biology (for reviews cf. [35, 36]). Of particular interest are theoretical studies ofcirca-rhythms. Let us recall that circa-rhythms are “classes of rhythms that are capable of free-running in constantconditions with a period approximating that of the environmental cycle to which they are normally synchronized”[37]. The examples of the rhythms are the circadian (24 hours), the circalunar (28 days), or the circannual (365.25days) ones. What makes circa-rhythms interesting from a formal point of view is the interplay between the phaseswhere there exists an external forcing (light–darkness periods, say), and the rhythmicity that sustains even after theexternal driving is switched off (as in experiments in constant darkness). Formally, the two cases may correspond tononlinearities involving (or not) an explicit time dependence of some parameters.The models are typically finite-dimensional (as dynamical systems) and involve nonlinearities of a step-functiontype. The latter are convenient from the point of view of “engineering” of a nonlinear behavior. Having someexperience with nonlinear equations one can force a system to behave in a given way, at least within a given rangeof parameters. To give an example, the model of circadian rhythms in Neurospora [38] involves four variables andtwo types of Hill functions. The rhythmicity occurs in the model even without an external driving, but the case withexplicit time dependence of coefficients is treated as well. Similar constructions are given in [39, 40].As one of the first applications of (6) in a biological context we thus address the problem of circa-rhythmicity.The main idea is to consider a composite Hamiltonian system that, as a whole, is conservative but not isolated fromthe external world (let us term the system an organism; our organism will not “live” if we isolate it from the externalworld!). We take simple and generic polynomial nonlinearities, and do not try to force switching by means of Hill-typefunctions. The lack of isolation of the organism from the environment means that it has nontrivial correlations with theoutside world, a fact that allows for starting with an initial condition ρ(0) 6= ρ(0)2. Only in such a case the nonlinearityinherent to the organism is nontrivial. There are two sets of degrees of freedom corresponding to two subsystems ofthe organism (let us term them the “brain” and the “heart”). When we look at certain averages associated with one ofthe subsystems (the heart) we find that their time evolution can be determined by an effective density matrix whosedynamics is given again by an equation of the form (6), but now with explicitly time-dependent coefficients in thenonlinear function f (ρ). Therefore, the interaction between the brain and the heart effectively turns the brain into apacemaker that drives the heart and generates its evolution with characteristic bursting patterns.Although the “brain” and the “heart” may be treated as metaphores and the model given below is to a largeextent a toy one, we are inclined to defend a more straightforward interpretation of our construction. One has tokeep in mind that the reactor we have described in the previous section is a highly non-Kolmogorovian system dueto various interactions between its different parts. There is no reason to believe that a real biochemical system isless non-Kolmogorovian. The assumptions we make about the “brain” and the “heart” mean that the system is non-Kolmogorovian and Hilbertian. The latter is perhaps even too weak a postulate, but yet this is a toy model.Soliton Kinetic Equations with Non-Kolmogorovian Structure: A New Tool for Biological Modeling?October 13, 20056FORMAL DESCRIPTION: DYNAMICS IN A SUBSYSTEMSo let us consider a composite system consisting of two subsystems interacting with each other via a nonlinear couplingdefined by a function f (ρ) satisfying the “no-feedback, no nonlinearity” condition. As a simple example consideri ˙ϑ = [ω(b†1b1−b†2b2︸ ︷︷ ︸J)+(1+X)a†a,ϑ ]− [Xa†a, f (ϑ)] (9)where [bk,b†k ] = 1 = [a,a†] and X = b1 + b†1 + b2 + b†2. For ϑ2 = ϑ we find i ˙ϑ = [ωJ + a†a,ϑ ] describing threeindependent degrees of freedom: Harmonic oscillation with unit frequency (the heart) combined with rotation in aplane with frequency ω (the brain). For ϑ 2 6= ϑ rotation and oscillation get nonlinearly coupled. Let us now eliminatethe rotation by switching to a rotating reference frame ρ = eiωJtϑe−iωJt where the density matrix satisfiesiρ˙ = [(1+X(t))a†a,ρ]− [X(t)a†a, f (ρ)]. (10)Denoting Y =−i(b1 +b†2−b†1−b2) we findX(t) = eiωJtXe−iωJt = X cosωt +Y sinωt, Y (t) = eiωJtYe−iωJt = Y cosωt−X sinωtand, for any t and t ′, [X(t),X(t ′)] = [Y (t),Y (t ′)] = [X(t),Y (t ′)] = 0. Since X , Y , and a†a commute we can introducetheir joint eigenvectorsX |x,y,n〉= x|x,y,n〉, Y |x,y,n〉= y|x,y,n〉, X(t)|x,y,n〉= x(t)|x,y,n〉, Y (t)|x,y,n〉= y(t)|x,y,n〉, (11)a†a|x,y,n〉= n|x,y,n〉, x(t) = xcosωt + ysinωt, y(t) = ycosωt− xsinωt. (12)Now take any time-independent normalized vector |ψ〉 = ∫ dxdyψ(x,y)|x,y〉 and make the Ansatz ρ(t) = |ψ〉〈ψ| ⊗w(t) where w is a density matrix acting only on the oscillator degress of freedom. Denoting g(ρ) = ρ − f (ρ) =|ψ〉〈ψ|⊗g(w) we rewrite (10) as|ψ〉〈ψ|⊗ iw˙ = |ψ〉〈ψ|⊗ [a†a,w]+X(t)|ψ〉〈ψ|⊗a†ag(w)−|ψ〉〈ψ|X(t)⊗g(w)a†a (13)Taking matrix elements of both sides of (13) between arbitrary 〈x,y| and |x′,y′〉 we obtainψ(x,y)ψ(x′,y′)iw˙ = ψ(x,y)ψ(x′,y′)[a†a,w]+ψ(x,y)ψ(x′,y′)(x(t)a†ag(w)− x′(t)g(w)a†a). (14)The Ansatz is internally consistent only ifψ(x,y)=√δ (x− x0)δ (y− y0), and then the diagonal x= x′= x0, y= y′= y0leads toiw˙ = [a†a,w]− x0(t)[a†a,g(w)] = [a†a, ˜f (w)], x0(t) = x0 cosωt + y0 sinωt, (15)˜f (w) = w− x0(t)g(w) =(1− x0(t))w+ x0(t) f (w). (16)If f satisfies the “no feedback, no nonlinearity” condition f (ρ) = ρ for ρ2 = ρ , the same holds for the effective ˜f .Eq. (15) determines the effective dynamics of the subsystem, and belongs to the class of Darboux-integrable nonlinearvon Neumann equations. It can be explicitly integrated by means of the techniques introduced in [14, 32].QUANTUM PACEMAKER AND FEEDBACKThe pacemaker is an oscillatory system whose state can be modified by an external entraining agent, a zeitgeber [37].In our example the pacemaker corresponds to the part of the composite system described by the Hamiltonian ωJ, i.e.to the brain, and the zeitgeber can be associated with the world external to the organism. Our main interest in thisexample is in the free-running oscillation of the pacemaker and the free-running rhythms of the heart it induces.The next level is the coupling between the pacemaker, which defines the clock “mechanism” of the brain, andthe observable-level circa-rhythms which define the “hands” of the clock (center of mass of the heart). We assumea nonlinear feedback between the hands and the mechanism, but the dynamics is nondissipative in the sense that theSoliton Kinetic Equations with Non-Kolmogorovian Structure: A New Tool for Biological Modeling?October 13, 20057energy of the hands averaged over a single cycle of the oscillation is constant. Motivated by the analysis of the previoussection we concentrate on the following class of equationsiw˙(t) = f1(t)[H,w(t)]+ f2(t)[H, f(w(t))] (17)which define the state of the hands of the clock. The “no-feedback, no-nonlinearity” condition reads w(t)= f1(t)w(t)+f2(t) f(w(t))whenever w(t)2 = w(t). Now assume we know a solution w0(t) of (6). Thenw(t) = e−iH∫ t0 f1(x)dxw0(∫ t0 f2(x)dx)eiH∫ t0 f1(x)dx (18)is a solution of (17) as can be verified by a direct calculation. The whole problem of solving (17) reduces to finding asolution of (6), which can be performed by soliton techniques [14, 32].Let us consider a simple but generic example where f (w) = (1− s)w+ sw2 andf1(t) = 1+ ε cosωt, f2(t) =−ε cosωt (19)The parameter s allows us to compare situations where the driven dynamics is linear (s = 0) and purely nonlinear(s = 1), and for the two cases investigate the role of the εs. Varying s we can also investigate stability properties of therhythms under fluctuations of the feedback.THE HEART: HANDS OF THE CLOCKThe hands of the clock are described by the Hamiltonian H = a†a = ∑∞n=0 n|n〉〈n| of a harmonic oscillator type. Thefrequency of the oscillator is equal to unity, meaning that this is a reference frequency for all the other frequenciesfound in the system. Modelling the heart by an oscillator is quite natural for obvious reasons. The quantum oscillatorhas in addition the appealing property of being delocalized, or extended in some formal sense. Intuitively, in our model,the hands move at time t to the region of space of greatest concentration of probability density. The probability densitymay be regarded as a measure of state of the heart. In the absence of a feedback between the hands and the pacemaker,the hands oscillate harmonically with their own internal frequency (unless damping is added). We shall see later thatthe nonlinear coupling may practically suppress the internal oscillations of the hands at certain intervals of time. Whatwill remain are the sudden bursts (or “heart-beats”) occuring, roughly, with the period of the pacemaker.Quantum harmonic oscillator is an infinite dimensional dynamical system and, hence, a solution of von Neumannequations may be characterized by an arbitrary number of parameters determining the initial state of the heart. Thereexists a simple trick allowing to construct infinitely-dimensional solutions on the basis of a single finite-dimensionalone. The trick exploits equal spacing of eigenvalues of H. With our choice of units the eigenvalues are given simply bynatural numbers. Let us divide them into sets containing N elements: {0,1, . . . ,N−1}, {N,N +1, . . . ,N +N−1}, andso on. Each such a subset corresponds to a block in the Hamiltonian, and each block can be represented by a N×Ndiagonal matrix of the formHk = k1+diag(0,1, . . . ,N−1) = k1+H0. (20)As a consequence, in each block we have to solve the same matrix equation since a restriction wk of w to the k-thsubspace satisfiesiw˙k = [Hk, f (wk)] = [H0, f (wk)]. (21)The job can be reduced to finding a sufficiently general solution of a N×N problem. In each subspace we can take adifferent initial condition and a different normalization of trace. The whole infinite-dimensional solution will take theform of a direct sumw(t) =⊕∞k=0 pkwk(t, pk), (22)∑∞k=0 pk = 1. The k-th part depends on pk in a complicated way since the function f (w) is not 1-homogeneous,i.e. f (λw) 6= λ f (w). The inhomogeneity implies that change of normalization simultaneously rescales time; thenormalization of probability implies that a change of pk in a k-th subspace influences all the other subspaces bySoliton Kinetic Equations with Non-Kolmogorovian Structure: A New Tool for Biological Modeling?October 13, 20058making their dynamics faster or slower. In this sense the solution, in spite of its simplicity, is not a simple direct sumof independent evolutions.In order to illustrate the possible effects we can use the solutions derived in [15] for the simplest nontrivial caseinvolving self-switching, i.e. for N = 3 and quadratic nonlinearity (nonlinearity that did not explicitly depend on time).We select a subspace spanned by three subsequent vectors |k〉, |k+1〉, |k+2〉. The family of interest is parametrizedby α ∈ R controlling the “moment” and type of switching between bursts. The parameter naturally occurs at thelevel of the Darboux transformation, where it characterizes an initial condition for the solution of the Lax pair. Thedensity matrices wk(t) = ∑2m,n=0 wmn|k+m〉〈k+n| are completely characterized by the k-independent matrix of time-dependent coefficients wmn. The reader may check by a straightforward substitution 6 that the matrix w00 w01 w02w10 w11 w12w20 w21 w22= 115+√5 5 ξ (t) ζ (t)¯ξ (t) 5+√5 ξ (t)¯ζ (t) ¯ξ (t) 5 (23)withξ (t) =(2+3i−√5i)√3+√5α√3(eγt +α2e−γt) eiω0t , ζ (t) =−9e2γt +(1+4√5i)α23(e2γt +α2) e2iω0tis indeed a normalized (Trw = 1) solution of the von Neumann equation iw˙ = [H,(1− s)w+ sw2], H = diag(0,1,2).The parameters are ω0 = 1− 5+√515+√5 s, γ =215+√5 s. Now let us rescale the trace. We do it in three steps. The modifieddensity matrix w1(t) = ei(1−s)Htw(t)e−i(1−s)Ht is a solution of iw˙1(t) = [sH,w1(t)2]. Thereforew2(t) = Λei(1−s)ΛHtw(Λt)e−i(1−s)ΛHt (24)is also a solution of iw˙2(t) = [H,sw2(t)2] and w3(t) = e−i(1−s)Htw2(t)ei(1−s)Ht solvesiw˙3(t) = [H,(1− s)w3(t)+ sw3(t)2]. (25)Performing these operations on our explicit solution we findw3(t) =Λ15+√5 5 ξ3(t) ζ3(t)¯ξ3(t) 5+√5 ξ3(t)¯ζ3(t) ¯ξ3(t) 5 (26)withξ3(t) =(2+3i−√5i)√3+√5α√3(eγΛt +α2e−γΛt) ei(ω0Λ+(1−s)(1−Λ))t , ζ3(t) =−9e2γΛt +(1+4√5i)α23(e2γΛt +α2) e2i(ω0Λ+(1−s)(1−Λ))t ,Now∫ t0 f1(x)dx = t + εω sinωt,∫ t0 f2(x)dx =− εω sinωt. We finally obtain the solution w(t) ofiw˙(t) =(1+ ε cosωt)[H,w(t)]− ε cosωt[H,(1− s)w(t)+ sw(t)2]. (27)Explicitly,w(t) =Λ15+√5 5 ξ (t) ζ (t)¯ξ (t) 5+√5 ξ (t)¯ζ (t) ¯ξ (t) 5 (28)ξ (t) =(2+3i−√5i)√3+√5α√3(e−γΛεω sinωt +α2eγΛεω sinωt)e−i(εω0Λ+ε(1−s)(1−Λ)−ε) sinωtω eitζ (t) = −9e−2γΛ εω sinωt +(1+4√5i)α23(e−2γΛεω sinωt +α2) e−2i(εω0Λ+ε(1−s)(1−Λ)−ε) sinωtω e2it6 The solutions were checked by means of Mathematica 4.2.Soliton Kinetic Equations with Non-Kolmogorovian Structure: A New Tool for Biological Modeling?October 13, 20059-100 -50 50 100 t-0.2- 2. Bursts of average position Tr qˆw(t) of the heart for ε = 2, ω = 0.08, α = 1, s =−1.1, Λ = 1. The brain pacemakersinusoidal oscillation is shown as a reference.-100 -50 50 100 t-0.2- 3. The same parameters as in Fig. 2 but with s =−10, i.e. for stronger brain-heart coupling.Since TrHw(t) = TrHw(0), TrHw(t)2 = TrHw(0)2, the internal energy of the system, averaged over one period Tof the pacemaker oscillation, isE =1T∫ t+Ttdt ′TrH f (w(t ′))= TrHw(0), (29)and does not depend on t. In this sense the subsystem is conservative.If one does not integrate over the pacemaker period, one finds that the internal energy of the hands harmonicallyoscillates with the pacemaker frequency ω . What is characteristic, however, the hands do not oscillate harmonicallybut behave as if they were accumulating energy during the phases of quiescence in order to suddenly release it inviolent bursts.In Figs. 2–5 we plot the dynamics of the hands q¯(t) = Tr qˆw(t) =∫∞−∞ dqq〈q|w(t)|q〉 as functions of time fordifferent parameters characterizing the nonlinearity, and for different initial conditions. In Fig. 6 we analyze stabilityof the bursts of the heart under changes of the brain-heart coupling. The system is very stable: Even very large changesof the parameter s do not change the qualitative structure of the bursts. Only for s sufficiently close to 0 the burstsdecay into periodic oscillations whose frequency is unrelated to the one of the brain pacemaker.SCENES FROM THE LIFE OF A TWO-QUBIT ORGANISMIn this example there is no external driving by a pacemaker. The system consists of two subsystems that as a wholeinteract via nonlinearity. The system is conservative, i.e. both its average energy and entropy are constant. We willsee that one can split the history of the system into three effective phases. First, the phase where the two subsystemsbehave as if they were completely uncoupled. Then, the two parts start to exhibit certain joint activity, there are flowsSoliton Kinetic Equations with Non-Kolmogorovian Structure: A New Tool for Biological Modeling?October 13, 200510-100 -50 50 100 t-0.2- 4. The same parameters as in Fig. 2 but with α = e−4.-7.5 -5 -2.5 2.5 5 7.5t-0.1- 5. Tr qˆw(t) for ε = 2, ω = 0.9, α = e2, s = 1, Λ = 1. The sinusoidal oscillation is the pacemaker.of energy between the subsystems, and the entropies of the subsystems change. Finally, the activity dies out and thesystem becomes indistinguishable from the one that never involved any internal activity. Analogies to “birth”, “life”,and “death” are so striking that the name “organism” becomes even more justified.The details of the model are taken from [15]. The two-qubit system is described by the HamiltonianH = 2σx⊗1+1⊗σz. (30)We start with the unnormalized density matrixρ(0) = 125+√7 0 0 00 5−√7 0 00 0 5+√15 00 0 0 5−√15 (31)which is written in such a basis thatH = 1 2 0 02 1 0 00 0 −1 20 0 2 −1 . (32)The density matrixρ(t) = exp[−5iHt]ρ(0)exp[5iHt] (33)Soliton Kinetic Equations with Non-Kolmogorovian Structure: A New Tool for Biological Modeling?October 13, 200511-100-50050100t-10-8-6-4-20s-0.2- 6. Stability of the heart-beats under changes of nonlinear coupling with the brain. Average position Tr qˆw(t) for ε = 2,ω = 0.08, α = 1, Λ = 1, and −10 ≤ s ≤ 0. For s sufficiently far from 0 the bursts do not qualitatively change with fluctuations ofs. Fine details of the bursts are smeared out by coarse-graining of the plot.is a solution of (6) with f (ρ) = ρ2. Such a ρ(t) describes simultaneously a dynamics of two non-interacting systemssatisfying the linear von Neumann equationiρ˙ = 5[2σx⊗1+1⊗σz,ρ]. (34)The environment does not trigger in this solution any switching, but only makes its evolution five times faster than inthe absence of the feedback. The Darboux transformation when applied to ρ(t) produces the solutionρ1(t) = exp[−5iHt]ρint(t)exp[5iHt] (35)whereρint(t) =125−√7tanh2t 0 −13i−3√7−√15−i√1058cosh2t−7i+3√7−3√15+i√1058cosh2t0 5+√7tanh2t 15i+√7−√15−i√1058cosh2t√7+√152cosh2t13i−3√7−√15+i√1058cosh2t−15i+√7−√15+i√1058cosh2t 5+√15tanh2t 07i+3√7−3√15−i√1058cosh2t√7+√152cosh2t 0 5−√15tanh2t . (36)The switching between the two asymptotic evolutions is triggered in the neighborhood of t = 0.If we look at the subentities forming the organism we notice that they do not evolve independently. The easiest wayof seeing this is to compute eigenvalues of reduced density matrices. Both subsystems are two-dimensional so thereare two eigenvalues for each reduced density matrix. They readp±(1) =12±√15−√720 tanh2t, qubit1 (37)p±(2) =12±√26+2√10540cosh2t , qubit2. (38)Soliton Kinetic Equations with Non-Kolmogorovian Structure: A New Tool for Biological Modeling?October 13, 200512-4 -2 2 4t0.640.650.660.670.680.69entropyFIGURE 7. Entropies of the two parts of the two-qubit organism as functions of time. The organism lives, approximately, for−2 < t < 2.The asymptotics areρint(−∞) = 125−√7 0 0 00 5+√7 0 00 0 5−√15 00 0 0 5+√15 , (39)ρint(+∞) =125+√7 0 0 00 5−√7 0 00 0 5+√15 00 0 0 5−√15= ρ(0), (40)and therefore the dynamics represents asymptotically two non-interacting subentities. It is also interesting that the +∞asymptotics is ρ1(t)≈ ρ(t). At large times the “organism” that “dies” becomes practically indistinguishable from theone that never “lived”.The “life” of the organism is the period of time when the two subentities exhibit certain joint activity. Computingthe von Neumann entropies of reduced density matrices of the two subentities we can introduce a quantitative measureof this activity. The entropies of the two particles are shown in Fig. 7. The organism lives several units of time. Similarare the scales of time when the off-diagonal matrix elements of ρint(t) become non-negligible.Although it is clear that the “organism” behaves during the evolution as an indivisible entity, one should not confusethis indivisiblility with the so-called nonseparability discussed in quantum information theory. The organism weconsider in the example is a two-qubit system and therefore one can check the separability of ρ1(t) by means ofthe Peres-Horodecki partial transposition criterion [41, 42]: A two-qubit density matrix ρ is separable if and only ifits partial transposition is positive. It turns out that partial transposition leaves ρ1(t) unchanged and thus is positive forany t. ρ1(t) is in this sense separable (has “zero entanglement”). This formal separability does not contradict the factthat the two-qubit organism is clearly an undivisible entity.EXAMPLES OF KINETIC EQUATIONS ASSOCIATED WITH THEIR LAX-VONNEUMANN FORMLet us illustrate the properties of von Neumann kinetic equations on our explicit solution (28). In our case ˜f (w) =(1−s)w+sw2 with s= s(t)= ε cosωt [cf. Eq. (15)]. Since our solution (28) satisfies the constraint x12 = x23, y12 = y23,p1 = p3, we can reduce the number of equations from six to four for the probabilities pA = x12 −α , pB = x13 −β ,Soliton Kinetic Equations with Non-Kolmogorovian Structure: A New Tool for Biological Modeling?October 13, 200513-20 20 40 60 t0. 8. Plot of pA +1/2 (upper, shifted by 1/2 for clarity of the plot) and pC (lower), for the same parameters as in Fig. 2.-20 20 40 60 t0. 9. Plot of pB +1/2 (upper) and pD (lower), for the same parameters as in Fig. 2.pC = y12−α , pD = y13−β , α =−(1− p1)/2, β =−p1, satisfyingp˙A =1− p12− pC + s(− 1− p12p1 + p1 pA− 1− p12 pB +1− p12pD + pB pC− pD pA)(41)p˙B = 2p1−2pD +2s(− p1− (p1 +1)(1−3p1)2 +(1− p1)pA +(1− p1)pC + p2 pD−2pA pC)(42)p˙C = −1− p12 + pA + s(31− p12p1−2p1 pA− 1− p12 pB− p1 pC−1− p12pD + pA pB + pC pD)(43)p˙D = −2p1 +2pB +2s(p2 p1− (1− p1)pA− p2 pB +(1− p1)pC + p2A− p2C)(44)As we have explained, the time dependence of s(t) is a result of interaction between two parts of the system. Thewhole system is described by a greater number of variables, but the type of nonlinearity is qualitatively the same asfor the subsystem.Fig. 8 and Fig. 9 show that pA, pB, pC, pD, are greater than 0 and smaller than 1, as expected on general grounds.These four probabilities correspond to noncommuting propositions and this is why their sum is not equal to 1 (actually,pA + pB + pC + pD is time-dependent).CONCLUSIONSFormally nonclassical probabilistic and logical systems occur in domains that have nothing to do with quantummechanics. Also the forms of dynamics usually known under the names of von Neumann or Heisenberg equationsare typical of all the systems that posess the so called Lax representations. Therefore, one should not be surprized thatSoliton Kinetic Equations with Non-Kolmogorovian Structure: A New Tool for Biological Modeling?October 13, 200514the mathematical structures one knows from quantum mechanics can be encountered in problems that are “classical”or “macroscopic”. In particular, the basic equations of chemistry or quantitative biology are typically nonlinear andkinetic and one can ask if these equations possess a “Lax form” that would make them into a kind of quantum lookingdynamics? The answer is: Yes, sometimes they can be mapped into nonlinear von Neumann equations. The vonNeumann equations are solitonic and integrable and therefore the kinetic equations are also solitonic and integrable.Obviously, not all the kinetic equations one writes down in biochemistry are integrable. So the link betweenkinetic and von Neumann equations requires further studies. Among other aspects that are worth mentioning hereare the links of von Neumann-type kinetic equations to DNA. For example, even the simple linear kinetic schemesas those discussed in the Introduction naturally lead to helical structures. The self-switching whose examples havebeen discussed in the context of oscillations has then an interesting reinterpretation in terms of formation of openand closed states of the helices (for details cf. [16]). Moreover, it is known that that DNA consists of two strandsthat evolve, in certain sense, in opposite directions (leading and lagging strands). This leads to another intriguing linkwith von Neumann equations since their solutions possess a natural tensor product structure and may be regarded ascomposite systems whose parts evolve in mutually time-reflected ways. This problem will be discussed on explicitexamples in a forthcoming paper.ACKNOWLEDGMENTSThe work of MC is a part of the Polish Ministry of Scientific Research and Information Technology (solicited) projectPZB-MIN-008/P03/2003. We acknowledge the support of the Flemish Fund for Scientific Research (FWO Project No.G.0335.02).REFERENCES1. J. von Neumann, Grundlagen Der Quantenmechanik, Springer-Verlag, Berlin, Heidelberg, New York, 1932.2. D. Aerts, Lett. Nuovo. Cim. 34, 107 (1982).3. D. Aerts, “The physical origin of the EPR paradox and how to violate Bell inequalities by macroscopical systems", inSymposium on the Foundations of Modern Physics: 50 years of the Einstein-Podolsky-Rosen Gedankenexperiment, edited by P.Lathi and P. Mittelstaedt, World Scientific, Singapore, 1985, pp. 305-320.4. I. Pitowsky, Quantum Probability - Quantum Logic, Springer Verlag, New York, 1989.5. D. Aerts, J. Math. Phys., 27, 202 (1986).6. M. Czachor, Found. Phys. Lett. 5, 249 (1992).7. D. Aerts, “The origin of the non-classical character of the quantum probability model", in Information, Complexity, and Controli n Quantum Physics, edited by A. Blanquiere, S. Diner and G. Lochak, Springer-Verlag, Wien-New York, 1987, pp. 77-100.8. C. Piron, Helvetica Physica Acta, 42, 330 (1969).9. L. Gabora and D. Aerts, Journal of Experimental and Theoretical Artificial Intelligence, 14, 327, (2002).10. D. Aerts and L. Gabora, Kybernetes, 34, 151 (2005).11. D. Aerts and L. Gabora, Kybernetes, 34, 176 (2005).12. L. Gabora, E. Rosch and D. Aerts, “How context actualizes the potentiality of a concept", in Worldviews, Science and Us:Bridging Knowledge and Its Implications for Our Perspectives of the World, World Scientific, Singapore, 2005.13. D. Aerts and M. Czachor, Journal of Physics A-Mathematical and General, 37, L123-L32, 2004.14. S. B. Leble, and M. Czachor, Phys. Rev. E 58, 7091 (1998).15. D. Aerts, M. Czachor, L. Gabora, M. Kuna, A. Posiewnik, J. Pykacz, and M. Syty, Phys. Rev. E 67, 051926 (2003).16. D. Aerts, and M. Czachor, “Abstract DNA-type Systems”, submitted to Nonlinearity, 2005.17. L. Gabora and D. Aerts, Interdisciplinary Science Reviews, 30, 69 (2005).18. S. Kauffman, Origins of Order, Oxford University Press, New York, 1993.19. J. Boone, E. Smith, D. Dennett and T. Earle, Current Anthropology, 39, S141, 1998.20. J. H. Schwartz, Sudden Origins, Wiley, New York, 1999.21. F. Varela, Principles of Biological Autonomy, Elsevier/North-Holland, New York, 1979.22. J. Chandler and G. Van de Vijver, Closure: Emergent Organizations and their Dynamics. Vol. 901 of Annals of the New YorkAcademy of Sciences, 1999.23. G. Kampis, Self-Modifying Systems in Biology and Cognitive Science: A New Framework for Dynamics, Information andComplexity, Pergamon Press, New York, 1991.24. R. Rosen, Anticipatory Systems: Philosophical, Mathematical & Methodological Foundations, Oxford, New York, 1985.25. L. Margulis and R. Fester, Symbiosis as a Source of Evolutionary Innovation, MIT Press, Cambridge, MA, 1991.26. N. Eldridge and S. J. Gould, “Punctuated Equilibrium: An Alternative to Phyletic Gradualism”, in Models in Paleobiology,edited by T. Schopf, Freeman, Cooper and Co., San Francisco, 1973, pp. 82-115.Soliton Kinetic Equations with Non-Kolmogorovian Structure: A New Tool for Biological Modeling?October 13, 20051527. S. A. Newman and G. B. Müller, “Morphological Evolution: Epigenetic Mechanisms”, in Embryonic Encyclopedia of LifeSciences, Nature Publishing Group, London, 1999.28. J. Cairns, J. Overbaugh and J. Miller, Nature 335, 142 (1988).29. C. B. Krimbas, Biology and Philosophy 19(2), 185 (2004).30. M. Czachor, Phys. Lett. A 225, 1 (1997).31. M. Czachor, and J. Naudts, Phys. Rev. E 59, R2497 (1999).32. N.V. Ustinov, M. Czachor, M. Kuna, and S.B. Leble, Phys. Lett. A 279, 333 (2001).33. N. V. Ustinov and M. Czachor, “Darboux-Integrable Equations with Non-Abelian Nonlinearities”, in Probing the Structure ofQuantum Mechanics: Nonlinearity, Nonlocality, Computation, and Axiomatics, edited by D. Aerts, M. Czachor, and T. Durt,World Scientific, Singapore, 2002, pp. 335–353.34. J. L. Cies´lin´ski, M. Czachor, and N. V. Ustinov, J. Math. Phys. 44, 1763 (2003).35. A. Goldbeter, Biochemical Oscillations and Cellular Rhythms. The Molecular Bases of Periodic and Chaotic Behaviour,Cambridge University Press, Cambridge, 1996.36. A. Goldbeter, Nature 420, 238 (2002).37. M. C. Moore-Ede, C. A. Fuller, F. M. Sulzman, The Clocks That Time Us: Physiology of the Circadian Timing System, HarvardUniversity Press, Cambridge, 1982.38. D. Gonze, J.-C. Leloup, A. Goldbeter, C.R. Acad. Sci. Paris. 323, 57 (2000).39. D. Gonze, M. R. Roussel, A. Goldbeter, J. Theor. Biol. 214, 577 (2002).40. D. Gonze, J. Halloy, P. Gaspard, J. Chem. Phys. 116, 10997 (2002).41. A. Peres, Phys. Rev. Lett. 77, 1413 (1996).42. M. Horodecki, P. Horodecki, R. Horodecki, Phys. Lett. A 223, 1 (1996).Soliton Kinetic Equations with Non-Kolmogorovian Structure: A New Tool for Biological Modeling?October 13, 200516


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