2.1 Compartmental models

Throughout the paper we consider systems containing a set of interconnected compartments and objects (such as ribosomes, particles, molecules, vehicles etc.) moving between them. We assume that the rate of transfer between compartments depends on the amount of objects in the source compartment as well as on the amount of free space in the target compartment. This naturally implies that each compartment has a well-defined finite capacity that limits the amount of modeled quantities that can be contained in the given compartment. We also allow explicit time dependence and in some cases dependence on the amount of objects and free space in other compartments.

For the formal definition, let us consider the set Q = {q₁, q₂, …, q_m} of compartments and the set A ⊂ Q × Q of transitions, where (q_i, q_j) ∈ A represents the transition from compartment q_i into q_j. Then, the directed graph D = (Q, A) is called the compartmental graph and it describes the structure of the compartmental model. The transitions are assumed to be immediate, thus loop edges are not allowed in the model since they do not introduce additional dynamical terms. Similarly, we do not allow parallel edges between two compartments in the same direction since they can be replaced by a single transition. We say that a (compartmental) graph is strongly connected if there exists a directed path between any two vertices in both directions, and we say that a graph is weakly reversible if it is a collection of isolated strongly connected subgraphs.

For each compartment q_i we introduce the sets of donors and receptors, respectively, as (1) that is, the set of donors of a given compartment are the compartments where an incoming transition originates from and the set of receptors are the compartments where an outgoing transition terminates in.

2.2 Chemical reaction networks (kinetic systems)

In this subsection we give a brief introduction of kinetic systems based on [18, 19], where more details can be found. A chemical reaction network (CRN) contains a set of species Σ = {X₁, X₂, …, X_N} and the corresponding species vector is given by X = [X₁ X₂ … X_N]^T. The species of a CRN are transformed into each other through elementary reaction steps of the form (2) where and are the source and product complexes, respectively, the vectors are stoichiometric coefficient vectors and functions are the rate functions with denoting the set of nonnegative real numbers. The matrix Y containing the stoichiometric coefficient vectors as columns is called the stoichiometric matrix. The subspace spanned by the so-called reaction vectors y_j′ − y_j is called the stoichiometric subspace of the CRN.

The CRN structure can be uniquely described by a directed graph as follows. For each complex we assign a vertex in the graph and for each elementary reaction step of the form C_j → C_j′ we assign a directed edge between the corresponding vertices. We call the resulting graph the reaction graph of the CRN. The deficiency of the CRN is defined as δ = m − ℓ − s, where m is the number of distinct complexes, ℓ is the number of linkage classes (graph components) in the reaction graph and s is the dimension of the stoichiometric subspace.

Let denote the state vector of the species as a function of time for t ≥ 0. Based on the above, the dynamics of the CRN is given by (3)

We assume that a reaction can only take place if each species of the given reaction have nonzero concentration; that is, we assume that whenever there exists k ∈ supp(y_j) such that x_k(t) = 0, where we say that k ∈ supp(y_j) if [y_j]_k > 0. This property ensures the invariance of the nonnegative orthant (or a part of it). We also presume standard regularity assumptions of the rate functions that guarantee local existence and uniqueness of solutions. Different results in this paper require different sets of such assumptions, thus for the sake of generality they will be specified later. Dynamics of the form of (3) is called persistent if no trajectory that starts in the positive orthant has an omega-limit point on the boundary of .

We note that for any (where denotes the stoichiometric subspace) we have that (4) and thus 〈x, v〉 is constant. Since was arbitrary we have that . This shows that the translates of define invariant linear manifolds for the system. We further define for each a positive stoichiometric compatibility class .

A set of ODEs of the form is called kinetic if it can be written in the form (3) with appropriate rate functions and stoichiometric coefficient vectors.

3.1 Kinetic modeling of compartmental transitions

Let us consider a compartmental model D = (Q, A). Let the set of species be Σ = {N₁, N₂, …, N_m} ∪ {S₁, S₂, …, S_m} where N_i and S_i represent the number of particles and available spaces in compartment q_i, respectively. To each transition (q_i, q_j) ∈ A we assign a reaction of the form (see, also [24]) (5) where is the rate function of the transition. Such a reaction represents that during the transition from compartment q_i to compartment q_j the number of items decreases in q_i and increases in q_j, while the number of available spaces increases in q_i and decreases in q_j. Let n_i and s_i denote the continuous amount of particles and free space in q_i, respectively.

Based on (3) the dynamics of the system is given by (6) where n and s denote the vectorized form of the variables n_i and s_i, respectively. It is easy to check that the model class in Eq (6) contains ribosome flow models described in [23] or [15], and extends them in two ways: firstly, the reaction rate function is not necessarily mass-action type and moreover, is time-varying, and secondly, the compartmental graph of the system can be arbitrary (i.e., there can be transitions between any two compartments). Note, that we also allow the transition rates to depend on the amount of objects and free space in other compartments as well, possibly describing inhibitory phenomena. Therefore, we call (6) a generalized time-varying ribosome flow model. Thus, our novel results not only extend the theory of ribosome flow models, but can be applied to other TASEP based transport models [30–34] and other flow models, such as the Traffic Reaction Model of [25] or the Nonlocal Flow Reaction Model of [26]. Finally, we note, that while more complicated network structures may not be biologically relevant in the case of ribosome flows, but can serve as a great tool for the analysis of other flow based physical models, e.g. traffic flows.

Clearly the reaction graph of the assigned CRN of a compartmental model is generally not strongly connected nor weakly reversible even if the compartmental graph is strongly connected. In fact, the reaction graph is weakly reversible if and only if each transition in the compartmental system is reversible. Even though the reaction graph, in some sense, loses the regularities of the compartmental graph, we can explicitly determine its deficiency from the compartmental topology and, as described in [27], CRNs of the form (6) exhibit persistence and stability properties in various senses in the time-invariant case.

3.2 Deficiency of CRNs realizing compartmental models

For a compartmental system D = (Q, A) let denote the undirected graph where the parallel edges are merged.

Theorem 3.1 The deficiency of a CRN assigned to a compartmental model D = (Q, A) is equal to the number of chordless cycles in the undirected graph .

Proof. For each transition between q_i and q_j we assign two complexes, namely N_i + S_j and S_i + N_j, regardless of the transitions’ direction, so reversible reactions do not introduce additional complexes, and thus the number of stoichiometrically distinct complexes is . A complex of the form N_i + S_j is only connected with the complex S_i + N_j, and thus we have linkage classes each consisting of exactly two complexes. To find the dimension of the stoichiometric subspace, denoted by , observe that the reaction vector of a reaction of the form N_i + S_j → N_j + S_i is (7) where denotes the kth unit vector. Again, since y_i→j = −y_j→i it suffices to consider the undirected graph |D|. Assume that y_i→j is such that (8)

Then by (7) we have that for each non-zero term of the form c_.→l′y_.→l′ the right-hand side also contains at least one non-zero term c_l′→.y_l′→., including the terms c_i→.y_i→. and c_.→jy_.→j. This shows that the edges corresponding to the reaction vectors of the right-hand side form possibly multiple cycles in |D|. Without the loss of generality we may assume that this subgraph does not contain cycles isolated from (q_i, q_j). We have to consider the following cases:

1. First, we assume that the right-hand side is a single chordless cycle and contains the transitions (9)
   Taking the inner product of unit vectors and (10) yields the system of linear equations: (11) which clearly has one solution where each weight is equal to one.
2. If the right-hand side consists of multiple cycles, then repeatedly using the previous argument we can replace the arcs not containing (q_i, q_j) with chords. Note, that if the reaction vector corresponding to the chord is already on the right-hand side, then we just have to modify its coefficient. This method decomposes the right-hand side and will leave us with one chordless cycle containing (q_i, q_j), leading back to the previous case with exactly one solution. Repeating the arc substitutions we can see that each arc becomes a chordless cycle with the reintroduced edges and the arising systems of linear equations have exactly one solution.

The first case above shows that the dimension of the stiochiometric subspace reduces by one for each set of reaction vectors that correspond to edges forming a chordless cycle in |D| and the second case shows that is reduced by that exact amount. If σ denotes the number of chordless cycles in , then the deficiency of the reaction network can be computed as .

3.3 Linear conservation laws

System (6) exhibits conservation in several senses. First of all, we have that (12) thus the sum of modeled quantities and free spaces in the system is constant along the trajectories of (6); that is, the function defined for as (13) is a first integral, where x₁, x₂, …, x_m and x_m+1, x_m+2, …, x_2m correspond to the variables n₁, n₂, …, n_m and s₁, s₂, …, s_m, respectively. Our next observation is that holds for each compartment, thus c_i ≔ n_i + s_i is the constant capacity of compartment q_i. Let c^(m) be a vector such that its ith coordinate is c_i. Substituting s = c^(m) − n we can rewrite (6) in a reduced state space as (14) or after an analogous substitution, as (15)

As a consequence of the preceding observations, the function , defined for as (16) is a first integral for (14), in which case each x_i = n_i (and similarly for (15) if each x_i = s_i). This shows that while the state space of the decomposed systems is , for a given initial condition the trajectories are contained in the (m − 1)-dimensional manifold (hyperplane) defined by (17)

For a generalized ribosome flow define and for r ∈ [0, c] let be the level set of H corresponding to r; that is, (18)

Example 1.1: (generalized) RFMR.

As a small example let us consider a Ribosome Flow Model on a Ring (RFMR) [13] with three sites. The underlying compartmental model is given by D = (Q, A), where (19)

The topology is shown in Fig 1.

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Fig 1. Compartmental graph of a three-dimensional RFMR.

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The corresponding CRN has the following species and reactions: (20)

It is easy to see that, indeed, the reaction graph is not weakly reversible and its deficiency is one. The dynamics of the model in the full state space is given by (6) as (21) which can be rewritten in the reduced state space based on (14) as (22)

In a classical RFMR each c_i = 1 and each transition-rate depends only on n_i and c_i − n_i = 1 − n_i, and follows the mass-action law. In an RFMR with different site sizes (RFMRD) [15] we allow arbitrary site sizes, in which case the above equation can be written as (23)

4.1 Equilibria of time-invariant systems

In this subsection we assume that the rate functions are continuously differentiable and only depend on the variables n_i and s_j in a nondecreasing manner; that is, we assume that for each i and j. Then the results [27] show that a system of the form (14) is cooperative (the name also highlights the importance of the exclusion of inhibitory phenomena), is (strongly) monotone and each level set L_r contains a unique globally (relative to its level set) asymptotically stable steady state. This implies that the steady states form a linearly ordered set. For i = 1, 2, …, m let e_i: [0, c] ↦ [0, c_i] denote the ith coordinate function of the steady state; that is, let (24) where n(0) ∈ L_r is arbitrary and ρ(t, n(0)) denotes the solution at time t with ρ(0, n(0)) = n(0). Clearly each e_i is continuous and the monotonicity of the system also shows that each e_i function is strictly increasing; that is, they are differentiable almost everywhere and their derivative are positive.

Example 1.2: Equilibria of generalized RFMR.

Let us consider a generalized version of the RFMR in Fig 1. Its time evolution in the reduced state space is given in the form (14) as (25) and for the simulations we set capacities c₁ = 5, c₂ = 25, c₃ = 50. The rate functions in the different cases are assumed to have the form (corresponding to the classical RFMRD) or to be rational functions of the form for some l > 0 with k₁₂ = 100, k₂₃ = 40, k₃₁ = 60. Fig 2 shows the equilibrium curves for these rate functions with various l values.

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Fig 2. Loci of equilibria of a generalized RFMR as a function of the total number of ribosomes for different l saturation parameters.

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Example 2: Not strongly connected model.

Let us consider consider a not strongly connected compartmental model given by D = (Q, A), where (26)

The topology is shown in Fig 3.

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Fig 3. Compartmental graph of a not strongly connected model.

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The corresponding CRN has the following species and reactions: (27)

The dynamics of the system in the reduced state space is given by (28)

Since the compartmental graph is not strongly connected the persistence and stability results of [27] are not applicable. However, empirical results show that the long-time behaviour of the system still exhibits some regularity, which can be divided into two cases base on the initial values of the system:

1. If r ≔ H(n(0)) ≤ c₁, then
2. If r ≔ H(n(0)) > c₁, then and n₁(t) and n₂(t) will converge to the unique equilibrium on the level set of the reduced compartmental model D′ = (Q′, A′) given by Q′ = {q₂, q₃}, . Note that since D′ is strongly connected, the results of [27] and the above investigation can be applied.

For the simulations we set c₁ = c₂ = c₃ = 100. The rate functions in the different cases are assumed to have form (corresponding to mass-action kinetics) or to be rational functions of the form for some l > 0 with k₂₃ = 15, k₃₂ = 25, k₃₁ = 35. Fig 4 shows the equilibrium curves for these rate functions with various l values. As described by the above cases we see that until the sum of the initial value exceed the capacity of the q₁ compartment the equilibrium lies on the n₁ axis. After that the equilibrium lies on the plane and since D′ is strongly connected we have that the coordinate functions of the equilibria e₂(r) and e₃(r), restricted to the set [c₁, c], are continuous and strictly increasing. We note that while the system is not strongly connected it exhibits many similar qualitative properties as strongly connected models. For example, for initial values satisfying H(n(0)) > c₁ the system is Lyapunov stable as described in [29].

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Fig 4. Loci of equilibria of a not strongly connected model as a function of the amount of modeled quantities for different l saturation parameters.

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Remark 4.1. The authors hypothesize that the long-time behaviour of a compartmental model with arbitrary compartmental structure can be similarly described. Recall that a (compartmental) graph D = (Q, A) can be written as a directed acyclic hypergraph of strongly connected components. The hypergraph will then contain three types of components:

1. we call a component trap if it does not have any outgoing edges,
2. we call a component source if it does not have any incoming edges,
3. we call a component intermediate if it is not a trap and not a source.

Based on the initial value and the exact compartmental structure the following phenomena can be observed:

• Traps (and only traps) can become full, thus possibly creating new traps.
• Sources (and only sources) can become empty, thus possibly creating new sources.
• After a sufficient number of traps are filled and sources are emptied, the compartmental graph D is decomposed into isolated strongly connected components; that is, the resulting graph is weakly reversible, in which case the results of [27] can be applied.

While these observations are elementary and show that the system is stable, the equilibria are clearly non-unique with respect to the total mass of the network and in general it is not straightforward to predict from the initial value which components will fill and empty.

4.2 Persistence

In this subsection we consider time-varying generalized ribosome flows of the form (6) only under mild regularity assumptions described by the following theorem, which is based on the results of [35] but the statements are rephrased to be more aligned with our framework. For the definition of notions related to Petri nets (e.g. siphons) and their exact connection with CRNs we refer to [27, 35].

Theorem 4.2. [35] The dynamics of a CRN of the form (3) is persistent if

1. (i) Each siphon of the CRN contains a subset of species which define a positive linear conserved quantity for the dynamics.
2. (ii) There exists a positive linear conserved quantity c^Tx for the dynamics.
3. (iii) There are nonnegative, continuous functions , such that
   1. (a) if for each k ∈ supp(y_j), then (and similarly for ) holds for each j = 1, 2, …, R, and
   2. (ii) for each j = 1, 2, …, R, for all and for all t ≥ 0 we have .

To verify condition (i) we would, in general, need to enumerate all siphons of the CRN, which is well-known to be an NP-hard problem. However, in our recent paper [27] we explicitly characterized the siphons of a CRN assigned to a strongly connected compartmental models in the time-invariant case. However, one can observe that conditions (i) and (ii) of 4.2 are independent of the choice of transition rates and even independent from whether the system is time-invariant or not; that is, our results, formulated in the following theorem, hold for time-varying compartmental systems as well.

Theorem 4.3. [27, Corollary 4.6] A siphon in the Petri net of a strongly connected compartmental graph either contains the vertices N_i and S_i corresponding to the same compartment q_i, or it contains all the vertices N₁, N₂, …, N_m or S₁, S₂, …, S_m.

Then the conclusions of Section 3.3 show that conditions (i) and (ii) are satisfied by virtue of the first integrals (16) and (13), respectively.

It is not straightforward to determine exactly what types of reaction rates satisfy condition (iii). For the sake of specificity, we characterize a class of reaction rates of special interest which can be written in the following form (29) where we assume that the transformations are nondecreasing, have θ_i(0) = ν_j(0) = 0 and satisfy and for each i, j = 1, 2, …, m. We also assume that the functions Ψ_ij take the form (30) where and . We further assume that for k_ij(t) there exist such that for all t ≥ 0. In this case we have (31) which are clearly monotonous in the sense of Theorem 4.2, and thus condition (i) is satisfied and the system is persistent.

Remark 4.4. The above investigation and, in particular, condition (iii) of Theorem 4.2 shows that Lemmata 5.1, 5.2 and Remark 5.3 of [27] can be modified to the time-varying case; that is, for a system of the form (6) with strongly connected compartmental graph and reaction rates of the form (29), for each τ > 0 there exists ϵ(τ) > 0 with ϵ(τ) → 0 as τ → 0 such that n_i(t), s_i(t) ∈ [ϵ, c_i − ϵ] holds for each i = 1, 2, …, m and t ≥ τ.

The denominator of (29) contains positive terms which can be interpreted as the inhibitory effect of other species, and the time-varying coefficient k_ij(t) introduces the dependence of the system parameters on various factors such as temperature or the dynamical behaviour of other species that are not explicitly modeled as state variables. This class of rate functions contains many well-known examples, demonstrating the range and flexibility of reaction rates of the above form:

1. Setting each θ_i(n_i) = n_i and ν_j(s_j) = s_j and Ψ_ij(n, s) = 0 we obtain the case of classical mass-action kinetics with time-varying rate coefficients: .
2. Setting each θ_i(n_i) = n_i and ν_j(s_j) = s_j and Ψ_ij(n, s) = l² − 1 + ln_i + ls_j + n_is_j for some l > 0 yields (32) corresponding to simple saturating kinetics described by the Monod equation.
3. The previous example can also be obtained by setting and and Ψ_ij(n, s) = 0, showing that (29) is not unique. Notice however, that for fixed θ_i, ν_j transformations the function Ψ_ij, and thus the fraction itself, is unique.
4. Setting each and for some l > 0 yields the classical Hill kinetics.

Example 1.3: Time-varying generalized RFMR.

Let us again consider a generalized version of the RFMR from Fig 1. For this example we set c₁ = c₂ = c₃ = 100, l = 100 and (33) where the coefficient functions are considered to be exponentially decaying perturbations of the nominal values (34) of the form (35)

As a comparison let us consider the solution of the time-invariant system with the above nominal values. Fig 5a shows the phase portrait of the perturbed and the original systems starting from various initial conditions with H(n(0)) = 150. Fig 5b shows the time evolution of the state variables with n(0) = [5 45 100]^T, where the state variables of the perturbed and the time-invariant system are depicted with blue lines and red lines, respectively. We can observe that since the time dependent terms are exponentially decaying and both systems evolve on the same linear manifold, the systems tend to the same equilibrium, as expected.

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Fig 5. Trajectories and time evolution of a generalized time-varying RFMR with decaying time dependence.

(a) Phase portrait of the system. (b) Time evolution of state variables.

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4.3 Stability of the solutions for periodic transition rates

In this section we investigate the periodic behaviour of the generalized ribosome flows based on the ideas of [36]. Let us consider a generalized ribosome flow in the reduced state space of the form (14) with transition rates of the form (29) and assume that the transition functions are and periodic with the same period (but having possibly different phases). Write (14) as and assume that the right-hand side satisfies the following monotonicity condition: F_i(t, x) ≤ F_i(t, y) for any two distinct points such that x_i = y_i and x_j ≤ y_j for j ≠ i. This condition is satisfied if, for example, the transition rates are such that Ψ_ij ≡ 0; that is, if there are no inhibitory phenomena. Then the system phase locks (or entrains) with the periodic excitations.

Theorem 4.5. Consider a system of the form (14) satisfying the above monotonicity assumption, where each is periodic with a common period T. Then for each r ∈ [0, c] there exists a unique periodic function with period T such that for all a ∈ L_r we have that (36)

Proof. The properties of the rate functions and the fact that ∇H is positive implies the result via [37, 38].

Remark 4.6. Since, in some sense, time-invariant systems can be seen as periodic, the stability result [27, Proposition 5.5] is a special case of the above theorem, where ϕ_r is reduced to a single point of the manifold L_r.

Example 1.4: Entrainment of generalized RFMR.

Let us again consider a generalized version of the RFMR from Fig 1. For this example we set c₁ = c₂ = c₃ = 100 and (37) which clearly have the same period T = 2π. Fig 6a and 6b show the phase portrait of the system starting from various initial conditions with l = 100, H(n(0)) = 150 and the time evolution of the state variables with n(0) = [5 45 100]^T, respectively.

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Fig 6. Entrainment of a generalized RFMR with periodic transition rates.

(a) Phase portrait of the system. (b) Time evolution of state variables.

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5.1 Factorization of the transition rates

Let us consider a generalized ribosome flow in the reduced state space of the form (14), in this case given by (46)

Notice that we can naturally factor some terms of the transition rates into the time-varying coefficient as (47)

Then (46) can be rewritten as (48)

This equation can be clearly embedded into the class of strongly connected systems of the form (38), since the reaction graph of (48) consists of species Σ = {N₁, N₂, …, N_m}, has the m × m identity matrix as its stoichiometric matrix and for each transition (q_i, q_j) ∈ A we assign a reaction of the form (49) and thus the system of differential equations can be written as (50) where the elements of are given by (51)

Note that the fractions are differentiable (and thus Lipschitz) and each k_ij(t) is piecewise locally Lipschitz, hence each is piecewise locally Lipschitz. This shows that generalized ribosome flows can be embedded into the class of rate-controlled biochemical networks described in [22] in a way that preserves the compartmental structure; that is, the reaction graph of (50) is topologically isomorph to the compartmental graph. In particular if the compartmental model is strongly connected, then the reaction graph of the reduced system (50) is strongly connected as well. Furthermore, combining the persistence of the system with Remark 4.4 we find that , and thus Theorem 5.3 ensures input-to-state stability.

5.2 Quasi-LTV factorization

A classical argument shows that the model reduction above can result in a Linear Time-Varying (LTV) system [6]. Consider an nonnegative function such that F(0) = 0, where k ≥ 1. Then for the function F(rx) we have (52) and thus (53) and since F(0) = 0, we find that F(x) = xf(x). Note, that the calculation also shows that . Since θ_i is real analytic we have that for some real analytic function. Then (48) can be rewritten as (54) where (55)

Similarly as before, the reaction graph of (54) consists of species Σ = {N₁, N₂, …, N_m}, has the m × m identity matrix as its stoichiometric matrix and for each transition (q_i, q_j) ∈ A we assign a reaction of the form (56) and thus the system of differential equations can be written as (57) where the elements of are given by (58)

Again, each is piecewise locally Lipschitz, thus for strongly connected compartmental models Theorem 5.3 ensures input-to-state stability via Remark 4.4.

5.3 Factorization of Monod kinetics

Let us consider a generalized version of the RFMR in Fig 1 with rational rate functions corresponding to Monod kinetics of the form (59) for some l > 0. As discussed before, the corresponding CRN is not strongly connected. However, using the functions (60) we can to rewrite (59) as (61)

Then the CRN corresponding to (64) has the following species and reactions: (62) which is strongly connected and isomorph to the compartmental model in Fig 1. We arrive at the same conclusion if we instead use the functions (63) to rewrite (59) as (64)

Note that the quasi-LTV factorization might be more complicated in some cases, but the construction described in Section 5.2 guarantees its existence.

5.4 Induced family of Lyapunov functions

The above investigation demonstrates that generalized ribosome flows can be embedded into rate-controlled biochemical networks in at least two different ways, where each embedding induces a different Lyapunov function of the form (45). Thus, in general, we may use at least two different Lyapunov functions governing the same dynamics. To characterize their exact relation, consider a factored system of the form (50) with its ISS-Lyapunov function . The quasi-LTV representation of the system admits an ISS-Lyapunov function of the form (65) so that we can write (66)

Remark 5.5. Since we have that which is exactly the Kullback-Leibler divergence . It is important to note that the Kullback-Leibler divergence is not a metric, since and it does not satisfy the triangle inequality. However, it is a nonnegative measure, meaning that it is nonnegative and zero if and only if and it is often used to measure the “distance” of probability distributions for example in information theory and machine learning [43].

While in general we are restricted to the above factorizations, in some special cases we may use a whole family of factorizations and corresponding Lyapunov functions. To illustrate this, consider an example when each for some l > 0 and , , a_i ≥ b_i (these properties ensure that the functions θ_i are nondecreasing). Then, after the factorization described in Section 5.1, the Lyapunov function (45) becomes (67)

We emphasize that (45) only depends on the θ_i functions, in this case parametrized with the l, a_i, b_i values; that is, it is independent of the network structure and transition rate coefficients. We can also perform the factorization with , , , yielding the Lyapunov function of the same form as in (67). This shows that the parameters a and b can be freely (apart from the constraints above) chosen in (67). We may also observe some interesting behaviour at the extrema of the parameters and l, namely, that if we choose each then the Lyapunov function in (67) is independent of l; that is, we have that (68)

Moreover, letting l→∞ yields the convergence (69) where is defined in (65).

Example 1.5: Family of Lyapunov functions of a generalized RFMR.

Let us again consider a generalized version of the RFMR in the reduced state space from Fig 1. For a given initial condition n₀ we can substitute n₃ = H(n₀) − n₁ − n₂, and thus the Lyapunov function restricted to the manifold can be seen as a two dimensional function with local coordinates n₁ and n₂.

We set the capacities as c₁ = c₂ = c₃ = 100 and k₁₂ = 100, k₂₃ = 60, k₃₁ = 20. The system has transition rates as described above with each a_i = b_i = 3; that is, we have that (70)

The simulations were performed with H(n₀) = 150. Fig 7a–7c show the Lyapunov function for various choices of and with l = 25 fixed. The second and third rows demonstrate the convergence characterized in (69); Fig 7d–7f show for increasing l values Fig 7g–7i shows for the same increasing l values.

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Fig 7. Comparison of Lyapunov functions for a generalized RFMR.

(a) l = 25, . (b) l = 25, . (c) l = 25, . (d) l = 25, . (e) l = 100, . (f) l = 200, . (g) l = 25, . (h) l = 100, . (i) l = 200, .

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Example 3: Family of Lyapunov functions for a larger network.

Let us consider a compartmental system with m = 100 compartments in the reduced state space. We assume that the transition rate functions are corresponding to Hill kinetics (modified intentionally to have different powers in the numerator and the denominator) and are of the form (71) with l = 350. We assume that the only nonzero coefficients are (72) for i = 1, 2, …, m, where indices are understood as modulo m. Clearly this compartmental graph is strongly connected. Finally, we set capacities (73)

Then the Lyapunov function (45) takes the form (74)

We can also factorize as , when (45) becomes (75)

Fig 8 shows the time evolution of Lyapunov functions , and V^LTV and their time derivatives.

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Fig 8. Time evolution and time derivative of Lyapunov functions obtained from various factorizations of the transition rates.

(a) Time evolution of Lyapunov functions. (b) Derivative of Lyapunov functions.

https://doi.org/10.1371/journal.pone.0288148.g008

Remark 5.6. In the above examples we restricted the factorizations to integer exponents so that we have real analytic transformations. However, the underlying dynamics is not changed through the factorizations and real analyticity is not directly used in the investigation of the ISS-Lyapunov function (45). Thus, as long as the factored is piecewise locally Lipschitz (which holds after an arbitrarily short time in virtue of Remark 4.4), we can generalize (67) for other values as well; to be precise, we can use any and real numbers.

Next, focusing on the Hill kinetics in (71), we note that while the denominator of the transformation in (71) cannot be factorized we can rearrange the transformation as (76) where choosing 0 < a_i ≤ 3 and 0 ≤ b_i ≤ a_i ensures that the time-varying coefficient functions are piecewise locally Lipschitz. In this case the exact value of the integral in (45) involves the generalized hypergeometric function and generally cannot be expressed in a closed form. However, in some special cases (such as b_i = 2 above) we can calculate the integral explicitly; for example setting a_i = 1.5 and b_i = 0.5 yields (77)

Example 4: Competition for ribosomes in the cell.

In this example we introduce a set of generalized ribosome flows connected by a finite pool of ribosomes to model competition in the cell. We follow [14], where the authors introduced a model for simultaneous translation and [16], where the authors generalized the model to include premature drop-off and attachment effects modeled with Langmuir kinetics. We will focus on the latter case and show that with a slight modification it can be formalized as a generalized ribosome flow model with a clear and natural compartmental interpretation. This demonstrates the usefulness and modeling power of generalized ribosome flows as one can prove various properties of many existing models of different conceptual levels. Moreover, our results show that many qualitative properties of the system carry over to more general settings, e.g. when the translation, drop-off and attachment rates are modeled with more sophisticated functions or when some (or all) rates are time-dependent.

For the sake of simplicity we will present this example in the reduced state space. Let us consider N mRNAs consisting of m₁, m₂, …, m_N number of sites. Let denote the continuous amount of ribosomes in the ith site of the j mRNA stand and let denote its capacity. Let c_z denote the capacity of the pool and n_z denote the amount of ribosomes in the pool. For the sake of notational simplicity let and also denote n_z and similarly for the capacities. Let the translation rate functions from the ith site the to (i + 1)th site on the jth mRNA be denoted as . Finally, let the detachment and attachment rates at the ith site of the jth mRNA be denoted respectively as and . The attachment rate to the first site and the detachment rate from the last site will be called initiation rate and production rate, respectively. Then the dynamics of the model is given by: (78)

Thus, indeed, simultaneous translation with a finite pool can be described by a generalized ribosome flow. Clearly the following function defines a linear first integral and is a crucial factor in the dynamical analysis of the system.

Remark 5.7. In [16] the authors consider the following special case:

• the capacity of each site is one; that is, each ,
• the translation rate are time-invariant and obey the mass-action law; that is, each for some ,
• the initiation and attachment rates are time-invariant and are given by for some and G_j(z) continuously differentiable strictly increasing function with G_j(0) = 0,
• the drop-off rates are time-invariant and are given by for some .

Since the drop-off rates are donor controlled the pool does not have a predefined capacity and the amount of ribosomes in the pool are only bounded by H(n(0)). Therefore, this special case does not fit in our compartmental framework (although, as most of our results are a consequence of the linear first integral combined with the cooperativity of the system they can be generalized to include donor controlled terms as well). It is assumed that the authors consider this case to capture the fact that the capacity of the pool might be several orders higher than the actual number of ribosomes, and thus the dependence on the available space in the pool may be negligible. However, some physical meaning is lost with this assumption and it might in fact lead to less precise simulations.

To see this, let us consider a network with N = 10 mRNAs with m = 5 sites. For the sake of simplicity let for each i and j, and assume that there are no premature drop-offs and attachments. We consider initation rates G_j(z) = z, G_j(z) = tanh(z) and G_j(z) = z² and set c_z = 10⁴. Since the equilibrium is unique on the level sets of the first integral we set each and we only change n_z(0). Fig 9 shows the ratio of the steady state of the pool in the case of donor controlled and mass-action production rates as we increase the ratio from 5 ⋅ 10⁻² to 1. As expected, the steady state ratio is close to one for saturating rate functions and for n_z(0) ≪ c_z. However, the ratio can get higher when the total number of ribosomes have the same magnitude as the capacity; that is, the inaccuracy of the donor controlled kinetics increases. While this assumption might be valid for realistic parameters of ribosome flows in other TASEP based flow models (especially with non-saturating kinetics) it might be crucial to model these transitions accurately.

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Fig 9. Steady state ratio of the donor controlled and the mass-action production rate for various initiation rates as a function of the ratio of the total number of ribosomes and the capacity of the pool.

https://doi.org/10.1371/journal.pone.0288148.g009

Effect of the total number of ribosomes. In the next simulation we follow [16] and we consider a single mRNA strand with m₁ = 3 sites. The initiation rate is set to while the attachment rates are and . The drop-off rates and production rate are set as , , . We assume that the translation rates obey the mass-action law with each . We set the initial values to and n₀(z) = c_z as before. Fig 10 shows the steady state of the system as we increase c_z from 0 to 5 for various rate functions. One can see that in each case the mRNA saturates as we increase the number of ribosomes and the rest of the ribosomes are accumulated in the pool. Finally, the same effect as in Fig 9 can be observed; that is, the donor controlled detachment rates shift the steady state of the pool to higher values.

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Fig 10. Steady state of a single mRNA strand in a pool modeled with an RFM and a GRFM with mass-action translation rate and drop-off rates, and attachment rate corresponding to different G₁(z) functions.

(a) RFM, G₁(z) = z. (b) RFM, G₁(z) = tanh(z). (c) RFM, . (d) GRFM, G₁(z) = z. (e) GRFM, G₁(z) = tanh(z). (f) GRFM, .

https://doi.org/10.1371/journal.pone.0288148.g010

We again emphasize the versatility of generalized ribosome flows as the initiation, translation, production, attachment and detachment rate function can be different on each site. For example let us consider a particular mRNA strand with saturating initation and attachment rates given by , with mass-action translation rates and with production and drop-off rates given by . Fig 11 shows evolution of the steady states as we increase n_z(0) = c_z as before. As expected the steady states of the mRNA sites are moved to lower values.

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Fig 11. Steady state of a single mRNA strand in a pool modeled with a GRFM with mass-action translation rate, rational fraction drop-off rates, and saturating attachment rates.

https://doi.org/10.1371/journal.pone.0288148.g011