Parameter Synthesis and Robustness Analysis of Rule-Based Models

Abstract

We introduce the Quantitative Biochemical Space Language, a rule-based language for a compact modelling of probabilistic behaviour of complex parameter-dependent biological systems. Application of rules is governed by an associated parametrised rate function, expressing partially known information about the behaviour of the modelled system. The parameter values influence the behaviour of the model. We propose a formal verification-based method for the synthesis of parameter values (parameter synthesis) which ensure the behaviour of the modelled system satisfies a given PCTL property. In addition, we demonstrate how this method can be used for robustness analysis.

This work has been supported by the Czech Science Foundation grant 18-00178S.

A Tumour Growth

Tumour growth is based on mitosis (i.e. cell division). The cell cycle is the process between two mitoses and it consists of four phases: the resting phase $G_1$, the DNA replication phase S, the resting phase $G_2$, and the mitosis phase M in which the cells segregate the duplicated sets of chromosomes between daughter cells. The three phases $G_1$, S, and $G_2$ constitute the pre-mitotic phase, also called interphase.

We have adopted the model of tumour growth  [44] to our language. It considers two populations of tumour cells: those in interphase and those in mitosis. We represent the tumour cell as an agent $\mathsf {T}$. The current phase is expressed with an atom $\mathsf {phase}$ in its composition, which can have two different states – $\mathsf {i}$ for interphase and $\mathsf {m}$ for mitosis. For simplicity, we omit the compartment from the rules since it does not change and plays no important role in this model.

Fig. 5.

Rules of the tumour growth model. The first rule describes the change of the phase of a cell from interphase to mitosis. The second rule describes the duplication of the cell to two daughter cells. Note that both start in interphase. The last two rules describe the death of cells in both possible states.

Fig. 6.

Visualisation of results of parameter synthesis (left) and quantitative model checking using sampling (right) for property $\phi$ for tumour growth model. The horizontal axis represents values of the parameter $a_1 \in [0, 3]$ and the vertical axis represents values of the parameter $d_2 \in [0.001, 0.5]$. The probability threshold 0.5 from the property $\phi$ is visible in both sampling (approximately the yellow line) and parameter synthesis (the grey line). It shows that the parameter synthesis method gives us a very precise result and is in agreement with quantitative model checking. (Color figure online)

The rules of the model are available in Fig. 5. Note that this model is a demonstration where all rules are reaction-based, i.e. they do not describe an abstract rule, only modification of concrete agents.

Given rate functions of rules are parametrised. Parameters $\mathsf {a_1}$ and $\mathsf {a_2}$ are present in rules responsible for change of phase and cell division, while parameters $\mathsf {d_1}$ and $\mathsf {d_2}$ are in the rules where the cell disappears or dies. The values $\mathsf {a_2} = 0.5$ and $\mathsf {d_1} = 0.3$ are constant the other two parameters are given by admissible ranges: $\mathsf {a_1} \in [0 ; 3]$ and for $\mathsf {d_2} \in [0.001; 0.5]$.

For the initial state, we assume a single agent $\mathsf {T}^{1}(\mathsf {phase}\{\mathsf {i}\})$. Please note that the model gives rise to infinite pMC since the second rule can generate additional agents. To obtain a finite abstract probabilistic model, we have heuristically limited the number of states of the model. Particularly, we generate all the states having the number of individuals of both species less or equal to 5 and we introduce a special abstract state which represents all the other states, which limits the size of possible state space to $6^2$. This approximation is incorrect only in cases when one wants to reach a state which is represented by the special state.

We are interested in property whether the population of tumour cells will reach almost its maximum with the probability higher than 0.5, meaning that the growth is not random but has rather tendency to grow without limitations. This property can be expressed as . In Fig. 6, there is a visualisation of parameter synthesis. The results show that the higher values of the parameter $a_1$ (cell division) and the lower values of the parameter $d_2$ increase the probability of property satisfaction. This result is quite expected, because both parameters directly influence cell division ($a_1$) and degradation ($d_2$) of cells. We have also computed the global robustness degree of the property, which is approximately 0.24. It can be interpreted as 24% of parameter space satisfies the property $\mathsf {True}~\mathbf {U}~\mathsf {T}^{j}(\emptyset ) > 8$.