Unifying mass-action kinetics and Newtonian mechanics by means of Nambu brackets

Abstract

We demonstrate that elementary biochemical reactions defined by mass-action kinetics satisfy a particular Nambu structure. To this end, we express biochemical reaction equations in terms of Nambu brackets and certain ω-factors. The ω-factors account for the fact that mass-action kinetics exhibits in general flow fields with finite divergence. The proposed approach by means of Nambu brackets and ω-factors unifies divergence freeflow fields of Newtonian mechanics and flow fields with finite divergence of mass-action kinetics.

Introduction

Hamiltonian mechanics can be defined by means of Poisson brackets [1]. In the case of a two-dimensional phase space (x,p), the Poisson bracket {A,B} of two functions A(x,p) and B(x,p) is defined by . In this two-dimensional case, the evolution of the state vector r = (x,p) of a dynamical system is given by , where H(x,p) is the Hamiltonian of the system. Moreover, any function defined on (x,p) evolves as . If we put , then we obtain , which states that H is constant. Nambu mechanics [2] can be introduced by generalizing the Poisson brackets such that they apply to systems that are characterized by a set of Hamiltonian functions H₁,...,H_n − 1 and evolve on n-dimensional state spaces with state vectors r = (r₁,...,r_n). Explicitly, the Nambu bracket {A₁,...,A_n} for the functions A₁(r),...,A_n(r) is defined by

1

where is the n-dimensional epsilon tensor. Nambu mechanics [2]

2

can then be alternatively expressed in terms of . Moreover, any function evolves as (in this context note that for Nambu brackets we have ). If we set for any k = 1,...,n − 1, then it follows that , i.e., the Hamiltonian-like functions are invariants of the dynamics (Eq. 2). The force I defined by the n-dimensional vector I(r) = { r,H₁,...,H_n − 1} is divergence free, i.e., we have with . This property implies that the density function ρ(r,t) for a many particle system with particles evolving like satisfies the Liouville equation: .

In the field of classical physics, Nambu mechanics has found several applications. Rigid body rotations satisfying Euler equations [2–8] and particles moving on two spheres [9–13] can be described in terms of Eq. 2. Certain electrodynamic problems (e.g., a charged particle moving in a magnetic field) have been addressed by means of Nambu mechanics [7, 14–18]. The Kepler problem (with its six-dimensional phase space and five invariants) exemplifies another Nambu system [19]. Chiral models [10] as well as the Calogero–Moser system [20, 21] have been investigated. The harmonic oscillator of classical Hamiltonian mechanics has been generalized to yield an elliptic oscillator [22] and multi-oscillator dynamics have been examined from the perspective of Nambu mechanics [19, 23] (for further oscillatory Nambu systems, see also [14, 16]). Likewise, the motion of a free particle from a Nambu perspective has been examined [14, 16]. Some suggestions concerning perturbation theoretical calculations of energy levels can be found in the literature as well [7]. Finally, it has been shown that Nambu mechanics, just as Hamiltonian mechanics, can be generalized to address so-called canonical-dissipative systems [3, 12] (see also [24]).

Our objective is to exploit Nambu brackets in order to study the evolution of biochemical reactions defined by mass-action kinetics. As we will show in Section 2, biochemical reactions exhibit invariants that can be regarded as counterparts to Hamiltonian functions occurring in Nambu mechanics. However, in general, the state space flows representing biochemical reaction kinetics are not divergence free. Therefore, we will need to introduce a factor into Eq. 2 that accounts for the flow field divergence. The motivation for our approach is at least threefold. First, our approach allows the identification of commonalities and differences between the dynamics of Newtonian (Nambu) systems and the dynamics of biological systems. Second, approaching biochemical reactions from a classical mechanics perspective facilitates the introduction of concepts and tools from classical mechanics that might yield new insights into the behavior and characteristic properties of biological systems. Third, the classical mechanics perspective may be used as departure point for introducing analysis methods that simplify the study of reaction kinetics in biology.

Nambu bracket formulation of elementary reactions

Mass-action kinetics for single reactions

The evolution of chemical and biochemical reactions are typically formulated in terms of dynamic systems [25]. As an example, we may consider a chemical reaction of two molecules A and B producing a molecule C: , where, as indicated, we are dealing with a reversible reaction such that both forward and reverse reactions are possible. In general, we consider a single (reversible) reaction that involves L + R molecules (macromolecules, biomolecules, or species). We have L components that react together in order to produce another R components such that

3

Note that we consider only elementary reactions for which stoichiometric coefficients are either unity or zero [25]. In Eq. 3 the parameters k₁ > 0 and k₂ ≥ 0 denote the reaction rates for the forward and reverse reactions, respectively. In the special case k₂ = 0, we are dealing with a nonreversible reaction (for an example, see Section 2.3). Let a_i and b_j denote the concentrations of the molecules A_i and B_j, respectively. From the law of mass action, it follows that the reaction kinetics reads [25]

4

with i = 1,...,L and j = 1,...,R. For example, the reaction yields

5

where a, b, and c denote the concentrations of molecules A, B, and C.

Equation 4 is a n = L + R dimensional dynamical system with state vector r = (r₁,...,r_n) = (a₁,...,a_L,b₁,...,b_R). Most importantly, there are n − 1 independent invariants. In view of our objective to take advantage of Nambu brackets, we will consider them as counterparts to Hamiltonian functions and denote them by H₁,...,H_n − 1. Let us illustrate the existence of n − 1 independent invariants. To this end, we use a first set of L invariants defined by

6

for i = 1,...,L. Clearly, they are linearly independent of each other. In addition, we define another set of R − 1-independent invariants: H_L + 1 = b₂ + a₁ , ... , H_{L + R − 1} = b_R + a₁. That is, we use

7

for j = 2,...,R. Again, it is obvious that the invariants in Eq. 7 are linearly independent from each other. Moreover, the invariants in Eq. 7 are linearly independent from those defined in Eq. 6. All other invariants of the reaction kinetics (Eq. 4) can be expressed in terms of the invariants H₁,...,H_n − 1 (e.g., the invariant a_i − a_i* with i ≠ i^* can be expressed by a_i − a_i* = H_i − H_i*).

Having defined the invariants H₁,...,H_n − 1, we evaluate next the vector-valued Nambu bracket I = {r,H₁,...,H_n − 1}. We first consider I_k for k = 2,...,L (the case k = 1 requires a special treatment). In this case, we have

8

We are interested in identifying the nonzero terms occurring on the right-hand side (we will see that there is exactly one such term). In order to obtain such a nonvanishing term, the indices i₂,...,i_n must be different from k. Consequently, (which is for k = 2,...,L) is not at our disposal. The only way to obtain, for H_k = a_k + b₁, a factor is to differentiate H_k with respect to b₁. Consequently, the index i_k + 1 is given by i_k + 1 = L + 1 and we obtain:

9

with . For the sake of readability, we replace in what follows integer values such as k and L + 1 by their corresponding coordinates r_k = a_k and r_L + 1 = b₁:

10

Next, we note that in any nonvanishing term all partial derivatives must be present. If we do not use all coordinates b₂,...,b_R as partial derivatives, then one of the remaining coordinates r_k would occur at least two times in the expression

11

The ϵ-tensor for such an expression would be zero. In view of this remark, we are next looking for R − 1 indices w₁,...,w_R − 1 such that the expression does not vanish. Moreover, all indices w₁,...,w_R − 1 must be different. There is only one solution: (w₁,...,w_R − 1) = (L + 1,...,L + R − 1). That is,

12

Substituting this result into Eq. 9, we can fix R − 1 indices and obtain

13

with

14

The remaining L − 1 Hamiltonian functions H₁,...,H_k − 1,H_k + 1,...,H_L depend on the variables b₁ and a_j (j ≠ k). In order to obtain a nonvanishing expression Z, we cannot use b₁ for one of the r_j coordinates (the ϵ-tensor would become zero). Consequently, we need to use factors of the form . In doing so, we fix all remaining indices:

15

We exchange the indices a_k and b₁ by means of one transposition:

16

We need to carry out L exchanges of index pairs (transpositions) in order to rearrange the sequence b₁,a₁,...,a_L into a₁,...,a_L,b₁. That is, we have

17

The special case k = 1 can be treated by similar arguments and we find the same result as in Eq. 17: if L is odd (even), we have (). We determine next the coefficients I_k for k = L + 2,...,L + R related to the coordinates r_k = b₂,...,b_R (the case k = L + 1 for b₁ requires a separate treatment). That is, we consider

18

and identify all nonzero terms on the right-hand side (as we will see, there is only one such term). In analogy to the previous case, the variable b_k cannot be used as partial derivative, which implies that, for the invariant H_{L − 1 + k} = b_k + a₁, we can obtain a nonvanishing factor only by putting r_w = a₁. Equation 18 becomes

19

with

20

By analogy to the previous case, we next note that a necessary condition for a nonvanishing expression in the sum of Eq. 19 is that the expression involves all partial derivatives . If we do not use up all coordinates a₂,...,a_L as partial derivatives, then we need to use another coordinate at least two times, which would make the ε-tensor zero. The only way to use up all variables a₂,...,a_L and to obtain a nonzero term is to construct the factor

21

Consequently, Eq. 19 reduces to

22

with

23

For H₁ = a₁ + b₁ we obtain a nonvanishing term Z′′ involving the factor only if we put . This implies

24

The remaining R − 2 Hamiltonian functions H_L + 1,...,H_{L − 2 + k},H_L + k,...,H_{L + R − 1} depend on the variables a₁ and b_j (j ≠ k). In order to obtain a nonvanishing term in the sum of Eq. 24, we cannot use a₁ for any of the coordinates r_j (the ε-tensor would become zero). Consequently, we need to use factors of the form . In doing so, we fix all remaining indices:

25

We exchange the indices a₁ and b_k by means of one transposition such that

26

By means of L − 1 transpositions, we rearrange the index-sequence b₁,a₂,...,a_L into a₂,...,a_L,b₁. In doing so, we obtain

27

For the special case , a similar calculation reveals that we obtain the same result as shown in Eq. 27: if L is even (odd), we have (). Combining the results from Eqs. 17 and 27, we eventually find

28

where the plus (minus) sign holds for odd (even) L (and the superscript T means that we take the transpose, i.e., we have a column vector). Equation 4 can equivalently be expressed in terms of

29

with the factor

30

Since we have I = {r,H₁,...,H_n − 1}, Eq. 29 can be cast into the form

31

Equation 31 provides us with a formulation of mass-action kinetics (Eq. 4) in terms of Nambu brackets. This formulation highlights two different aspects. First, the Nambu bracket in Eq. 31 points out that the reaction dynamics involves invariants. Second, the ω-factor indicates that the reaction dynamics in general will not be divergence free. In particular, for the force term g(r) = ±ω( r) { r,H₁,...,H_n − 1 }, we obtain

32

The divergence can inform us to a certain extent about the stability of attractors (e.g., fixed points) of biochemical reactions. Let V(t) ≥ 0 denote a volume in the state space spanned by r₁,...,r_n. Then, the integral over the divergence of g determines to rate of change of that volume [26, 27]:

33

Recall that for Nambu systems we have , which reflects the fact that state space volumes are preserved under the dynamics of Nambu systems. In contrast, for biochemical reactions we may observe contractions of state space volumes.

Let us illustrate Eq. 31 with an example. For , we have L = 2 and R = 1. The state vector is r = (a,b,c) and the invariants read H₁ = a + c and H₂ = b + c, such that the Nambu bracket (Eq. 28) becomes . Alternatively, we first note that and (with ) and then compute . From Eq. 30, we obtain ω(r) = − k₁ab + k₂c . Consequently, Eq. 31 reads

34

which is equivalent to Eq. 5 and gives us the dynamics of the reaction in terms of a Nambu bracket. In what follows, we assume k₂ > 0. The divergence of the force term is and depends in general on the evolution of the concentrations a(t) and b(t). In any case, the divergence is negative, that is, we have for all r = (a,b,c). From Eq. 33, it follows that

35

Consequently, we have with

36

In words, any state space volume decays to zero, which indicates that the biochemical reaction approaches an attractor in the limit t → ∞. For the reaction, this attractor is given by a fixed point that satisfies ω = 0. Equation (34) highlights commonalities and differences between the reaction and conservative systems of classical (Nambu) mechanics. Just as can be observed in Nambu systems, the reaction corresponds to a dynamical system of order n that involves n − 1 invariants or Hamiltonians (here, n = 3). Unlike systems of classical mechanics, the system is nonconservative, which is indicated by the ω-factor in Eq. 34. This implies that, in general, perturbations will not persist. Moreover, this implies that the biological system can exhibit more than neutral stability. It can exhibit an attractor. Equations 35 and 36 illustrate that the second observation (the violation of the classical mechanics scenario) allows us to introduce a concept of nonlinear physics that has been developed for systems that violate Hamiltonian dynamics: the concept of the contraction of state space volumes. For the biochemical reaction, we find that state space volumes V(t) shrink in the limit t → ∞ to a single point such that V →0.

Coupled biochemical reactions

For coupled biochemical reactions, similar considerations can be carried out. In the general case, we consider M reversible reactions involving n species (i.e., n different types of biomolecules) X₁,...,X_n. These reactions read

37

for j = 1...,M. The parameters and correspond to stoichiometric coefficients equal to unity or zero for the elementary reactions under consideration [25]. In our context, the stoichiometric coefficients indicate whether or not in the reaction j a molecule occurs on the left or right-hand side of the reaction dynamics. We introduce the index sets of all left and right participating molecules and . Let |{·}| denote the size of a set. Accordingly, we define the set sizes , , and . That is, is the total number of different components (or reagents) involved in the reaction j. We have and define the difference . That is, there are components not involved in the reaction j. Each reaction contributes to a change of the n-dimensional state vector r = (x₁,...,x_n), where x_i is the concentration of molecules X_i. In order to describe each reaction separately by means of a Nambu bracket, let us define next n − 1 independent invariants. The first (or ) invariants are those defined in the context of the single reaction (Eq. 3). In doing so, we obtain the functions . In addition, we define independent invariants on the basis of the components not involved in the reaction j:

38

As a result, we obtain a total of independent invariants. Formally, we may define the indices k_u in Eq. 38 as follows. First, we define the index set of all indices of reagents not involved in the reaction j. Subsequently, we sort the index set in ascending order and thus obtain . The indices in (38) can then be identified with the indices in such that for and . The reaction equations of the coupled reactions (Eq. 37) can then be written by means of Nambu brackets as

39

where the factors are determined from the single reactions (in analogy to the results derived in Section 2.1). From a mathematical perspective, Eq. 39 describes n-coupled first-order differential equations that involve polynomial functions. In doing so, there is a striking similarity with Haken networks studied in the context of pattern recognition and multistable competitive systems [28–32].

Further generalization

Our proposed approach may be generalized in various ways. For example, we may consider so-called clamped variables. That is, we may consider substances whose concentrations are determined externally (and are held constant over time). Let A′_i and B_i′ denote such substances and a_i′ and b_i′ their concentrations. Then Eq. 3 becomes

40

Likewise, (4) reads

41

The formulation in terms of Nambu brackets defined by Eq. 31 still holds, provided we generalize Eq. 30 as

42

Analogous considerations can be made for coupled reaction equations. We restrict ourselves to giving an example in this regard. We consider a cyclic of three nonreversible reactions that involves the substances X, Y, Z, and three molecules C₁′, C₂′, and C₃′ whose concentrations are held constant. The reactions are

43

Let x, y, z denote the concentrations of X, Y, Z. Then the invariants of the first reaction are and . Likewise, for the second and third reactions, we obtain , , , and . For the reaction dynamics in Eq. 43, we obtain from Eq. 39

44

where ω^(k) can be determined from the single reactions listed in Eq. 43 and r = (x,y,z). We find , , and . Here, the parameters c_k are the concentrations of the substances C_k′. Substituting the variables ω^(k) and the Hamiltonian invariants into Eq. 44 gives us

45

The divergence of the force vector g is given by such that any state space volume V(t) decays as with b = k₁c₁ + k₂c₂ + k₃c₃ > 0.

Conclusions

We established to some extent a unification of classical Newtonian mechanics and mass-action kinetics. Newtonian mechanics can be expressed in terms of Hamiltonian mechanics. The latter is based on Poisson brackets which can be generalized to Nambu brackets in order to address multi-Hamiltonian systems. We showed that biochemical mass-action kinetics in turn can be described in terms of such multi-Hamiltonian Nambu brackets. This line of arguments puts Newtonian mechanics and reaction kinetics into a unified framework based on Nambu brackets.

We considered single and coupled reactions. Our main results in terms of Eqs. 31 and 39 point out two key aspects of biochemical reaction kinetics. On the one hand, biochemical reactions exhibit invariants as indicated by the Nambu brackets. On the other hand, when interpreting biochemical reactions in terms of dynamical systems, the respective state space flows in general are not divergence free, as indicated by the state-dependent ω-factors. In doing so, we pointed out commonalities and differences between systems of classical mechanics and biological systems satisfying mass-action kinetics.

The framework involving Nambu brackets provides a basis for introducing concepts and tools of classical mechanics into biology. For example, when force fields are not divergence free, then the deviation from the divergence free condition has both a qualitative and quantitative aspect.

First, at a fixed point r, we have , where λ_k are Lyapunov exponents. Consequently, a necessary condition for the existence of a stable fixed point is that the divergence is negative [26]. Second, when then determines the rate of volume contraction: . Quantitatively, χ may be used as a measure for dissipation rate [27]. In our context, χ quantifies how far away a system is from being an ordinary Nambu system with χ = 0 and (i.e., χ serves as a distance measure).

Finally, as mentioned in the introduction, the classical mechanics perspective may be exploited in order to develop analysis methods that simplify the study of reaction kinetics in biology. For example, Eq. 39 illustrates that M biochemical reactions involve M different ω-factors irrespective of the number n of species involved in those reactions. In particular, when we have more species than reaction steps (n > M), then, from Eq. 39, we may derive a dynamic description for the variables . If the system under consideration is such that, from Eq. 39, a closed description in can be obtained, then the dynamical description in terms of ω-factors reduces the dimensionality of the problem at hand from n to M. It might be beneficial to study the system in the (lower) M-dimensional space rather than in the original (higher) n-dimensional space.