Oscillations and bistability in a model of ERK regulation

Abstract

This work concerns the question of how two important dynamical properties, oscillations and bistability, emerge in an important biological signaling network. Specifically, we consider a model for dual-site phosphorylation and dephosphorylation of extracellular signal-regulated kinase (ERK). We prove that oscillations persist even as the model is greatly simplified (reactions are made irreversible and intermediates are removed). Bistability, however, is much less robust—this property is lost when intermediates are removed or even when all reactions are made irreversible. Moreover, bistability is characterized by the presence of two reversible, catalytic reactions: as other reactions are made irreversible, bistability persists as long as one or both of the specified reactions is preserved. Finally, we investigate the maximum number of steady states, aided by a network’s “mixed volume” (a concept from convex geometry). Taken together, our results shed light on the question of how oscillations and bistability emerge from a limiting network of the ERK network—namely, the fully processive dual-site network—which is known to be globally stable and therefore lack both oscillations and bistability. Our proofs are enabled by a Hopf bifurcation criterion due to Yang, analyses of Newton polytopes arising from Hurwitz determinants, and recent characterizations of multistationarity for networks having a steady-state parametrization.

Appendices

Files in the Supporting Information

Table 5 lists the files in the Supporting Information, and the result/proof each file supports. All files can be found at the online repository: https://github.com/neeedz/ERK.

Table 5 Supporting Information files and the results they support

Newton-polytope method

Here we show how analyzing the Newton polytopes of two polynomials can reveal whether there is a positive point at which one polynomial is positive and simultaneously the other is zero (Proposition 10 and Algorithm 1). In Appendix C, we show how we used this approach, which we call the Newton-polytope method, to find a Hopf bifurcation leading to oscillations in the reduced ERK network (in Theorem 6).

Notation 13

Consider a polynomial $f = b_1 x^{\sigma _1} + b_2 x^{\sigma _2} + \dots + b_{\ell } x^{\sigma _{\ell }} \in {\mathbb {R}}[x_1,x_2,\dots , x_s]$, where the exponent vectors $\sigma _i \in {\mathbb {Z}}_{\ge 0}^s$ are distinct and $b_i \ne 0$ for all i. A vertex $\sigma _i$ of $\mathrm {Newt}(f)$, the Newton polytope of f, is a positive vertex (respectively, negative vertex) if the corresponding monomial of f is positive, i.e., $b_i >0$ (respectively, $b_i<0$). Also, $N_{f}(\sigma )$ denotes the outer normal cone of the vertex ${\sigma }$ of $\mathrm {Newt}(f)$, i.e., the cone generated by the outer normal vectors to all supporting hyperplanes of $\mathrm {Newt}(f)$ containing the vertex $\sigma$. Finally, for a cone C, let $\mathrm {int}(C)$ denote the relative interior of the cone.

For an extensive discussion on polytopes and normal cones, see the book of Ziegler (1995).

Proposition 10

Let $f,g \in {\mathbb {R}}[x_1,x_2,\ldots x_s]$. Assume that $\alpha$ is a positive vertex of $\mathrm{Newt}(f)$, $\beta _+$ is a positive vertex of $\mathrm{Newt}(g)$, and $\beta _-$ is a negative vertex of $\mathrm{Newt}(g)$. Then, if $\mathrm {int} (N_{f}(\alpha )) \cap \mathrm {int} (N_g(\beta _+))$ and $\mathrm {int}(N_{f}(\alpha )) \cap \mathrm {int} (N_g(\beta _-))$ are both nonempty, then there exists $x^* \in {\mathbb {R}}^s_{>0}$ such that $f(x^*)>0$ and $g(x^*)=0$.

To prove Proposition 10 we use the following well-known lemma and its proof.

Lemma 2

For a real, multivariate polynomial

$$\begin{aligned} f = b_1 x^{\sigma _1} + b_2 x^{\sigma _2} + \dots + b_{\ell } x^{\sigma _{\ell }} \in {\mathbb {R}}[x_1,x_2,\dots , x_s], \end{aligned}$$

if $\sigma _i$ is a positive vertex (respectively, negative vertex) of $\mathrm{Newt}(f)$, then there exists $x^* \in {\mathbb {R}}^s_{>0}$ such that $f(x^*)>0$ (respectively, $f(x^*)<0$).

Proof

Let $\sigma _i$ be a vertex of $\mathrm{Newt}(f)$. Pick $w=(w_1,w_2,\ldots ,w_s)$ in the relative interior of the outer normal cone $N_f(\sigma _i)$, which exists because $\sigma _i$ is a vertex. Then, by construction, the linear functional $\langle w, - \rangle$ is maximized over the exponent-vectors $\sigma _1, \sigma _2, \dots , \sigma _{\ell }$ at $\sigma _i$. Thus, we have the following univariate “polynomial with real exponents” in t:

$$\begin{aligned}&f(t^{w_1},t^{w_2},\ldots ,t^{w_s}) ~=~ b_1 t^{\langle w, \sigma _1\rangle } + b_2 t^{\langle w, \sigma _1\rangle } + \dots + b_{\ell } t^{\langle w, \sigma _{\ell } \rangle } ~=~ b_i t^{\langle w, \sigma _i \rangle } \\&+ \text {lower-order terms}~. \end{aligned}$$

So, for t large, $\text {sign}(f(t^{w_1},t^{w_2},\ldots ,t^{w_s}))=\text {sign}(b_i)$. Note that $(t^{w_1},t^{w_2},\ldots ,t^{w_s})\in {\mathbb {R}}^s_{>0}$. $\square$

Our proof of Proposition 10 is constructive, through the following algorithm, where we use the notation $f_w(t):=f(t^{w_1}, t^{w_2},\ldots ,t^{w_s})$, for $t\in {\mathbb {R}}$ and $w=(w_1, w_2, \dots , w_s)\in {\mathbb {R}}^s$.

Proof of Proposition 10

Let $a_+x^{\alpha }$ be the term of f corresponding to the vertex $\alpha$ of $\mathrm {Newt}(f)$, and similarly let $b_+x^{\beta _+}$ (respectively, $b_-x^{\beta _-}$) be the term of g corresponding to the vertex $\beta _+$ (respectively, $\beta _-$) of $\mathrm {Newt}(g)$. Thus, $a_+>0$, $b_+>0$, and $b_-<0$. Let $\{a_1,a_2, \ldots ,a_{d}\}\subseteq {\mathbb {R}}$ denote the remaining set of coefficients of f, so that $f= a_+ x^{\alpha } + (a_1 x^{\sigma _1}+ a_2 x^{\sigma _2}+ \dots + a_d x^{\sigma _d})$, for some exponent vectors $\sigma _i \in {\mathbb {Z}}_{\ge 0}^s$.

Algorithm 1 terminates: First, $\ell$ and m in line 2 exist by hypothesis. Also, $\tau _{\ell }$ and $\tau _m$ in lines 4–5 exist by the proof of Lemma 2 and by construction. Next, $\min h(r)$ in line 8 exists because h is a continuous univariate function defined on a compact interval.

By construction and because cones are convex, the vector $r \ell +(1-r) m$, which is a convex combination of $\ell$ and m, is in the relative interior of $N_{f}(\alpha )$ for all $r\in [0,1]$. Thus, $\langle r \ell + (1-r) m, ~\alpha - \sigma _i \rangle >0$ for all $i=1,2,\dots , d$ and for all $r \in [0,1]$. This (together with a straightforward argument using continuity and compactness) implies the following:

$$\begin{aligned} \delta ~:=~ \inf _{r \in [0,1]} \min _{i=1,2,\dots , d} \langle r \ell + (1-r) m, ~ \alpha - \sigma _i \rangle ~>~0. \end{aligned}$$

Next, let $\beta := \inf _{r\in [0,1]} \langle r \ell + (1-r) m, ~ \alpha \rangle$. Then, for all $r \in [0,1]$ and $t>0$,

$$\begin{aligned} \nonumber f_{r \ell +(1-r) m}(t) ~&=~ a_+ t^{\langle r \ell +(1-r) m), \alpha \rangle } + \left( a_1 t^{\langle r \ell +(1-r) m), \sigma _1 \rangle } + \dots + a_d t^{\langle r \ell +(1-r) m), \sigma _d \rangle } \right) \\ ~&>~ a_+ t^{\beta } - (|a_1|+|a_2|+ \dots + |a_d|) t^{\beta - \delta } ~=:~ {\widetilde{f}}(t)~. \end{aligned}$$

(29)

In ${\widetilde{f}}(t)$, the term $a_+ t^{\beta }$ dominates the other term, for t large, so there exists $T^*>0$ such that ${\widetilde{f}}(t)\ge 0$ when $t \ge T^*$. So, by (29), the while loop in line 8 ends when $T\ge T^*$ (or earlier).

Algorithm 1 is correct: For T fixed, the minimum of $\psi (r):=\left( g(T^{r\ell +(1-r)m})\right) ^2$ over the compact set [0, 1] is attained, because $\psi$ is continuous. Next we show that this minimum value is 0, or equivalently that for $\chi (r) := g(T^{r \ell +(1-r)m})$ there exists some $r^*\in (0,1)$ such that $\chi (r^*) =0$. Indeed, this follows from the Intermediate Value Theorem, because $\chi$ is continuous, $\chi (0) =g(T^m)<0$ (because $T > \tau _m$), and $\chi (1) =g(T^{\ell })>0$ (because $T>\tau _{\ell })$.

Finally, the inequality $f(T^{r^*\ell +(1-r^*)m})>0$ holds by construction of T, so defining $x^*:=T^{r^*\ell +(1-r^*)m} \in {\mathbb {R}}^s_{>0}$ yields the desired vector satisfying $f(x^*)>0$ and $g(x^*)=0$. $\square$

Using the Newton-polytope method

Here we show how we used Algorithm 1 to find the Hopf bifurcation in Theorem 6. (For details, see the supplementary files reducedERK-hopf.mw and reducedERK-cones.sws). Recall from the proof of that theorem, that our goal was to find some $x^* \in {\mathbb {R}}^{10}_{>0}$ and ${\hat{\kappa }}^* = (k_{\mathrm {cat}}^*, k_{\mathrm {off}}^*, {\ell }_{\mathrm {off}}^*) \in {\mathbb {R}}^3_{>0}$ satisfying the following conditions from Proposition 5:

$$\begin{aligned} {\mathfrak {h}}_4 ({\hat{\kappa }}^*; x^*)>&0~, ~ {\mathfrak {h}}_5 ({\hat{\kappa }}^*; x^*) >0~, ~ {\mathfrak {h}}_6 ({\hat{\kappa }}^*; x^*) =0~, \nonumber \\&~\mathrm{and} ~ \frac{\partial }{\partial k_{\mathrm {cat}}} {\mathfrak {h}}_6 ({\hat{\kappa }}; x) |_{({\hat{\kappa }}; x)=({\hat{\kappa }}^*; x^*)} \ne 0~. \end{aligned}$$

(30)

Step One Specialize some of the parameters: set $k_{\mathrm{off}}=\ell _{\mathrm{off}}=1$ and $x_3=x_4=x_6 = x_7= x_9= x_{10} = 1$. (Otherwise, ${\mathfrak {h}}_5$ and ${\mathfrak {h}}_6$ are too large to be computed.)

Step Two Do a change of variables: let $y_i = 1/x_i$ for $i=1,2,5,8$. These variables $x_i$ were in the denominator, so switching to the variables $y_i$ yield polynomials.

Let ${\mathcal {H}}_4$, ${\mathcal {H}}_5$, and ${\mathcal {H}}_6$ denote the resulting polynomials in ${\mathbb {Q}}[k_{\mathrm {cat}}, y_1, y_2, y_5, y_8]$ after performing Steps One and Two. Accordingly, our updated goal is to find $(k_{\mathrm {cat}}^*, y^*_1, y^*_2, y^*_5, y^*_8) \in {\mathbb {R}}^{5}_{>0}$ at which ${\mathcal {H}}_4$ and ${\mathcal {H}}_5$ are positive and ${\mathcal {H}}_6$ is zero. (In a later step, we must also check the partial-derivative condition in (30).)

Step Three Apply (a straightforward generalization of) Algorithm 1 as follows.

• (i) Find a positive vertex of ${\mathcal {H}}_4$ and a positive vertex of ${\mathcal {H}}_5$ whose outer normal cones intersect (denote the intersection by C), and a positive vertex and a negative vertex of ${\mathcal {H}}_6$ (denote their outer normal cones by $D_+$ and $D_-$, respectively) for which:

  • (a) the intersection $D_+ \cap D_-$ is 4-dimensional, and

  • (b) the intersections $C \cap D_+$ and $C \cap D_-$ are both 5-dimensional.

• (ii) By Proposition 10, a vector $(k_{\mathrm {cat}}^*, y^*_1, y^*_2, y^*_5, y^*_8)$ that accomplishes our updated goal, is guaranteed. To find such a point, we follow Algorithm 1 to obtain $k_{\mathrm {cat}}^*=729,y_1^*\approx 16.79978292, y_2^*\approx 453.5941389, y_5^*\approx 6.587368051$, and $y_8^*\approx 80675.77181$.

Recall the specializations in Step One and change of variables in Step Two, to obtain ${\hat{\kappa }}=(729,1,1)$ and

$$\begin{aligned}&x^*~\approx ~ ( 0.05952457867,~ 0.002204614024,~ 1,~ 1,\\&0.1518056972,~ 1,~ 1,~ 0.00001239529511,~ 1,~ 1 )~. \end{aligned}$$

Step Four Verify that the conditions in (30) hold.