Analysis of a Post-translational Oscillator Using Process Algebra and Spatio-Temporal Logic

Abstract

We describe the modelling of a post-translational oscillator using a process algebra and the specification of complex properties of its dynamics using a spatio-temporal logic. We show that specifications in the Logic of Behaviour in Context can be seen as hypotheses about oscillations and other biochemical behaviours, to be tested automatically by model-checking software. By using these techniques we show that the theoretical model behaves in a manner in keeping with known properties of biological circadian oscillators.

Appendices

A Basic Jolley Model

The basic Jolley PTO model is constructed in c$\pi$ as follows:

$$\begin{aligned} E &\triangleq e(x).x.E\\ F &\triangleq f(x).x.F \end{aligned}$$

$$\begin{aligned} S00 \triangleq (\nu M_{00})&s00a \langle be \rangle .(u.S00 + ra.S01)\\&+ s00b \langle be \rangle .(u.S00 + rb.S10)\\ S01 \triangleq (\nu M_{01})&s01e \langle be \rangle .(u.S01 + r.S11)\\&+ s01f \langle bf \rangle .(u.S01 + r.S00)\\ S10 \triangleq (\nu M_{10})&s10e \langle be \rangle .(u.S10 + r.S11)\\&+ s10f \langle bf \rangle .(u.S10 + r.S00)\\ S11 \triangleq (\nu M_{11})&s11a \langle bf \rangle .(u.S11 + ra.S01)\\&+ s11b \langle bf \rangle .(u.S11 + rb.S10) \end{aligned}$$

$$\begin{aligned} \varPi \triangleq c_{S} \cdot S00 \parallel c_{E} \cdot E \parallel c_{F} \cdot F \end{aligned}$$

where

$$\begin{aligned} c_S = 10^5, c_E = 1, c_F = 1. \end{aligned}$$

B Coupled jPTOs Model

The coupled model is constructed from the same substrate and enzyme species as the basic model in Appendix A. The second jPTO is a copy of the original substrate, renamed so it forms a distinct species:

$$\begin{aligned} T00 \triangleq (\nu M_{00})&t00a \langle be \rangle .(u.T00 + ra.T01)\\&+ t00b \langle be \rangle .(u.T00 + rb.T10)\\ T01 \triangleq (\nu M_{01})&t01e \langle be \rangle .(u.T01 + r.T11)\\&+ t01f \langle bf \rangle .(u.T01 + r.T00)\\ T10 \triangleq (\nu M_{10})&t10e \langle be \rangle .(u.T10 + r.T11)\\&+ t10f \langle bf \rangle .(u.T10 + r.T00)\\ T11 \triangleq (\nu M_{11})&t11a \langle bf \rangle .(u.T11 + ra.T01)\\&+ t11b \langle bf \rangle .(u.T11 + rb.T10) \end{aligned}$$

The process term is the same as above, but with the addition of the new (copy) substrate:

$$\begin{aligned} \varPi \triangleq c_{S} \cdot S00 \parallel c_{T} \cdot T00 \parallel c_{E} \cdot E \parallel c_{F} \cdot F \end{aligned}$$

where

$$\begin{aligned} c_S = 10^5, c_T = 10^5, c_E = 1, c_F = 1, \end{aligned}$$

and the global affinity net is then extended to allow the new substrate to interact with the enzymes:

C Weaker Coupled jPTOs

For the weaker coupled model we have a separate phosphatase for each substrate. The model in Appendix B. is extended by replacing species F with the following:

$$\begin{aligned} F_S &\triangleq fs(x).x.F_S\\ F_T &\triangleq ft(x).x.F_S \end{aligned}$$

and the process term is extended:

$$\begin{aligned} \varPi \triangleq c_{S} \cdot S00 \parallel c_{T} \cdot T00 \parallel c_{E} \cdot E \parallel c_{F_S} \cdot F_S \parallel c_{F_T} \cdot F_T \end{aligned}$$

where

$$\begin{aligned} c_S = 10^5, c_T = 10^5, c_E = 1, c_{F_S} = 1, c_{F_T} = 1, \end{aligned}$$

and the affinity net is altered so each substrate only has affinity for one of the phosphatases:

D Driving Other Reactions

To construct the model which drives another phosphorylation reaction, we first construct P which is the molecule to be phosphorylated:

$$\begin{aligned} P &\triangleq (\nu M_{P}) p \langle x \rangle .(u.P + r.P')\\ P' &\triangleq \tau _{d}.P \end{aligned}$$

where $d = 10^{-4}$ and $M_P = \{ x \leftrightarrow u : 1, x \leftrightarrow r : 1 \}$.

The model is then the same as the basic model in Appendix A, but with a new site, which interacts with the P molecule, added to the $S11$ state of the substrate:

$$\begin{aligned} S11 \triangleq (\nu M_{11})&s11a \langle bf \rangle .(u.S11 + ra.S01)\\&+ s11b \langle bf \rangle .(u.S11 + rb.S10)\\&+ s11p(x).x. S11 \end{aligned}$$

the new molecule added to the process:

$$\begin{aligned} \varPi \triangleq c_{S} \cdot S00 \parallel c_{E} \cdot E \parallel c_{F} \cdot F \parallel c_{P} \cdot P \end{aligned}$$

where

$$\begin{aligned} c_S = 10^5, c_E = 1, c_F = 1, c_P = 10^5 \end{aligned}$$

and the affinity net is extended with

$$\begin{aligned} M = \{ s00a \leftrightarrow e&: 818.18, \\ s00b \leftrightarrow e&: 0, \\ s01e \leftrightarrow e&: 13.64, \\ s10e \leftrightarrow e&: 4903.17, \\ s01f \leftrightarrow f&: 4903.17, \\ s10f \leftrightarrow f&: 13.64, \\ s11a \leftrightarrow f&: 0, \\ s11b \leftrightarrow f&: 818.18, \\ s11p \leftrightarrow p&: 3\times 10^{-4} \}. \end{aligned}$$

E Perturbation

To construct the model with a pulse of inhibitor, we take the model in Appendix D and replace the driven species P with an inhibitor $In$ which decays and a species $ProdIn$ which autonomously produces the inhibitor:

$$\begin{aligned} In &\triangleq (\nu M_{In}) p \langle x \rangle u.In + \tau _d.0 \\ ProdIn &\triangleq \tau _{d}.P \end{aligned}$$

where $M_{In} = \{ x \leftrightarrow u : 0.1 \}$ and $d=5\times 10^{-3}$ and the inhibitor producer added to the process:

$$\begin{aligned} \varPi \triangleq c_{S} \cdot S00 \parallel c_{E} \cdot E \parallel c_{F} \cdot F \parallel c_{P} \cdot ProdIn \end{aligned}$$

where

$$\begin{aligned} c_S = 10^5, c_E = 1, c_F = 1, c_P = 10^5 \end{aligned}$$

In this model the inhibitor binds to the substrate in its S11 state. The models where the inhibitor binds to one or the other of the enzymes is constructed in a similar way, with a corresponding new site on the enzyme instead of the substrate. When binding to the enzyme, however the rate should be adjusted from $3\times 10^{-4}$ to 5.