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Journal ArticleDOI

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

01 Mar 2010-Journal of Computational Biology (J Comput Biol)-Vol. 17, Iss: 3, pp 325-336
TL;DR: An efficient algorithm is presented for solving the parameter synthesis problem to identify sets of parameters for which a given system of nonlinear ODEs does not reach a given set of undesirable states and is exact if the given model is affine.
Abstract: The dynamics of biological processes are often modeled as systems of nonlinear ordinary differential equations (ODE). An important feature of nonlinear ODEs is that seemingly minor changes in initial conditions or parameters can lead to radically different behaviors. This is problematic because in general it is never possible to know/measure the precise state of any biological system due to measurement errors. The parameter synthesis problem is to identify sets of parameters (including initial conditions) for which a given system of nonlinear ODEs does not reach a given set of undesirable states. We present an efficient algorithm for solving this problem that combines sensitivity analysis with an efficient search over initial conditions. It scales to high-dimensional models and is exact if the given model is affine. We demonstrate our method on different models of the acute inflammatory response to bacterial infection, and identify initial conditions consistent with three biologically relevant outcomes.

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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Parameter Synthesis in Nonlinear Dynamical Systems:
Application to Systems Biology
Alexandre Donz
´
e
1
, Gilles Clermont
2
,
Axel Legay
1
, and Christopher J. Langmead
1,3
?
1
Computer Science Department, Carnegie Mellon University, Pittsburgh, PA
2
Department of Critical Care Medicine, University of Pittsburgh, Pittsburgh, PA
3
Lane Center for Computational Biology, Carnegie Mellon University, Pittsburgh, PA
Abstract. The dynamics of biological processes are often modeled as systems of nonlinear ordinary differential
equations (ODE). An important feature of nonlinear ODEs is that seemingly minor changes in initial conditions
or parameters can lead to radically different behaviors. This is problematic because in general it is never possible
to know/measure the precise state of any biological system due to measurement errors. The parameter synthesis
problem is to identify sets of parameters (including initial conditions) for which a given system of nonlinear ODEs
does not reach a given set of undesirable states. We present an efficient algorithm for solving this problem that
combines sensitivity analysis with an efficient search over initial conditions. It scales to high-dimensional models
and is exact if the given model is affine. We demonstrate our method on a model of the acute inflammatory response
to bacterial infection, and identify initial conditions consistent with 3 biologically relevant outcomes.
Key words: Verification, Nonlinear Dynamical Systems, Uncertainty, Systems Biology, Acute Illness
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. Regardless of its mathematical form, a model is an invaluable
tool for thoroughly examining how the behavior of a system changes when the initial conditions are altered.
Such studies can be used to generate verifiable predictions, and/or to address the uncertainty associated with
experimental measurements obtained from real systems.
In this paper, we consider the parameter synthesis problem which is to identify sets of parameters for
which the system does (or does not) reach a given set of states. Here, the term “parameter” refers to both
the initial conditions of the model (e.g., bacterial load at time t = 0) and dynamical parameters (e.g., the
bacterium’s doubling rate). For example, in the context of medicine, we might be interested in partitioning
the parameter space into two regions those that, without medical intervention, deterministically lead to
the patient’s recovery, and those that lead to the patient’s death. The parameter synthesis problem is relatively
easy to solve when the system has linear dynamics, and there are a variety of methods for doing so (e.g.,
[6–8]). Our algorithm, in contrast, solves the parameter synthesis problem for nonlinear dynamical systems.
That is, for systems of nonlinear ordinary differential equations (ODEs). Moreover, our approach can also be
extended to nonlinear hybrid systems (i.e., those containing mixtures of discrete and continuous variables,
see [11] for details). Nonlinear ODE and hybrid models are very common in the Systems Biology, Synthetic
Biology, and in Medical literature but there are very few techniques for solving the parameter synthesis
problem in such systems. This paper’s primary contribution is a practical algorithm that can handle systems
of this complexity.
Our algorithm combines sensitivity analysis with an efficient search over parameters. The method is
exact if the model has affine dynamics. For nonlinear dynamical systems, we can guarantee an arbitrarily
?
Corresponding Author: cjl@cs.cmu.edu

high degree of accuracy with respect to identifying the boundary delineating reachable and non-reachable
sets. Moreover, our method runs in minutes, even on high-dimensional models. We demonstrate the method
by examining two models of the inflammatory response to bacterial infection [20, 26]. In each case, we
identify sets of initial conditions that lead to each of 3 biologically relevant outcomes.
The contributions of this paper are as follows:
An algorithm for computing parameter synthesis in nonlinear dynamical systems. This work builds on
and extends formal verification techniques that were first introduced in the context of continuous and
hybrid nonlinear dynamical systems [13].
The results of two studies on two different models of the inflammatory response to bacterial infection.
The first model is a 4-equation model, the second is a 17-equation model.
This paper is organized as follows: We outline previous work in reachability for biological systems
in Sec. 2. Next, we present our algorithm in Sec. 3. We demonstrate our method on two models of acute
inflammation in Sec. 4. We finish by discussing our results and ideas for future work in Sec. 5.
2 Background
Our 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. Formal verification methods can
be characterized by the kind of system they consider (e.g., discrete-time vs continuous-time, finite-state vs
continuous-state, linear vs non-linear dynamics, etc), and by the kind of properties they can verify (e.g,
reachability the system can be in a given state, liveness the system will be in a given set of state infinetly
often, etc). The algorithm presented in this paper is intended for verifying reachability properties under pa-
rameter uncertainty in nonlinear hybrid systems. 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]). Symbolic
methods for hybrid systems have the advantage that they are exhaustive, but in general only scale to sys-
tems of small size (< 10 continuous state variables). Another class of techniques invokes abstractions of
the model [2]. Such methods have been applied to biological systems whose dynamics can be described by
multi-affine functions. Here, examples include applications to genetic regulatory networks (e.g., [6–8]). Batt
and co-workers proposed an approach to verify reachability and liveness properties written in the linear tem-
poral logic (LTL) [24] (LTL can be used to check assumptions about the future such as equilibrium points)
of genetic regulatory networks under parameter uncertainty. In that work, the authors show that one can
reduce the verification of qualitative properties of genetic regulatory networks to the application of Model
Checking techniques [10] on a conservative discrete abstraction. Our method is more general in the sense
that we can handle arbitrary nonlinear systems but a limitation is that we cannot handle liveness properties.
However, we believe that our algorithm can be extended to handle liveness properties by combining it with
a recent technique proposed by Fainekos [15]. We note that there is also work in the area of analyzing piece-
wise (stochastic) hybrid systems (e.g., [14, 16, 18, 9]). Our method does not handle stochastic models at the
present time.
Several techniques relying on numerical computations of the reachable set apply to systems with general
nonlinear dynamics ([5, 28, 22]). In [5], the authors presents an hybridization technique, which consists in
approximating the system with a piecewise-affine approximation to take advantage of the wider family of
methods existing for this class of systems. In [21], the authors reduce the reachability problem to a partial
differential equation which they solve numerically. As far as we know, none of these techniques have been
2

applied successfully to nonlinear systems of more than a few variables. By contrast, our method builds on
techniques proposed in [12, 13] which can be applied to significantly larger models.
A more “traditional” tool used for the analysis of nonlinear ODEs is bifurcation analysis, which was
applied to the biological models used in our experiments ([26, 20]). Our approach deviates from bifurcation
analysis in several ways. First, it is simpler to apply since it only relies on the capacity to compute numerical
simulations for the system, avoiding the need of computing equilibrium points or limit cycles. Second, it
provides the capacity of analyzing transient behaviors. Finally, when it encounters an ambiguous behavior
(e.g., bi-stability) for a given parameter set, it reports that the parameter has uncertain dynamics and can
refine the result to make such uncertain sets as small as desired.
3 Algorithm
In this section, we give a mathematical description of the main algorithm used in this work.
3.1 Preliminaries
The set R
n
and the set of n × n matrices are equipped with the infinite norm, noted k · k. We define the
diameter of a compact set R to be kRk = sup
(x,x
0
)∈R
2
kx x
0
k. The distance from x to R is d(x, R) =
inf
y∈R
kx yk. The Haussdorf distance between two sets R
1
and R
2
is:
d
H
(R
1
, R
2
) = max( sup
x
1
∈R
1
d(x
1
, R
2
), sup
x
2
∈R
2
d(x
2
, R
1
)).
Given a matrix S and a set P, SP represents the set {Sp, p P}. Given two sets R
1
and R
2
, R
1
R
2
is
the Minkowski sum of R
1
and R
2
, i.e., R
1
R
2
= {x
1
+ x
2
, x
1
R
1
, x
2
R
2
}.
3.2 Simulation and Sensitivity Analysis
We consider a dynamical system Sys = (f, P) of the form:
˙x = f(t, x, p), p P, (1)
where x R
n
, p is a parameter vector and P is a compact subset of R
n
p
. We assume that f is continuously
differentiable. Let T R
+
be a time set. For a given p, a trajectory ξ
p
is a function of T which satisfies the
ODE (Eq. 1), i.e., for all t in T ,
˙
ξ
p
(t) = f(t, ξ
p
(t), p). For convenience, we include the initial state in the
parameter vector by assuming that if p = (p
1
, p
2
, . . . , p
n
p
) then ξ
p
(0) = (p
1
(0), p
2
(0), . . . , p
n
(0)). Under
these conditions, we know by the Cauchy-Lipshitz theorem that the trajectory ξ
p
is uniquely defined.
The purpose of sensitivity analysis techniques is to predict the influence on a trajectory of a perturbation
of its parameter vector. A first order approximation of this influence can be obtained by a Taylor expansion
of ξ
p
(t) around p. Let δp R
n
p
. We have:
ξ
p+δp
(t) = ξ
p
(t) +
ξ
p
p
(t) δp + O
kδpk
2
. (2)
The second term in the right hand side of Eq. (2) is the derivative of the trajectory with respect to p. Since
p is a vector, this derivative is a matrix, which is called the sensitivity matrix. We denote it as: S
p
(t) =
ξ
p
p
(t)
3

The sensitivity matrix can be computed as the solution of a system of ODEs. Let s
i
=
ξ
p
p
i
(t) be the i
th
column of S
p
. If we apply the chain rule to its time derivative, we get:
(
˙
s
i
(t) =
f
x
(t, x(t), p)s
i
(t) +
f
p
i
(t, x(t), p),
s
i
(0) =
x(0)
p
i
.
(3)
Here
f
x
(t, x(t), p) is the Jacobian matrix of f at time t. The equation above is thus an affine, time-varying
ODE. In the core of our implementation, we compute ξ
p
and the sensitivity matrix S
p
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 reachable set induced by the set of parameters P at time t is:
R
t
(P) =
[
p∈P
ξ
p
(t).
The set R
t
(P) can be approximated by using sensitivity analysis. Assume that for a given parameter p in P
we computed a trajectory ξ
p
and the sensitivity matrix S
p
associated with it. Given another parameter vector
p
0
in P, we can use this matrix to get an estimate
ˆ
ξ
p
p
0
(t) of ξ
p
0
(t). This is done by dropping higher order
terms in the Taylor expansion given in Equation 2. We have:
ˆ
ξ
p
p
0
(t) = ξ
p
(t) + S
p
(t)(p
0
p). (4)
If we extend this estimation to all parameters p
0
in P, we get the following estimate of the reachable set
R
t
(P):
ˆ
R
p
t
(P) =
[
p
0
∈P
ˆ
ξ
p
0
(t) = {ξ
p
(t) S
p
(t)p} S
p
(t)P. (5)
Thus
ˆ
R
p
t
is an affine mapping of the initial set P into R
n
(see Figure 1).
It can be shown that if the dynamics are affine, i.e., if f(t, x, p) = A(t, p)x + b(t, p), then the estimation
is exact. However, in the general case,
ˆ
R
p
t
(P) is different from R
t
(P). Since the estimation is based on a
first order approximation around parameter p, it is local in the parameter space and its quality depends on
how “big” P is. In order to improve the estimation, we can partition P into smaller subsets P
1
, P
2
, . . . , P
l
and compute trajectories using new initial parameters p
1
, p
2
, . . . , p
l
to get more precise local estimates. As
a practical matter, we need to be able to estimate the benefit of such a refinement. To do so, we compare
ˆ
R
p
t
(P
j
) the estimate we get when using the “global” center, p; to
ˆ
R
p
j
t
(P
j
) the estimate we get when
using the “local” center, p
j
, and p
0
i
P
j
. We do this for each P
j
. Figure 1 illustrates the essential features
of the algorithm.
Proposition 1. We have
d
H
(
ˆ
R
p
t
(P
j
),
ˆ
R
p
j
t
(P
j
)) Er r(P, P
j
), (6)
where
Err(P, P
j
) = kξ
p
j
(t)
ˆ
ξ
p
p
j
(t)k + kS
p
j
(t) S
p
(t)kkP
j
k. (7)
4

Fig. 1. Comparison between a “global” and a “local” estimate of the reachable set. The large square on the left hand side represent a
region of parameter space, P. The oval-shaped region on the right hand side corresponds to the true reachable set, R
t
(P), induced
by parameters P at time t. The large parallelogram on the right hand side corresponds to the estimated reachable set,
ˆ
R
p
t
(P), using
a sensitivity analysis based on trajectory labeled ξ
p
which starts at point p P. The point labeled
ˆ
ξ
p
p
j
, for example, is an estimate
of where a trajectory starting at point p
j
would reach at time t. If we partition P and consider some particular partition, P
j
, we
can then compare the estimated reachable sets
ˆ
R
p
t
(P
j
) and
ˆ
R
p
j
t
(P
j
), which correspond to the small light-gray and small dark
gray parallelograms, respectively. We continue to refine until the distance between
ˆ
R
p
t
(P
j
) and
ˆ
R
p
j
t
(P
j
) (Eq. 7) falls below some
user-specified tolerance.
In other words, the difference between the global and the local estimate can be decomposed into the
error introduced in the estimation ξ
p
p
j
(t) of the state reached at time t using p
j
(first term on RHS of Eq.
7), and another term involving the difference between the local and the global sensitivity matrices and the
distance from local center (second term on RHS of Eq. 7).
Proof. let y be in
ˆ
R
p
t
(P
j
). There exists p
y
in P
j
such that y =
ˆ
ξ
p
p
y
(t). We need to compare
ˆ
ξ
p
p
y
(t) = ξ
p
(t) + S
p
(t)(p
y
p) (8)
with
ˆ
ξ
p
j
p
y
(t) = ξ
p
j
(t) + S
p
j
(t)(p
y
p
j
). (9)
By introducing
ˆ
ξ
p
p
j
(t) = ξ
p
(t) + S
p
(t)(p
j
p) (10)
and after some algebraic manipulations of (8), (9), and (10), we get
ˆ
ξ
p
p
y
(t)
ˆ
ξ
p
j
p
y
(t) = ξ
p
j
(t)
ˆ
ξ
p
p
j
(t) + (S
p
j
(t) S
p
(t))(p
y
p
j
)
kξ
p
j
(t)
ˆ
ξ
p
p
j
(t)k + kS
p
j
(t) S
p
(t)kkP
j
k = Er r(P, P
j
). (11)
Let x =
ˆ
ξ
p
j
p
y
(t) which is in
ˆ
R
p
j
t
(P
j
), then it can be shown that ky xk Err(P, P
j
) and so
d(y,
ˆ
R
p
j
t
(P
j
)) Er r(P, P
j
). This is true for any y P
j
, thus
sup
y
ˆ
R
p
t
(P
j
)
d(y,
ˆ
R
p
j
t
(P
j
)) Er r(P, P
j
).
Similarly, we can show that
sup
x
ˆ
R
p
j
t
(P
j
)
d(x,
ˆ
R
p
t
(P
j
)) Er r(P, P
j
)
which proves the result. ut
5

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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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Book ChapterDOI
23 Mar 2000
TL;DR: This work presents an implementation of an exact reachability operator for nonlinear hybrid systems, and an equivalent formulation is developed of the key equations governing the continuous state reachability.
Abstract: Reachability analysis is frequently used to study the safety of control systems We present an implementation of an exact reachability operator for nonlinear hybrid systems After a brief review of a previously presented algorithm for determining reachable sets and synthesizing control laws--upon whose theory the new implementation rests--an equivalent formulation is developed of the key equations governing the continuous state reachability The new formulation is implemented using level set methods, and its effectiveness is shown by the numerical solution of three examples

279 citations

Frequently Asked Questions (2)
Q1. What have the authors contributed in "Parameter synthesis in nonlinear dynamical systems: application to systems biology" ?

The authors present an efficient algorithm for solving this problem that combines sensitivity analysis with an efficient search over initial conditions. The authors demonstrate their method on a model of the acute inflammatory response to bacterial infection, and identify initial conditions consistent with 3 biologically relevant outcomes. 

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.