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In distributive phosphorylation catalytic constants enable non-trivial dynamics

Carsten Conradi, Maya Mincheva

TL;DR

It is argued that in distributive double phosphorylation (sequential or distributive) the catalytic constants enable non-trivial dynamics and if the rate constant values in a network of cyclic distributive double phosphorylation are such that Hopf bifurcations and sustained oscillations can occur, then a network of sequential distributive double phosphorylation with the same rate constant values will show multistationarity.

Abstract

Ordered distributive double phosphorylation is a recurrent motif in intracellular signaling and control. It is either sequential (where the site phosphorylated last is dephosphorylated first) or cyclic (where the site phosphorylated first is dephosphorylated first). Sequential distributive double phosphorylation has been extensively studied and an inequality involving only the catalytic constants of kinase and phosphatase is known to be sufficient for multistationarity. As multistationarity is necessary for bistability it has been argued that these constants enable bistability. Here we show for cyclic distributive double phosphorylation that if its catalytic constants satisfy the very same inequality, then Hopf bifurcations and hence sustained oscillations can occur. Hence we argue that in distributive double phosphorylation (sequential or distributive) the catalytic constants enable non-trivial dynamics. In fact, if the rate constant values in a network of cyclic distributive double phosphorylation are such that Hopf bifurcations and sustained oscillations can occur, then a network of sequential distributive double phosphorylation with the same rate constant values will show multistationarity -- albeit for different values of the total concentrations. For cyclic distributive double phosphorylation we further describe a procedure to generate rate constant values where Hopf bifurcations and hence sustained oscillations can occur. This may, for example, allow for an efficient sampling of oscillatory regions in parameter space. Our analysis is greatly simplified by the fact that it is possible to reduce the network of cyclic distributive double phosphorylation to what we call a network with a single extreme ray. We summarize key properties of these networks.

In distributive phosphorylation catalytic constants enable non-trivial dynamics

TL;DR

It is argued that in distributive double phosphorylation (sequential or distributive) the catalytic constants enable non-trivial dynamics and if the rate constant values in a network of cyclic distributive double phosphorylation are such that Hopf bifurcations and sustained oscillations can occur, then a network of sequential distributive double phosphorylation with the same rate constant values will show multistationarity.

Abstract

Ordered distributive double phosphorylation is a recurrent motif in intracellular signaling and control. It is either sequential (where the site phosphorylated last is dephosphorylated first) or cyclic (where the site phosphorylated first is dephosphorylated first). Sequential distributive double phosphorylation has been extensively studied and an inequality involving only the catalytic constants of kinase and phosphatase is known to be sufficient for multistationarity. As multistationarity is necessary for bistability it has been argued that these constants enable bistability. Here we show for cyclic distributive double phosphorylation that if its catalytic constants satisfy the very same inequality, then Hopf bifurcations and hence sustained oscillations can occur. Hence we argue that in distributive double phosphorylation (sequential or distributive) the catalytic constants enable non-trivial dynamics. In fact, if the rate constant values in a network of cyclic distributive double phosphorylation are such that Hopf bifurcations and sustained oscillations can occur, then a network of sequential distributive double phosphorylation with the same rate constant values will show multistationarity -- albeit for different values of the total concentrations. For cyclic distributive double phosphorylation we further describe a procedure to generate rate constant values where Hopf bifurcations and hence sustained oscillations can occur. This may, for example, allow for an efficient sampling of oscillatory regions in parameter space. Our analysis is greatly simplified by the fact that it is possible to reduce the network of cyclic distributive double phosphorylation to what we call a network with a single extreme ray. We summarize key properties of these networks.
Paper Structure (23 sections, 14 theorems, 72 equations, 4 figures, 5 tables)

This paper contains 23 sections, 14 theorems, 72 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Biochemical reaction networks with mass action kinetics
  3. Reaction networks with $n$ species and $r$ reactions
  4. Steady states
  5. The Jacobian at steady state
  6. The Jacobian of networks with a single extreme vector
  7. Simple Hopf bifurcations in networks with a single extreme ray
  8. Analysis of networks of cyclic distributive double phosphorylation
  9. Steady states of network (\ref{['net:cyc_simple']})
  10. Simple Hopf bifurcations for network (\ref{['net:cyc_simple']})
  11. A procedure to locate simple Hopf bifurcations in network (\ref{['net:cyc_simple']})
  12. Lifting to the full network (\ref{['eq:MA_cyc']})
  13. Discussion
  14. Cyclic versus sequential distributive double phosphorylation
  15. Cyclic and distributive: emergence of oscillations
  16. ...and 8 more sections

Key Result

Corollary 2.2

Suppose the matrix $E$ consists of a single positive column vector. Let $J_{\lambda}(h)$ be as in (eq:Jac_E_one-D) with $\mathop{\mathrm{rank}}\nolimits(J_{\lambda}(h))=\mathop{\mathrm{rank}}\nolimits(J_1(h)) =s <n$ and let $a_i(\lambda,h)$, $b_i(h)$ be the coefficients of the characteristic polynom Moreover, the polynomial $\det(\mu I - J_{\lambda}(h))$ is given by the following formula:

Figures (4)

  • Figure 1: Reaction schemes describing distributive double phosphorylation of a protein $S$ at two binding sites by a kinase $K$ and a phosphatase $F$. Panel (\ref{['fig:scheme_cyc']}) cyclic double phosphorylation, panel (\ref{['fig:scheme_dis']}) distributive. In panel (\ref{['fig:scheme_cyc']}) the subscript $_{ij}$ denotes the state of the phosphorylation sites: $0$ unphosphorylated, $1$ phosphorylated (e.g. $S_{10}$ denotes those molecules of $S$, where the first site is phosphorylated and the second site is unphosphorylated). In panel (\ref{['fig:scheme_dis']}) the subscript $_i$ denotes the number of attached phosphate groups (e.g. $S_0$ denotes unphosphorylated protein).
  • Figure 2: Numerical verification of Hopf bifurcations (panel (\ref{['fig:S11_vs_c1']}), labeled $H$) and a limit cycle (panel (\ref{['fig:Sxx_vs_t']}) and (\ref{['fig:S11_vs_S00']})). Rate constants $k$ as in Table \ref{['tab:rc_cyc_exa']} with $\lambda=1$. Initial value $x(0)$ as in (\ref{['eq:h_x0_exa']}) -- apart from $x_3(0)$ and $x_6(0)$: to obtain an initial value near the steady state $x$ given in (\ref{['eq:h_x0_exa']}) we choose $x_3(0) = 1.1\cdot\frac{1}{15}$ and $x_6(0)=0.9\cdot\frac{1}{15}$).
  • Figure 3: Simulation of network (\ref{['eq:MA_cyc']}) for $\kappa_2 = \kappa_5 = \kappa_8 = \kappa_{11} = k_b$ and different values $k_b$ (ODEs have been solved with ode15s (Mathworks) for $x(0)$ as in the table of panel (\ref{['tab:ini_on_orbit']}) and $\kappa_i$, $c_i$ as in Table \ref{['tab:rc_cyc_exa']} (on page \ref{['tab:rc_cyc_exa']} with $\lambda=1$). Panel (\ref{['fig:small_kb']}): $k_b=0$ corresponds to the ODEs derived from network (\ref{['net:cyc_simple']}), $k_b=0.05$ to the ODES (\ref{['eq:sys_cyc_full_1']}) -- (\ref{['eq:sys_cyc_full_10']}) for $k_b=0.05$. The oscillations indicate for $k_b=0.05$ a stable limit cycle close to the stable limit cycle for $k_b=0$. Panel (\ref{['fig:larger_kb']}): the stable limit cycle does not seem to exist for larger values of $k_b$.
  • Figure 4: Oscillations and multistationarity in distributive phosphorylation. Panel (a) & (b) Hopf bifurcations (H) and sustained oscillations in network (\ref{['eq:MA_cyc']}); panel (c) multistationarity in network (\ref{['eq:MA-seq']}). Rate constants for both networks as in (\ref{['eq:exa_k_seq_bistab_cyc_osci']}), total concentrations for network (\ref{['eq:MA_cyc']}) in eq. (\ref{['fig:tab_ci_cyc']}) and for network (\ref{['eq:MA-seq']}) in eq. (\ref{['fig:tab_ci_seq']}).

Theorems & Definitions (36)

  • Remark 2.1: The relative interior of $\ker(S)\cap\mathbb{R}_{\geq 0}^r$
  • Definition 1: Networks with a single extreme ray
  • Corollary 2.2
  • Remark 2.3
  • Definition 2
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Corollary 3.1
  • ...and 26 more