A Structural Classification of Candidate Oscillatory and Multistationary Biochemical Systems

Abstract

Molecular systems are uncertain: The variability of reaction parameters and the presence of unknown interactions can weaken the predictive capacity of solid mathematical models. However, strong conclusions on the admissible dynamic behaviors of a model can often be achieved without detailed knowledge of its specific parameters. In systems with a sign-definite Jacobian, for instance, cycle-based criteria related to the famous Thomas’ conjectures have been largely used to characterize oscillatory and multistationary dynamic outcomes. We build on the rich literature focused on the identification of potential oscillatory and multistationary behaviors using parameter-free criteria. We propose a classification for sign-definite non-autocatalytic biochemical networks, which summarizes several existing results in the literature. We call weak (strong) candidate oscillators systems which can possibly (exclusively) transition to instability due to the presence of a complex pair of eigenvalues, while we call weak (strong) candidate multistationary systems those which can possibly (exclusively) transition to instability due to the presence of a real eigenvalue. For each category, we provide a characterization based on the exclusive or simultaneous presence of positive and negative cycles in the associated sign graph. Most realistic examples of biochemical networks fall in the gray area of systems in which both positive and negative cycles are present: Therefore, both oscillatory and bistable behaviors are in principle possible. However, many canonical example circuits exhibiting oscillations or bistability fall in the categories of strong candidate oscillators/multistationary systems, in agreement with our results.

Appendices

Appendix 1: Non-dimensionalization and Boundedness of the Two-Gene Network

We will carry out the non-dimensionalization procedure for the toggle switch network, leaving the derivation for the other cases to the reader. We follow non-dimensionalization steps similar to those proposed in Elowitz and Leibler (2000); Franco et al. (2011) and Kim and Winfree (2011). Consider the simple (dimensional) model:

$$\begin{aligned}&\tau \,\dot{R}_1= c_{1}+a_1\frac{1}{K_{M1}^{n} + P_2^{n}} - R_1, \quad \dot{P}_1=k_p R_1 - k_d P_1, \end{aligned}$$

(13a)

$$\begin{aligned}&\tau \,\dot{R}_2 =c_{2}+a_2\frac{1}{K_{M2}^{n} + P_1^{n}} - R_2, \quad \dot{P}_2 =k_p R_2 - k_d P_2. \end{aligned}$$

(13b)

Here $c_{i}$ is the “leak” transcription of RNA. For simplicity, we assume that the translation and degradation rates for the proteins are the same. Constant $\tau$ is the mRNA half-life in the system. Constants $K_{Mi}$ represent the number of proteins necessary to half-maximally repress $R_i$. Finally, assume the translation efficiency of each RNA species is given by $\bar{p}_i$, which corresponds to the average number of proteins produced by a single RNA molecule. We define the non-dimensional variables: $r_i={R_i}/{\bar{p}_i}$, $p_i={P_i}/{K_{Mj}}$, $(i,j)\,\in \,\{(1,2),(2,1)\}$. We rescale time as $\tilde{t} = t / \tau$, and also define the non-dimensional parameters:

$$\begin{aligned}&\gamma _i=\frac{c_i}{\bar{p}_i},\quad \alpha _i=\frac{a_i}{\bar{p}_i\,K_{Mi}^n},\quad \beta _i=\frac{k_p\,\bar{p}_i}{k_d\,K_{Mj}},\quad T=\frac{1}{\tau k_d}. \end{aligned}$$

The resulting non-dimensional equations are:

$$\begin{aligned}&\dot{r}_1= \gamma _1+\alpha _1\frac{1}{1+p_2^{n}} - r_1, \quad T \dot{p}_1=\beta _1 r_1 - p_1, \end{aligned}$$

(14a)

$$\begin{aligned}&\dot{r}_2 =\gamma _2+\alpha _2\frac{1}{1+p_1^{n}} - r_2, \quad T \dot{p}_2 =\beta _2 r_2 - p_2, \end{aligned}$$

(14b)

Finally, if we assume $T\approx 1$, we get a system in the same form as Eq. (1).

We also report a brief evidence of the global boundedness of the two-gene system trajectories. The system expressed by Eq. (1) can be rewritten as

$$\begin{aligned} \left[ \begin{array}{c} \dot{r}_1 \\ \dot{p}_1 \\ \dot{r}_2 \\ \dot{p}_2 \end{array} \right]&= \left[ \begin{array}{cccc} ~-1~ &{} ~0~ &{} ~0~ &{} ~0~ \\ \beta _1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} -1 &{} 0 \\ 0 &{} 0 &{} \beta _2 &{} -1 \end{array} \right] \, \left[ \begin{array}{c} r_1 \\ p_1 \\ r_2 \\ p_2 \end{array} \right] + \left[ \begin{array}{c} \gamma _1 + H_1(p_2) \\ 0 \\ \gamma _2 + H_2(p_1) \\ 0 \end{array} \right] \\&= \tilde{M} \left[ \begin{array}{c} r_1 \\ p_1 \\ r_2 \\ p_2 \end{array} \right] + \tilde{H}(p_1, p_2). \end{aligned}$$

$\tilde{M}$ is an asymptotically stable matrix and $\tilde{H}(p_1, p_2)$ is a bounded quantity ($\Vert \tilde{H}(p_1, p_2) \Vert \le \eta$), because $\gamma _i$ are constants and $H_i(p_j)$ are Hill functions. The right-hand side of the equation is given by the sum of a linear term and a globally bounded nonlinear term. If we neglect the nonlinear part, we achieve a linear system which is asymptotically stable and admits a global quadratic Lyapunov function $V(x) = x^\top P x$, where $\tilde{M}^\top P + P \tilde{M} = - Q$, with $Q$ positive definite. Hence, all the system solutions will be globally ultimately bounded in a set of the form $x^\top P x \le k$, for some $k>0$ which depends on $P$, $Q$ and $\eta$ (see Khalil 2002 Section 5.2 and Blanchini and Miani 2008 Section 4.4.1).

Appendix 2: Critical Systems

In the main text, we have considered non-critical systems in order to provide unified results. Here we discuss our results and examine the validity of our characterization for possibly critical systems. Note that Propositions 3 and 4, concerning candidate multistationary systems, hold true for critical systems as well. The assumption comes into play when we consider the characterization of candidate oscillatory systems.

Validity of Proposition 1

For critical networks, the sufficiency part of Proposition 1 does not hold: The existence of negative cycles of order two does not assure that the system can oscillate. For example, consider the following system:

$$\begin{aligned} \dot{x}_1&=- b_1x_1 + H_{12}(x_2),\nonumber \\ \dot{x}_2&=- b_2x_2 + H_{21}(x_1), \end{aligned}$$

(15)

where $b_1$ and $b_2$ are positive constants, and $H_{ij}$ are positive, bounded, and monotone functions. To ensure the existence of a negative cycle, we assume that (for example) $H_{12}$ is non-increasing, while $H_{21}$ is non-decreasing. (We can take $H_{ij}$ as Hill functions.) For any choice of $b_i$ and $H_{ij}$, the system has a single, asymptotically stable equilibrium point. However, oscillations are possible if we take $b_1=b_2=0$, i.e., if we remove our assumption that negative self-loops must exist at each node.

The necessity part of Proposition 1 still holds, because if there are no negative cycles the system cannot have equilibria with oscillatory instability.

Validity of Proposition 2

In the presence of negative cycles only, instability must be oscillatory. Thus, Proposition 2 still holds. However, some critical systems cannot be destabilized at all, as just shown in example (15).

It is legitimate to ask whether a critical system with only negative cycles (all of order two) is necessarily stable. This would extend Proposition 1 as follows:

Conjecture

A system is a candidate oscillator in the weak sense if and only if it has at least a negative cycle of order greater than two.

While the “if” part of this conjecture is true, unfortunately the the “only if” part is false. This can be seen by counterexample. We modify example (15) by introducing a new variable $x_3$ and a positive cycle:

$$\begin{aligned} \dot{x}_1&= - b_{1}x_1 + H_{12}(x_2) \\ \dot{x}_2&= - b_{2}x_2 + H_{21}(x_1) + \mu x_3 \\ \dot{x}_3&= - b_{3}x_3 + {\nu x_2.} \end{aligned}$$

The system Jacobian is:

$$\begin{aligned} J = \left[ \begin{array}{ccc} -b_{1} &{} -\delta &{} 0 \\ \gamma &{} - b_{2} &{} \mu \\ 0 &{} \nu &{} - b_{3} \end{array} \right] \end{aligned}$$

This matrix may well have unstable complex eigenvalues. Let us assume $b_{1}= b_{2} = 0$. The characteristic polynomial is:

$$\begin{aligned} s^3 + b_{3}s^2 + (\gamma \delta -\mu \nu )s + \gamma \delta b_{3} \end{aligned}$$

Building the Routh–Hurwitz table for this polynomial, one can check that for $\mu ,\nu >0$ two roots are complex with positive real parts. For $b_{1}, b_{2} >0$ small enough, the property is preserved.

A property relevant to our discussion on critical cases is the following known result (see for instance Edelstein-Keshet 2005, Chapter 6.5):

Proposition 5

If all the self-loops are negative, and all the other cycles are negative and of order two, then any equilibrium of the system is stable.

Appendix 3: Systems Affected by Delays

Many molecular systems have been successfully modeled using delayed differential equations. A notable example is given by gene networks, where the transport of RNA and proteins across cellular membranes are well captured by explicit delays (Hori , Hara 2011; Lewis 2003). Delay differential equations are infinite dimensional systems and a formal, exhaustive treatment of this case would require a more sophisticated setup. However, we can show that local, cycle-based sufficient conditions for OTIs and RTIs can be stated in a wide class of systems with delays. Consider the delay differential equation model

$$\begin{aligned} \dot{x}(t) = f(x(t),x(t-\tau _1),x(t-\tau _2), \dots , x(t-\tau _M)). \end{aligned}$$

(16)

Under standard differentiability assumptions, the corresponding linearized system around an equilibrium point is

$$\begin{aligned} \dot{\xi }(t) = A_0 \xi (t) + \sum _{k=1}^M~A_k \xi (t-\tau _k). \end{aligned}$$

(17)

For simplicity assume that there are no delayed self-loops, i.e., that the matrices $A_k$ have zero diagonal entries for $k \ge 1$.

It is well known that the stability of the system above can be established by inspecting the roots of this equation:

$$\begin{aligned} \det \left[ A_0 + \sum _{k=1}^M~A_k e^{-\tau _k s}-sI\right] =0. \end{aligned}$$

(18)

Stability is ensured if the roots have negative real parts. We now consider the following auxiliary system, in which all delays are (fictitiously) set to $0$:

$$\begin{aligned} \dot{\zeta }(t) = \left[ A_0+\sum _{k=1}^M~A_k\right] \zeta (t) = \bar{A} \zeta (t). \end{aligned}$$

(19)

Matrix $\bar{A}$ is the same Jacobian we would obtain by setting the delays $\tau _i =0$ in system (16):

$$\begin{aligned} \dot{x}(t) = f(x(t),x(t),x(t), \dots , x(t)). \end{aligned}$$

Note that equilibria are delay independent, so the delay-free system above has the same equilibria as the delayed system (16).

As previously done in this paper, we can associate graphs with systems (17) and (19), where nodes correspond to species concentrations and signed, directed arcs correspond to the dynamic interactions among species (delayed or not), defined by matrices $A_k$, $k=0,\ldots ,M$. Delays do not change the sign of the cycles. Therefore, any positive/negative cycle of (17) corresponds to a positive/negative cycle of (19).

Proposition 6

Assume that system (17) has only negative cycles. Then, the system admits solely OTIs.

Proof

Ab absurdo, assume that the system admits a real transition to instability, and thus, one root of Eq. (18) crosses the imaginary axis with value $s=0$. This is equivalent to writing Eq. (18) as

$$\begin{aligned} \det \left[ A_0+\sum _{k=1}^M~A_k \right] =0. \end{aligned}$$

If this is true, then also the auxiliary system (19) must admit a zero eigenvalue. But this is impossible if the system has only negative cycles, according to Proposition 2. $\square$

Proposition 7

Assume that system (17) has only positive cycles. Then, the system admits solely RTIs.

Proof

To prove this proposition, we need two observations:

• (a) In the absence of negative cycles (with the exception of self-loops), system (17) is a linear positive system with delay. Therefore, matrix $\bar{A}$ in system (19) is a Metzler matrix.

• (b) We invoke a well-known property of positive linear systems with delay (with no delayed self-loops). System (18) is stable if and only if its delay-free, auxiliary counterpart (19) is stable (Haddad and Chellaboina 2004; Liu et al. 2010) (see also Wang et al. 2008, Theorem 6.5).

If all its cycles are positive, the auxiliary system (19) can transition to instability only by means of a pole in $s=0$. Again ab absurdo let us assume that the delay system (18) admits instability with a pair of dominant imaginary eigenvalues and no real eigenvalue crossing the origin. Then, the auxiliary system (19) would also be unstable. However, the dominant eigenvalue of the Metzler matrix $\bar{A}$ is real; therefore, the auxiliary system (19) would transition to instability with a pole at $s=0$. But this would also imply that Eq. (18) is satisfied with $s=0$, which is a contradiction. Hence, we conclude that if all its cycles are positive, a transition to instability with a pair of imaginary eigenvalues is impossible for system (17). $\square$

As a comment to the above proposition, we stress that equilibria are delay independent; therefore, if the auxiliary subsystem presents multiple equilibria, so does the delayed system.

We conclude this appendix by noting that Proposition 5 is not valid in the presence of delays. Indeed, the presence of a delay in a “second order negative cycle” may compromise stability, as it is well known in control theory.

Appendix 4: Structural Cross-Constraints Among Functions

We have not considered models presenting cross-linked dynamic terms in several equations. Cross-linked terms appear typically in models built using the mass-action kinetics formalism. For instance, consider the reactions

where $\emptyset$ indicates elimination of a species from the system (degradation or out-flow). Indicating the concentration of a species with the corresponding small letter, the differential equation model is:

$$\begin{aligned} \dot{x}&= x_0 - g_X(x) -pg_{XZ}(x,z) \end{aligned}$$

(20)

$$\begin{aligned} \dot{y}&= g_X(x) -g_{Y}(y) \end{aligned}$$

(21)

$$\begin{aligned} \dot{z}&= g_Y(y) -g_{XZ}(x,z). \end{aligned}$$

(22)

Here, all the reaction rates are increasing functions. Identical dynamic terms appear in the three equations, so the Jacobian has dependent entries:

$$\begin{aligned} J = \left[ \begin{array}{ccc} -(\alpha + p\mu ) &{} 0 &{} -p\nu \\ ~\alpha &{} -\beta &{} ~0 \\ -\mu &{} ~\beta &{} - \nu \end{array} \right] , \end{aligned}$$

where, for a fixed equilibrium, all Greek letters represent positive constants. This structure presents both negative and positive cycles, but it may only undergo oscillatory transitions to instability. Thus, it is a candidate oscillator in the strong sense. This fact can be seen by writing the characteristic polynomial:

$$\begin{aligned} p(s) = (s+\alpha ) (s+\beta ) (s+\nu ) + \mu p (s+\beta ) s +p\nu \alpha \beta , \end{aligned}$$

where all coefficients are positive, and thus, there cannot be real positive roots.

This would seem in contradiction with the presence of the positive cycle $1 \rightarrow 3 \rightarrow 1$. However, our results are not invalidated because we do not consider cross-constraints among entries in the Jacobian, such as the fact that $J_{22} = - J_{32}$. Clearly, if we could change all entries independently (without changing sign), this system could present a real transition to instability.

For systems with cross-constrained dynamics, we believe that algorithmic/numerical methods are the best approach to discriminate admissible transitions to instability.