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Motivated by the lack of a suitable constructive framework for analyzing popular stochastic models of Systems Biology, we devise conditions for existence and uniqueness of solutions to certain jump stochastic differential equations (SDEs). Working from simple examples we find
*reasonable* and
*explicit* assumptions on the driving coefficients for the SDE representation to make sense. By “reasonable” we mean that stronger assumptions generally do not hold for systems of practical interest. In particular, we argue against the traditional use of global Lipschitz conditions and certain common growth restrictions. By “explicit”, finally, we like to highlight the fact that the various constants occurring among our assumptions
*all can be determined once the model is fixed*. We show how basic long time estimates and some limit results for perturbations can be derived in this setting such that these can be contrasted with the corresponding estimates from deterministic dynamics. The main complication is that the natural path-wise representation is generated by a counting measure with an intensity that depends nonlinearly on the state.

The observation that detailed modeling of biochemical processes inside living cells is a close to hopeless task is a strong argument in favor of stochastic models. Such models are often thought to be more accurate than conventional rate-diffusion laws, yet remain more manageable than, say, descriptions formed at the level of individual molecules. Indeed, several studies [

In this work we shall consider some “flow” properties of a stochastic dynamical system in the form of a quite general continuous-time Markov chain. Since the pioneering work of Gillespie [

In the general literature, when discussing existence/uniqueness and various types of perturbation results, different choices of assumptions with different trade-offs have been made. One finds that the treatment often falls into one of two categories taking either a “mathematical” or a “physical” viewpoint. Both of the conditions are highly general but with subsequently less transparent proofs and resulting in more abstract bounds. Or the conditions are formed out of convenience, say, involving global Lipschitz constants, and classical arguments carry through with only minor modifications.

Protter ( [

A study of the flow properties of jump SDEs is found in [

In a more applied context, stability is often thought of as implied from physical premises and the solution is tactically assumed to be confined inside some bounded region ( [

Evidently, essentially no systems of interest satisfy global Lipschitz assumptions since the fundamental inte- raction almost always takes the form of a quadratic term. Interestingly, for ordinary differential equations, it has been shown [

Besides its expository material, the purpose of this paper is to devise simple conditions that imply stability for finite and, in certain cases, infinite times, and that, when applied to systems of practical interest, yield explicit expressions for the associated stability estimates. As a result the framework developed herein applies in a constructive way to any chemical network, of arbitrary size and topology, formed by any combination of the elementary reactions (2.3) to be presented in Section 2. Additionally, it will be clear how to encompass also other types of nonlinear reactions that typically result from adiabatic simplifications.

As an argument in favor of this bottom-up approach one can note that, for evolutionary reasons, biochemical systems tend to operate close to critical points in phase-space where the efficiency is the highest. Clearly, for such dynamical systems, an analysis by analogy might be highly misleading.

We also like to argue that our results are of interest from the modeling point of view. Due to the type of phenomenological arguments often involved, judging the relative effect of the (non-probabilistic) epistemic uncer- tainty is a fundamental issue which has so far not rendered a consistent analysis.

The expository material in Section 2 is devoted to formulating the type of processes we are interested in. We state the master equation as well as the corresponding jump SDE and we also look at some simple, yet informative actual examples. Since it is expected that the properties of the stochastic dynamics are somehow similar to those of the deterministic version, we search for a set of minimal assumptions in the latter setting in Section 3. Techniques for finding explicit values of the constants occurring among our assumptions are also devised. The main results of the paper are found in Section 4 where we put our theory together and prove existence and uniqueness, as well as long time estimates and limit results for perturbations. A concluding discussion is found in Section 5.

In this section we start with the physicist’s traditional viewpoint of pure jump processes and write down the governing master equations. These are evolution equations for the probability densities of continuous-time Markov chains over a discrete state space. Although the application considered here is mesoscopic chemical kinetics, identical or very similar stochastic models are also used in Epidemiology [

We then proceed with discussing a path-wise representation in terms of a stochastic jump differential equation. The reason the sample path representation is interesting is the possibility to reason about flow properties and thus compare functionals of single trajectories. This is generally not possible with the master equation approach.

For later use we conclude the section by looking at some prototypical models. A simple analysis shows, somewhat surprisingly, that an innocent-looking example produces second moments that grow indefinitely.

We consider a chemical network consisting of D different chemical species interacting according to R prescribed reaction pathways. At any given time t, the state of the system is an integer vector

where

where

The physical premises leading to a description in the form of discrete transition laws (2.1) often imply the existence of a system size

with the names of the species in capitals. These propensities are generally scaled such that

The models we consider here all have states in the positive integer lattice and the assumption that no transition can yield a state outside

Assumption 2.1 (Conservation and stability). For all propensities,

To state the chemical master equation (CME), let for brevity

The convention of the transpose of the operator to the right of (2.4) is the standard mathematical formulation of Kolmogorov’s forward differential system ( [

such that the propensities in (2.1) can be retrieved,

Under assumptions to be prescribed in Section 4.1 it holds that the dynamics of the expected value of some time-independent unknown function

We now consider a path-wise representation for the stochastic process

In the present context of analyzing models in stochastic chemical kinetics, the path-wise jump SDE representation seems to have been first put to use in [

We thus assume the existence of a probability space

The transition probability (2.1) defines a counting process

The counting processes are obtained from the transition intensities (cf. (2.1))

where by

The marked counting measure ( [

where

Using that the point processes are independent and therefore have no common jump times ( [

such that the total intensity is given by

Put

such that

where

The jump SDE (2.11) can now be written in terms of a scalar counting random measure

Equation (2.16) expresses exponentially distributed reaction times that arrive according to a point process of intensity

One frequently decomposes (2.16) into its “drift” and “jump” parts,

The second term in (2.17) is driven by the compensated measure

In analytic work it is often necessary to “tame” the process by deriving results under a stopping time

Although there are many general versions of Itô’s change of variables formula available in the setting of semi- martingales (see for example [

where the sum is over jump times

Alternatively, decomposing (2.18) into drift- and jump parts and taking expectation values we get, since the compensated measure is a local martingale,

This is Dynkin’s formula ( [

When considering stability properties we will need to compare different trajectories with respect to the same noise. The details of this coupling are not defined in either (2.11) or (2.16) and must in fact be chosen explicitly. Since this equality is easy to inspect for a unit-rate Poisson process, the viewpoint of local time expressed in (2.10) provides an answer; two processes

A refinement of this construction was devised, also by Kurtz, in ( [

To obtain such a coupled version of (2.16) we will have to make the thinning dependent on both trajectories. This is achieved by firstly replacing the cumulative intensities in (2.12) with the base (or minimal) intensities

and use the new total base intensity

Secondly, we also define the remainder intensity,

In analogy with the previous construction we have the associated total intensity

with the non-symmetricity due to

As a concrete example of how this comparative thinning might be used, consider the following variant of (2.19),

For this specific example, the terms governed by the base counting measure

We mention also that an equivalent construction, but one that leads to different algorithms, can be obtained via a thinning of a single measure [

implying the total intensity

In analogy to (2.26) we get

This time, however, the intensity of the counting measure is generally larger and the equivalence is obtained as a result of the thinning procedure.

With this much formalism developed, we may conveniently quote the following result:

Theorem 2.1 ( [

then (2.7) is valid for

Since the governing Equation (2.7) for the expected value of

Corollary 2.2 Under the assumptions of Theorem 2.1, and if, moreover, in an arbitrary norm

then (2.7) is valid for

In stating these results we have suppressed the conditional dependency on the initial state which we for simplicity consider to be some non-random state

Consider the bi-molecular birth-death system,

that is, the system is in contact with a large reservoir such that

and the vector propensity function

where

For a state

which upon a moments consideration is just the same thing as the model

that is, a constant intensity discrete random walk process. An explicit solution is the difference between two independent Poisson distributions,

where

From time to time below we shall be concerned with the following closed version of (2.32), consisting of a single reversible reaction,

This is clearly a finite system since the number

and will prove to be a useful example in the stochastic setting since formally, all states in

while (2.36b) is represented by

These examples, while very simple to deal with, will provide good counterexamples in both Sections 3 and 4.

In this section we shall be concerned with the deterministic drift part of the dynamics (2.17). We are interested in techniques for judging the stability of the time-homogeneous ODE (2.2), the so-called reaction rate equations implied by the rates (2.1). Stability and continuity with respect to initial data are considered in Sections 3.1 and 3.2. The main motivation for this discussion stems from the observation that assumptions that do not hold in this very basic setting are unlikely to hold in the stochastic case. In Section 3.3, techniques for explicitly obtaining all our postulated constants are discussed. A good point in favor of taking the time to describe these techniques is that we have not found such a discussion elsewhere.

Initially we will consider states

Many stability proofs can be thought of as comparisons with relevant linear cases. This is the motivation for the well-known Grönwall’s inequality which we state in the following two versions.

Lemma 3.1. Suppose that

The same conclusion holds irrespective of the differentiability of

The most immediate way of comparing the growth of solutions to the ODE (2.2) to those of a linear ODE is to require that the norm of the driving function is bounded in terms of its argument;

since then by the triangle inequality,

where Grönwall’s inequality applies. Unfortunately, (3.3) is a too strict requirement for our applications.

Proposition 3.2 The bi-molecular birth-death system (2.32) does not satisfy (3.3).

Proof. We compute

The problem with the simple condition (3.3) is that it does not take the direction of growth into account; the offending quadratic propensity in (2.32) actually decreases the number of molecules. To deal with this, let

from which one deduces

where Grönwall’s inequality applies anew. The assumption (3.5) is weaker than (3.3) since the former implies the latter by the Cauchy-Schwarz inequality. Indeed, as in the proof of Proposition 3.2 it is readily checked that for the bi-molecular birth-death system (2.32), we get

Unfortunately, in the case of an infinite state space and strong dependencies between the species the assumption (3.5) is also often unrealistic.

Proposition 3.3 Neither (2.36a) nor (2.36b) admits a bound of the kind (3.5).

Proof. As in the proof of Proposition 3.2 we look at a ray

This negative result can perhaps best be appreciated as a kind of loss of information about the dependencies between the species in the functional form of the condition (3.5). The number of

A way around this limitation can be found provided that we leave the general case

It then follows that

Again, in view of Grönwall’s inequality Lemma 3.1, we tentatively require that

implying the bounded dynamics

We remark in passing that the criterion (3.7) is sharp in the sense that if the reversed inequality can be shown to be true, then the growth of solutions can be estimated from below.

Example 3.1 As a point in favor of this approach we compute for the bi-molecular birth-death system (2.32),

The chosen “outward” vector

Example 3.2 For the reversible system (2.36a), we have already noted that

Example 3.3 A slightly more involved model reads as follows:

This example has been constructed such that the quadratic reaction increases

This example hints at a general technique for obtaining suitable candidates for the weight vector

For well-posedness of the ODE (2.2) we also need continuity with respect to the initial data. We cannot ask for uniform Lipschitz continuity since

presumably with some growth restrictions on

where the form of

Theorem 3.4 Suppose that the ODE (2.2) satisfies (3.7) and (3.10) and that initial data

Proof. Combining (3.7) with Grönwall’s inequality we get the a priori estimate

Hence the (bounded) solution to

is readily found through its integrating factor. The order estimate is a consequence of the fact that

since the trajectory is continuous.

As briefly discussed by the end of Section 3.1, finding bounds on A and

Proposition 3.5 (linear case) Write a set of

and the matrix

Proof. The first assertion is immediate since the smallest such constant

For non-zero

Example 3.4 The simple special case in Proposition 3.5 is generally sharp except for when there are linear reactions affecting all species considered in the model. For example, in a one-dimensional state space, the single decay

Other than for those special examples, for the most important linear cases,

Proposition 3.6 (quadratic case). Write a general quadratic propensity as

Proof. Since

The indicated upper bound is derived from the fact that

Example 3.5 The most important quadratic cases are summarized in

Example 3.6 The bi-molecular birth-death model (2.32) admits the constants

obtain a slightly smaller constant

Example 3.7 As a highly prototypical example we consider the following natural extension of the bi- molecular birth-death model (2.32),

where in this example it is informative to consider the dependence on the system’s size

We now consider the properties of the stochastic jump SDE (2.16). For convenience we start by collecting all assumptions in Section 4.1. In the stochastic setting the requirements for existence and uniqueness are slightly stronger than in the deterministic case such that the one-sided bound (3.10) needs to be augmented with an

Reaction | Bound on |
---|---|

0 | |

Reaction | Bound on | |
---|---|---|

0 | 2 | |

unsigned version, implying essentially the assumption of at most quadratically growing propensities. We demonstrate that this assumption is reasonable by constructing a model involving cubic propensities and with unbounded second moments. On the positive side we show in Section 4.2 that the assumptions are strong enough to guarantee finite moments of any order during finite time intervals.

We prove existence and uniqueness of solutions to the jump SDE (2.16) in Section 4.3. A sufficient condition for the existence of asymptotic bounds of the pth order moment is given in Section 4.4 where we also derive some stability estimates.

We state formally the set of assumptions on the jump SDE (14) as follows.

Assumption 4.9 For arguments

(1)

(2)

(3)

(4)

The parameters

After the original draft of the current paper was finished, the author became aware of two other papers discussing very similar conditions [

In Assumption 4.9 (2) the case

are impossible. Similarly, when

are excluded.

Note that (2) and (4) are redundant in the sense that they are both implied by (3). However, as we saw in Section 3.3, in (4) it is often possible to find sharper constants

Assumption 4.1 (2) specifies the discussion to propensities with at most quadratic growth, at least when measured in the direction of the weight vector

Example 4.2 Consider the model

such that the stoichiometric vector is given by

Proposition 4.1 For the model in Example 4.2, if

Proof. Assume that both the second and the third moment are bounded for

such that the growth of the second moment remains bounded only provided that the third moment remains finite. It is convenient to look at the cumulative third order moment. From (2.7),

By the arithmetic-geometric mean inequality,

We put

which can be integrated and rearranged to produce the bound

Hence the third, and consequently also the second moment explode for some finite

Interestingly, we note that if

In this section we consider general moment bounds derived from (2.20) using localization. To get some guidance, let us first assume that the differential form of Dynkin’s formula (2.7) is valid. Since any trajectory

by Assumption 4.1 (1). Clearly, the differential form of Grönwall’s inequality in Lemma 3.1 applies here. A correct version of this argument unfortunately looses the sign of

Proposition 4.2 If Assumption 4.1 (1) is true, then

where

Here and below we shall make use of the stopping time

Proof. From (2.20) with

By the integral form of Grönwall’s inequality in Lemma 3.1 we deduce in terms of

such that the same bound holds for

We attempt a similar treatment for obtaining bounds in mean square. Assuming tactically that (2.7) is valid, writing

where

However, this tentative condition is often violated in practice since the second term

More realistic conditions arise when seeking to bound

Proposition 4.3 If for some constants

The proof of Proposition 4.3 follows the same pattern as for Proposition 4.2, but using this time

Proposition 4.4 Under Assumption 4.1 (1) and (2) the condition (4.10) of Proposition 4.3 is true with

Proof. This is straightforward: we get by the assumptions and Hölder’s inequality,

where an application of Young’s inequality yields the indicated bounds.

As a strong point in favor of our running assumptions we now demonstrate that the above reasoning can be generalized: these two conditions implies finite time stability in any order moment. We note that in a recent manuscript [

Theorem 4.5 (Moment estimate). Under Assumption 4.1 (1) and (2), for any integer

where

The proof of Theorem 4.5 and some later results will simplify using the following bound.

Lemma 4.6 Let

Proof. Both results follow from Taylor expansions;

respectively, where

Proof of Theorem 4.5. Using (2.20) with

Using Lemma 4.6 (4.12) and Assumption 4.1 (1) and (2) we obtain

where

for some constant

We shall now prove that the jump SDE (2.16) under Assumption 4.1 has a uniquely defined and locally bounded solution. To this end and following ( [

Theorem 4.7 (Existence). Let

Then if

requirement that

Proof. Below we let

Since the propensities are bounded for bounded arguments (Assumption 2.1), using the stopping time we find that the jump part is absolutely integrable and hence a local martingale

where

first that

Taking supremum and expectation values we get from Burkholder’s inequality ( [

Writing

By Grönwall’s inequality we have thus shown that

Next assume that

where, although there is now a dependency on

For the case that the initial data

Theorem 4.8 (Uniqueness) Let Assumption 4.1 (1)-(4) hold true. Then any two paths

We shall be using the observation that, for

Proof. Under the same stopping time as before we get from Itô’s formula using the coupling described in Section 2.2.2 that

From the integer inequality we find that there is a constant

Using that

In a certain sense the previous result is trivial; from the Poisson representation (2.8) we see that up to the first explosion, a path is uniquely determined from an initial state and a series of Poisson distributed events. However, and as we shall see below, the above proof is prototypical for more involved situations. An example would be when devising hybrid approximations in continuous state space. Indeed, in the above proof, note that if the integer inequality did not hold we would naturally have to rely on the Cauchy-Schwartz inequality instead. With

for which

Although Theorem 4.5 shows that any moments are bounded in finite time, a relevant question from the modeling point of view is whether the first few moments remain bounded indefinitely. We give a result to this effect which relies on the existence of solutions in

Theorem 4.9 (Ergodicity) Under Assumption 4.1 (1), (2), suppose that

Then for integer

Proof. The case

bound

by Taylor’s formula for some

For the first term in (4.23) we get from the scaled Young’s inequality with exponent

for some

Next by the arithmetic-geometric mean inequality we get

provided that we choose

Taken together we thus have

Since

for a certain new constant

We next aim at deriving some stability estimates with respect to perturbations in the reaction coefficients. An early account of this was given by Kurtz in [

We make this formal by requiring that

where

The existence of an absolute constant

with

The starting point for the analysis will be Itô’s formula under the coupling described in Section 2.2.2. The techniques used below generalize well to pth order moment estimates, but for ease of exposition we let

Hence under the model for coefficient perturbations (4.26)-(4.27) we have that

Theorem 4.10 (Continuity). Let two trajectories

Proof. We use the stopping time

where (4.28) was used. Simplifying further for a bounded

Using that

To get rid of the stopping time we write in terms of indicator functions,

Using

Relying on the existence result in Theorem 4.7 we find that for any given

As a by-product of the proof we see that if the process is bounded, then for

Corollary 4.11 (Perturbation estimate, bounded version) If in Theorem 4.10, the processes

The constant

For an unbounded system it is apparently much more difficult to obtain explicit estimates. However, by controlling also the martingale part we can strengthen Theorem 4.10 in another direction.

Theorem 4.12 (Continuity/sup) Under the same assumptions as Theorem 4.10 we have that

Proof. The quadratic variation of the martingale part in (4.29) can be bounded as

after using (4.28) and the integer inequality anew. For the drift part we may use the corresponding bound developed in the proof of Theorem 4.10. After taking supremum and expectation values of (4.29) and using Burkholder's inequality we therefore arrive at

by Grönwall’s inequality and using the notation

and the conclusion follows as before.

We have proposed a theoretical framework consisting of a priori assumptions and estimates for problems in stochastic chemical kinetics. The assumptions are strong enough to guarantee well-posedness for a large and physically relevant class of problems. Long time estimates and limit results for perturbations in rate constants have been studied to exemplify the theory. The assumptions are constructive in the sense that explicit techniques for obtaining all postulated constants have either been worked out in detail or at least indicated. We have seen that the case

In the course of motivating our setup we have seen that most problems do not admit global Lipschitz constants and that one-sided versions do not provide a better alternative. Another conclusion worth highlighting is that it pays off to consider jump SDEs in a fully discrete setting in that there are potential complications in proving uniqueness in continuous state space. A practical implication is that care should be exercised when forming continuous approximations to these types of jump SDEs.

For future work we intend to re-visit certain classical results from the perspective of the framework developed herein; for example, thermodynamic limit results, time discretization strategies, and quasi-steady state approximations―all of which have a practical impact in a range of applications.

The author likes to express his sincere gratitude to Takis Konstantopoulos for several fruitful and clarifying discussions. Early inputs on this work were also gratefully obtained from Henrik Hult, Ingemar Kaj, and Per Lötstedt.

This work was supported by the Swedish Research Council within the UPMARC Linnaeus center of Excellence.