# Parameter synthesis in nonlinear dynamical systems: application to systems biology.

## Summary (2 min read)

### 1 Introduction

- The fields of Systems Biology, Synthetic Biology, and Medicine produce and use a variety of formalisms for modeling the dynamics of biological systems.
- Such studies can be used to generate verifiable predictions, and/or to address the uncertainty associated with experimental measurements obtained from real systems.
- The authors algorithm, in contrast, solves the parameter synthesis problem for nonlinear dynamical systems.
- The authors demonstrate the method by examining two models of the inflammatory response to bacterial infection [20, 26].
- This work builds on and extends formal verification techniques that were first introduced in the context of continuous and hybrid nonlinear dynamical systems [13].

### 2 Background

- The authors work falls under the category of formal verification, a large area of research which focus on techniques for computing provable guarantees that a system satisfies a given property.
- The most closely related work in this area uses symbolic methods for restricted class of models (e.g., timed automata [4], linear hybrid systems [1, 19, 17]).
- Several techniques relying on numerical computations of the reachable set apply to systems with general nonlinear dynamics ([5, 28, 22]).
- The authors approach deviates from bifurcation analysis in several ways.

### 3.2 Simulation and Sensitivity Analysis

- Under these conditions, the authors know by the Cauchy-Lipshitz theorem that the trajectory ξp is uniquely defined.
- (2) The second term in the right hand side of Eq. (2) is the derivative of the trajectory with respect to p.
- The equation above is thus an affine, time-varying ODE.
- In the core of their implementation, the authors compute ξp and the sensitivity matrix Sp using the CVODES numerical solver [27], which is designed to solve efficiently and accurately ODEs (like Eq. 1) and sensitivity equations (like Eq. 3).

### 3.3 Reachable Set Estimation Using Sensitivity

- The reachability problem is the problem of computing the set of all the states visited by the trajectories starting from all the possible initial parameters in P at a given time t. Definition 1 (Reachable Set).
- The set Rt(P) can be approximated by using sensitivity analysis.
- Figure 1 illustrates the essential features of the algorithm.
- The authors can show that the convergence is quadratic.

### 3.4 Parameter Synthesis Algorithm

- The authors state a parameter synthesis problem and propose an algorithm that provides an approximate solution.
- An approximate solution is a partition P = Psaf ∪ Punc ∪ Pbad where Psaf and Pbad are defined as before and Punc (i.e., uncertain) may contain both safe and bad parameters.
- Exact solutions cannot be obtained in general, but the authors can try to compute an approximate solution with the uncertain subset being as small as possible.
- The idea is to iteratively refine P and to classify the subsets into the three categories.

### 4 Application to Models of Acute Inflammation

- The authors applied their method to two models of the acute inflammatory response to infection.
- The pro-inflammatory elements are primarily responsible for eliminating the pathogen, but bacterial killing can cause collateral tissue damage.
- The authors can now define three sets of states corresponding to the three clinically relevant outcomes as follows: (i) Health = Falive∩Faseptic; (ii) Aseptic death = Fdead∩Faseptic; and (iii) Septic death = Fdead∩Fseptic.

### 4.1 Experiments

- In the first experiment, the authors validated their method by reproducing results previously obtained in [26] using bifurcation analysis.
- The region Fdeath given by D ≥ 5 was used in their algorithm and the authors checked the intersection with reachable set at time 300 hours.
- In their second experiment, the authors varied growth rate of pathogen, kpg, and NA.
- Figure 4-(B) shows that there are three distinct regions in the kpg-NA plane, corresponding to the three clinical outcomes.
- The authors then performed several experimentations with the 17-equation model.

### 5 Discussion and Conclusions

- Complex models are increasingly being used to make predictions about complex phenomena in biology and medicine (e.g., [3, 25]).
- Thus, it is important to have tools for explicitly examining a range of possible parameters to determine whether the behavior of the model is sensitive to those parameters that are poorly estimated.
- Performing this task for nonlinear models is especially challenging.
- Moreover, there are no known methods capable of providing provable bounds on numerical errors for general nonlinear differential equations.
- Temporal properties could easily be introduced in their framework.

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### Cites methods from "Parameter synthesis in nonlinear dy..."

...The use of the sensitivity-based analysis guided by the measure of the robustness has been successfully applied in several domains ranging from analog circuits [28] to systems biology [29,30], to study the parameter space and also to refine the uncertainty of the parameter sets....

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##### References

2,091 citations

### "Parameter synthesis in nonlinear dy..." refers background in this paper

...The most closely related work in this area uses symbolic methods for restricted class of models—e.g., timed automata (Annichini et al., 2000), linear hybrid systems (Alur et al., 1995; Henzinger, 2000; Frehse, 2008)....

[...]

893 citations

### "Parameter synthesis in nonlinear dy..." refers background in this paper

...Another class of techniques invokes abstractions of the model (Alur et al., 2000)....

[...]

397 citations

### "Parameter synthesis in nonlinear dy..." refers methods in this paper

...Numerical solvers for ordinary differential equations based on interval analysis (Nedialkov et al., 1999) or Taylor methods (Berz et al., 1996) exist that can compute rigorous enclosures of the solutions and they could be applied in our context, but their scalability for higher order systems and…...

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279 citations

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##### Frequently Asked Questions (2)

###### Q2. What are the future works mentioned in the paper "Parameter synthesis in nonlinear dynamical systems: application to systems biology" ?

There are several areas for future research. Their first order error control mechanism can be improved to make the refinements more efficient and more adaptive when nonlinear ( i. e. higher order ) behaviors dominate any linear dependance on parameter variations. Finally, the authors believe that the method could easily be used in the context of personalized medicine.