Hamilton Institute, National University of Ireland, Maynooth, Co. Kildare, Ireland

Department of Mathematics & Computer Science, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany

Centre for Systems Biology at Edinburgh, University of Edinburgh, Edinburgh EH9 3JD, UK

Institute of Mathematics, University of Potsdam, Am Neuen Palais 10, D-14469 Potsdam, Germany

Abstract

Background

In many applications, ordinary differential equation (ODE) models are subject to uncertainty or variability in initial conditions and parameters. Both, uncertainty and variability can be quantified in terms of a probability density function on the state and parameter space.

Results

The partial differential equation that describes the evolution of this probability density function has a form that is particularly amenable to application of the well-known method of characteristics. The value of the density at some point in time is directly accessible by the solution of the original ODE extended by a single extra dimension (for the value of the density). This leads to simple methods for studying uncertainty, variability and likelihood, with significant advantages over more traditional Monte Carlo and related approaches especially when studying regions with low probability.

Conclusions

While such approaches based on the method of characteristics are common practice in other disciplines, their advantages for the study of biological systems have so far remained unrecognized. Several examples illustrate performance and accuracy of the approach and its limitations.

Background

Ordinary differential equations (ODEs) are commonly used for modeling biological and biochemical systems. ODE models are often subject to considerable uncertainty and/or variability in both initial conditions and parameters

To study those effects globally the sensitivity analysis problem can be formulated in terms of an ODE with random initial conditions. The task then is to determine the probability density function (pdf), or some features of the pdf, at a time

To numerically solve the particular class of PDEs that arise in ODEs with random initial data, finite difference schemes are commonly applied

It is however well-known that first-order PDEs can be solved using the method of characteristics

Although the method of characteristics is known and widely used in other fields such as meteorology, e.g.

Results and Discussion

Methodology

ODEs with uncertain or variable input

Consider an ODE of the form

with ^{
d
}. Typically, ^{
n
}and the parameters ^{m }of the system with

where ^{
d
}→ ℝ^{
d
}is continuously differentiable with respect to

Uncertainty and variability in the model input can be modeled by assuming that _{0 }= _{0 }is a random variable with pdf _{0 }: ℝ^{
d
}→ ℝ. Consequently, the solution {_{
t
}}_{
t≥0 }of the initial value problem (1) is a stochastic (Markov) process. For any _{
t
}, i.e., _{0 }amounts to computing the density

where _{i }
_{i }

Computing the pdf along solutions of the ODE

It is well-known that first-order PDEs of the form (3) can be solved using the method of characteristics

Noting that

The PDE (3) can thus be solved pointwise for each _{0 }by solving the original ODE (1) together with an extra dimension for the density

with initial conditions _{0 }and _{0}(_{0}). Since ^{
d
}, the new system (6) has

**(I) **Discretize the region of interest in the state space ℝ^{
d
}, resulting in discretization points _{
i
}(0),

**(II) **For the initial values (_{
i
}(0), _{0}(_{
i
}(0))) ∈ ℝ^{
d+1}, _{
i
}(

This procedure directly yields the density values _{
i
}(_{
i
}(

In comparison, Monte Carlo-based methods require density estimation subsequent to solving the ODE for the sample points:

**(i) **Sample the initial distribution _{0}, resulting in sampling points _{
i
}(0) ∈ ℝ^{
d
},

**(ii) **For each sampling point _{
i
}solve the original ODE (1) to obtain _{
i
}(

**(iii) **Estimate the density from the propagated points _{
i
}(_{
i
}(

In contrast to the method of characteristics, the quality of the approximation

We illustrate the advantages of the method of characteristics for sensitivity analysis of ODE models by two examples in gene expression. We will see that descriptors such as mean and variance may provide only poor information about the pdf. Our first example demonstrates the benefits of the method of characteristics in terms of an efficient computation of the pdf in regions with low probability. In the second example, the pdf contracts onto a lower-dimensional manifold of the state space, and the method of characteristics provides the density values directly along that manifold. In a third example we further show how the method can be used to compute the likelihood of an ODE model and thus facilitates comparison to experimental data. For the sake of simplicity, we choose normal initial distributions to describe state and parameter uncertainty and variability. The method of characteristics, however, provides a general strategy to compute model uncertainty/variability and likelihood for arbitrary distributions of initial values and parameters with the only restriction that the associated pdf

Examples

Sensitivity analysis and the impact of variability

**Example 1 (analyzing regions of low probability) **Consider a protein _{
d
}> 0, the concentration

Autoregulation

**Autoregulation**. Illustration of a protein that activates its own expression by cooperatively binding to the promoter which regulates its transcription.

The first summand describes the activation of gene expression in terms of a Hill function _{max }> 0 denotes the maximal expression rate, _{0 }may, for example, represent the abundance of the protein

Figure _{0 }with mean ^{2 }= 0.2 (in [_{max }= 1 (in [_{d }

Example 1 (autoregulated gene expression)

**Example 1 (autoregulated gene expression)**. (a) For an initial uniform grid on [0, 5] with grid size

Most of the distribution is translated linearly, since for large

**(I) **Discretize the sub-interval of interest, and solve the original ODE (7) for each discretization point _{
i
}(_{
i
}(0).

**(II) **For the initial values (_{
i
}(0), _{0}(_{
i
}(0))),

The two-step procedure is illustrated in Figure

Two-step procedure

**Two-step procedure**. Illustration of the two-step procedure to compute the final pdf at specific points in space. (I) For given discretization/data points _{i}(_{i}(0) and their initial density values _{0}(_{i}(0)), the method of characteristics is used to compute the pdf along the trajectories forward in time (dotted arrow).

Apart from variability in the initial concentration _{max }and by both _{max}, i.e., with extended state space variables (_{max})' and (_{max})', respectively. The initial distribution was assumed to be a joint normal distribution, where _{max }and

Example 1 (autoregulated gene expression, extended state space)

**Example 1 (autoregulated gene expression, extended state space)**. (a) Marginal distributions of protein concentration _{max})' (solid gray), and (_{max})' (dotted) compared to the distribution without parameter variability (dashed-dotted gray) computed by the two-step procedure with a final uniform grid (_{max }∈ [0.5, 1.5], _{max})' and (

Numerically, we discretized the above integral using the midpoint rule.

Comparison of the distributions indicates that variability in the cooperativity _{max})' and (_{max}, respectively. Same as in the one-dimensional marginal distribution, it can be seen that the variability in protein concentration is mainly dominated by the variability in the maximal expression rate _{max}.

The numerical integration (compare eq. (8)), necessary to visualize multivariate densities for

**Example 2 (variability along lower-dimensional manifolds) **Consider the genetic toggle switch _{1 }and _{2 }mutually repress the other protein's expression (illustrated in Figure _{1 }and _{2 }of _{1 }and _{2}, respectively, are modeled by the two-dimensional system

Genetic toggle switch

**Genetic toggle switch**. Illustration of a genetic toggle switch. The two proteins _{1 }and _{2 }mutually repress their expression by cooperatively binding to the promoter regulating the transcription of the other protein.

The parameter _{1 }> 0 represents the effective expression rate of protein _{1}, and _{1 }> 0 describes the repression cooperativity of the promoter that regulates the expression of _{1 }by _{2}. Analogously, _{2 }and _{2 }describe the effective expression rate of protein _{2 }and its promoter's cooperativity of repression by _{1}, respectively. The parameter _{1 }(_{2}) corresponds to the concentration of _{2 }(_{1}) that represses the promoter activity of _{1 }(_{2}) by 50%. The parameter _{d }

Assume that the concentrations _{1 }and _{2 }at _{0 }is shown in Figure _{1 }= _{2 }= _{1 }= _{2 }= _{1 }= _{2 }= 1 (in [_{d }

Example 2 (genetic toggle switch)

**Example 2 (genetic toggle switch)**. (a) Initial density and the vector field defined by the two-state system (9). (b) At

Deterministic models in a likelihood setting for comparison with experimental data

So far we have discussed the use of the method of characteristics to study the sensitivity of ODE models. It however also offers benefits when comparing model output with experimental data for model assessment, such as validation/falsification and selection between different models, or parameter estimation. An exact match of the deterministic model with the data is unlikely, and a quantification of the mismatch remains a critical issue. Some numerical approaches are based on verifying that the experimental data lies within regions of the state space that are reachable with certain parameter sets of the (often linearized) ODE model

Stochastic approaches offer a natural way of assessing deviations of the model output from data based on the likelihood function. Given a set of data points _{1},...,_{N }}, the likelihood of a model is defined as the probability that the model predicts this data. For continuous state space models

where _{0}, and further given data points _{1}(_{N}(

**(I) **Solve the ODE (1) for each of the _{
i
}(^{
d
}, _{
i
}(0).

**(II) **For the initial values (_{
i
}(0), _{0}(_{
i
}(0))) ∈ ℝ^{
d+1}, _{
i
}(

In accordance with (10), the likelihood ℒ(

**Example 3 (parameter estimation) **Reconsider the ODE model (7) of autoregulated gene expression from Example 1. Assume that we want to estimate the maximal expression rate _{max }of the protein

As we are interested in the likelihood of different values of _{max}, we consider the autoregulation model (7) extended according to (2) by _{max}. We apply the above two-step procedure to a representative ensemble {_{1},...,_{
N
}} of values of _{max}. For each pair (_{
i
}) the backward-solution yields a different value (_{
i
}). Given prior knowledge in terms of a joint pdf _{0 }for _{0 }and _{max}, the forward-solution of (6) with initial conditions ((_{
i
}(0), _{
i
}), _{0}(_{
i
}(0), _{
i
})) then yields the likelihood values _{i})) associated with each _{i},

We computed the likelihood of a set of equidistant values of _{max }∈ [0, 2] using the same parameter values as in Example 1 (shown in Figure _{0 }and _{max }with parameters _{0 }and _{max}, where _{0 }is normally distributed with _{max }is independently uniformly distributed on the interval [0, 2] (dashed black line). The first scenario accounts for prior knowledge of _{0 }and _{max}, where a more or less precise knowledge of _{max }is given (since ^{2}(_{max}) is small). Accordingly, the maximum-likelihood estimate is _{max}. In the second scenario no prior information on _{max }was imposed (except for its constraint within [0, 2]). The maximum-likelihood estimate of _{max }is therefore solely determined by the value of _{max }that yields the initial value closest to _{0}) = 2. Since the data point _{max }(compare with Figure

Example 3 (parameter estimation)

**Example 3 (parameter estimation)**. For a fictional data point _{max }∈ [0, 2] (uniform discretization points with grid size _{max}. Shown are two different likelihood functions corresponding to (a) prior information imposed on both _{0 }and _{max }by means of a joint normal distribution (solid gray line), and (b) no prior information on _{max }in terms of _{max }assumed to be uniformly distributed on [0, 2] (dashed black line).

Conclusions

Studying the effects of uncertainty and variability in initial values or parameters of ODE models can be computationally intensive, since it generally involves solving the system a large number of times. The method of characteristics offers a simple yet accurate alternative to conventional approaches for small- and moderate-dimensional systems. The approach does not assume a particular shape of neither input nor output distribution, it only requires the pdf to be sufficiently smooth (continuously differentiable) and yields density values that are exact up the accuracy of the ODE solver used. Our first two examples illustrate how a precise characterization of the model uncertainty/variability can be obtained with only few trajectories. In this context we also demonstrated that the analysis can be efficiently restricted to certain sub-regions of the state/parameter space. One limitation of the two-step procedure used for the latter analysis is that for chaotic models the backward-forward solution of the ODE is ill-conditioned

We provide M

**M ATLAB files illustrating the usage of the method of characteristics**.

Click here for file

Authors' contributions

AYW, WH and RHM planned and performed the research, AYW performed the numerical simulations, all authors contributed to the design and the writing of the manuscript. All authors read and approved the final manuscript.

Acknowledgements

This work was supported in part by the Science Foundation Ireland, Grant SFI07/RPR/I177. We thank Ken R. Duffy (Hamilton Institute/NUIM) for his critical reading of the manuscript.