The use of mathematical models provides tools for the analysis of complex molecular interactions aiming at an understanding of processes occurring in living cells. For many problems in cellular control, stochastic effects and time-delays can be ignored and systems of first-order ordinary differential equations (ODEs) can adequately model the underlying processes. Denoting by x and p the biochemical concentrations and parameters, respectively, the instantaneous change in x is described by the vector field f:

= f(x, p).     (1)

In the study of such systems, an important goal is to understand how the observed physiological behavior arises out of gene network topology and parameters p. Some of these questions may be studied via examining the bifurcation manifolds Σ of the ODE system, which partition the parameter space into regions of different qualitative behavior (see e.g., 1 for a general overview to bifurcation theory). From ODE models and measured parameters, the forward problem of computing the bifurcation diagram has contributed significantly towards elucidating the complex mechanisms underlying cellular processes. For instance, mathematical and symbolic bifurcation analysis has led to an understanding of the possible dynamical behaviors that may arise out of simple gene systems (for a monograph, see 2, examples of more recent papers dealing with natural, designed, and model systems are 34567). For cell cycle models, bifurcation diagrams have given biologists a systems-level perspective of the roles played by the various constituent modules, as well as providing the ability to predict the behavior of mutant cells 89. The desire to locate regions in parameter space exhibiting interesting dynamics has led to the development of computational tools for the automatic discovery of bifurcation sets 10.

In contrast, inverse problems have only recently attracted attention in biology as a way to unravel the workings of cellular mechanisms. In inverse problems one looks for causes for observed or desired effects 11. Mathematically, such problems are typically ill-posed, in particular unstable; special mathematical techniques, called "regularization methods", have to be used to cope with this ill-posedness. Many variational and iterative regularization techniques have been developed over the years and applied to a variety of problems in science, engineering and finance (see 11 and some references quoted there, and 12).

In the current context of cell biology, one would like to address problems such as: which parameter configurations lead to an observed qualitative behaviour of the system ("identification")? How can one introduce a certain qualitative behaviour into the system via parameter variations ("design")? We summarize such problems under the name of inverse bifurcation problems, where the task is to map the space of bifurcation diagrams back to the space of parameters. In particular, we consider inverse bifurcation problems of two types: identification and design. For the former, one would like to infer parameter values from observed bifurcation diagrams and hence the issue of uniqueness is typically of concern. For the latter, one is interested in parameter values that produce the desired outcome, hence uniqueness is not an issue. The notion of "inverse bifurcation" was first introduced into biology in 13; more recently, another inverse bifurcation problem from mathematical ecology was studied in 14 by integral equation methods. These papers are concerned with the existence and uniqueness of nonlinear terms in equations realizing prescribed bifurcation diagrams and use analytical methods. In contrast, given the size and complexity of gene networks, we take a computational approach in addressing problems of inverse bifurcation. In this paper, we consider problems of moderate size for which the ill-posedness is not yet a crucial issue. Since ill-posedness increases with dimension, a major issue for larger problems will be to use appropriate regularization techniques.

For the design type of inverse bifurcation problems, there exists previous work in the engineering literature: The distance to bifurcation manifolds has been introduced to quantify the "parametric robustness" of system designs in 15; in the context of design of chemical processes, optimization problems with constraints involving distance to bifurcation manifolds have been treated in 16. For a recent review see 17. For biological applications as we have them in mind, this issue of parametric robustness is also important. In addition, other geometric properties of the bifurcation diagram are of interest. These include the size of the parameter region resulting in bistability of solutions and the parametric distance between regions of different qualitative behavior. We will develop methodolgies by which inverse bifurcation problems involving the optimization of such quantities can be solved.

In this paper, we show the applicability of our inverse bifurcation algorithms to low dimensional gene systems. We formulate the inverse problems as constrained optimization problems, whose objective function and constraints involve geometric properties of bifurcation diagrams. We demonstrate that these problems can be solved efficiently by applying gradient-based nonlinear optimization algorithms in combination with one-parameter continuation methods to locate bifurcation points. The latter is a standard capability provided by existing bifurcation analysis software (see 1 for references to state-of-the-art numerical implementations).

The ODE model (1) defines a mathematical relationship between the parameters p and the time course of biochemical concentrations, x(t). Of the set of all parameters, some describe the biochemical mechanisms constituting the machinery of the regulatory system, while others are parameters to whose variation the gene system should respond. In this paper, we refer to the former as the system parameters and the latter as the control inputs.

In situations where the qualitative behavior of the regulatory system changes in response to shifts in the control input, the corresponding bifurcation diagram is of importance. In particular, the proper working of the regulatory system may depend on the locations, shapes and sizes of the various regions of qualitative behavior. For instance, in the cell cycle model 8 the correct spatial relationship (in parameter space) of the transitions points is crucial for ensuring the survival of cells. In fact, the model predicts that "dynamically challenged" mutants with shifted bifurcation diagrams may suffer irrecoverably. The resulting effects include difficulties exiting mitosis and decreases in cell mass after each cell division. For this particular model, the inverse bifurcation problem is to map geometric as well as topological relationships in the bifurcation diagrams to conditions on the parameters. Another application of inverse bifurcation in the cell cycle model is for the so-called "Pinocchio effect" involving check points. This effect occur when the necessary conditions for safe progression to the next cell phase are not satisfied hence the regulatory system should delay the state transition as far as possible. Here, the inverse bifurcation problem is to find out how the system may be constructed so that, when triggered by the presence or absence of certain chemicals, the range of bistability is maximized.

In the case of system parameters, they are either regulated by other control mechanisms in the cell, or the behavior of the system is constructed to be insensitive to variation in these parameters 18. Within an operating region in the control input space, the minimal distance in the system parameter space to bifurcation manifolds provides a quantitative measure of the robustness of the system to environmental perturbations. Here, the inverse bifurcation problem would involve finding the parameter combinations so that the system bifurcation diagram is insensitive to the unregulated parameters. Biological applications of this class of problems include homeostasis and rhythmic pacemakers. Again, the problem of interest is to map some shape property of the bifurcation diagram to conditions on the parameter space.

For inverse bifurcation problems of biological interest such as those described above, the question arises as to how to formulate them mathematically so that the solution can be obtained in a computationally tractable and stable manner. Typically, in biological applications the parameter space is of high dimension. Furthermore, in carrying out inverse analysis, the (forward) bifurcation analysis usually has to be applied a large number of times. In most cases, the former condition precludes formulations based on the use of multi-parameter continuation techniques 19. Instead, formulations based on continuation along rays in parameter space are preferred. Below, we formulate distance to bifurcation manifolds (first introduced in 15) in terms of a forward operator and an l₂ functional. Subsequently, we describe the sensitivity of the minimal distance with respect to parameters by adjoint methods.

Consider the splitting of m-dimensional parameter space P ⊂ ℝ^m into input and system parameters, p = (p_i, p_s) ∈ P_i × p_s . For an ODE system, let Σ denote a bifurcation manifold of interest, consisting of sets in parameter space P for which structural stability breaks down 1. For a given system parameter p_s, we further define Σ(p_s) ≡ Σ ∩ {p_s} as the intersection of Σ with the p_s-plane. In Figure 2, the geometric relationship between Σ and Σ(p_s) is illustrated. Let the forward operator F :P → P be a mapping in parameter space, taking a given point to its orthogonal projection on Σ(p_s), assumed to be well-defined. That is,

F(p) ≡ (F(p)_i, F(p)_s)

The notation is illustrated in Figure 1.

Using F(p), the inverse bifurcation problems examined in this paper are mathematically formulated as:

Subject to: p_low ≤ p ≤ p_upp

0 ≤ c(F(p)_i),     (2)

where ||·|| denotes the l₂-norm and c : P_i → ℝ^k represents k-dimensional nonlinear constraints. In the rest of this section, we discuss adjoint sensitivity analysis of F(p), which is important for applying gradient-based optimization methods for solving (2) as well as for computing F(p) iteratively. Denote the linearization of the above map at p as F'(p). The adjoint operator F'*(p) is defined to satisfy the following: for all δp, δ ∈ ℝ^m,

<F'(p)δp, δ> = <δp, F'*(p)δ>,     (3)

where the notation <·, ·> denotes the l₂-inner product. Suppose <·, l> is a linearized functional of interest on P. Given a parameter perturbation δp, the forward functional sensitivity is then given by <F'(p)δp, l>. The same sensitivity can be obtained via the adjoint solution, ψ ≡ F'*(p)l. From the definition of the adjoint operator (3), it follows that:

<F'(p)δp, l> = <δp, ψ>.     (4)

That is, for all δp ∈ P the functional sensitivities can be computed via a single linear solution for ψ rather than repeated application of the linearized forward operator, F'(p)δp.

For the case where the functional of interest is the l₂-distance J(p) = ||F(p)_i - p_i||, the adjoint solution for the linearization J'(p)(·) is given in terms of the vector normal to the manifold Σ. Denoting N_i and N_s as the components of the normal vector in P_i and P_s respectively, it can be shown that (up to sign),

To fix the sign of ψ, the component N_i is chosen to be p_i - F(p)_i. Figure 2 illustrates the components of the vector N normal to the manifold Σ.

Thus, obtaining expressions for the adjoint vector for various bifurcations of interest reduces to the problem of deriving the associated normal vectors, N_s. Under certain transversality conditions, the normal vectors for several codimension-one and higher bifurcations have been derived 1617. Here we consider the generic codimension-one bifurcations, namely saddle-node and Hopf bifurcations. Let the left and right critical eigenvectors of f_x at the given bifurcation point be denoted by w and v respectively. That is, w and v solve the following eigen-systems for the critical eigenvalue ω_crit and its conjugate _crit,

f_xv = ω_crit v

.

The expressions for normal vectors are given as:

where superscript H denotes conjugate transpose and . These expressions above prescribe the components of the adjoint solution, thus enabling efficient gradient calculation via (4). Now, we briefly mention methods for computing the projection F(p). Methods of iterative and direct type for finding the (locally) closest bifurcation point have been derived 17. In the current work we use the former approach, based on using the component of the normal vector in the input plane, N_i. Provided certain conditions on the principal curvatures of Σ(p_s) are met, geometric convergence is assured. The algorithm is discussed in Section 3. Figure 3 illustrates the method in a simple example, producing a sequence of iterates (_i) converging to the point F(p)_i that is closest to a (non-convex) neighboring region with respect to p_i.

Finally, we mention that the minimal distance functional can be used to model many other problems of interest. For instance, it can serve as an estimate of the separation between a reference manifold Σ_ref and a given region of qualitative behavior, or the size of the region of a qualitative behavior via the maximum radius of the inscribed sphere.

Here we give an outline of the algorithm for general inverse bifurcation problems. The main ingredients are applications of the projection operator F(p), as well as the adjoint, F'*(p). The computation of the former is denoted by the routine APPLYF (see Figure 4). Each time APPLYF is called, corrector steps (using, for instance, Newton's method) have to be carried out on the previously computed x_init to find the initial solution for the current value of p_s. Once the corrected solution x_corr is computed, the iterative procedure LOCMINDIST (see Figure 5) is called to compute the nearest point on the bifurcation manifold of interest. This procedure is based on a series of one-parameter continuations and gradient calculations according to (6). The inputs include the following: v the initial search direction in parameter space; ε_proj the relative tolerance on the iterative proceure. In general, several search directions (denoted by V = {v₁,⋯, v_max}) have to be used to approximate the globally closest point. However, for the examples shown in the paper the initial search space V is only 1 dimensional. Once F(p) and the corresponding solution are obtained, F'*(p) is then computed. The derivative information, together with constraints (p_low, p_upp, c(p)) and Hessian approximation (HessA) are then used to compute an SQP step and update the Hessian. Figure 6 describes the inverse bifurcation algorithm.

Our Inverse Bifurcation Toolbox is an implementation of the above described algorithm. It combines the capability of several packages: the MATCONT package 20 for performing the underlying one-parameter continuations, the MATLAB Optimization Toolbox 21 for performing gradient-based constrained optimization, the MathSBML package 22 for reading in biological models in the SBML format and finally the Mathematica Symbolic Toolbox for MATLAB 23 for communications. The combination of MATLAB and Mathematica has the advantage of allowing building on existing freely-available software.

After the SBML model is read in via MathSBML, the vector fields are differentiated symbolically to obtain expressions for f_x, f_p, f_xp and f_xx. Subsequently, they are numerically evaluated when called by MATCONT. Reparametrization of parameters is done automatically to allow continuation in arbitrary directions. Presently, the toolbox is able to handle bifurcations of saddle-node and Hopf type. Augmentation to include bifurcations of limit cycles is currently underway.

Given the limitations of the current implementation, our software is suitable for handling problems involving tens of variables and parameters but possibly not suited for problem dimensionality on the order of hundreds or higher. Further algorithm development is needed to significantly upscale the method. In addition, in high dimensional applications instabilities may appear and hence appropriate regularization techniques need to be developed (e.g., stopping rules in the case of iterative methods).

We propose an inverse problems methodology to address a class of problems arising in the study and design of gene systems. The methodology is based on the (inverse) map from the space of bifurcation diagrams to the space of biochemical parameters. Within this general inverse bifurcation framework, we show that many questions of biological interest may be formulated as optimization problems involving minimal distances to bifurcation manifolds. Adjoint sensitivity analysis is carried out to allow for efficient solution using gradient-based optimization methods.

Several general questions remain for the methodology: what biological problems can be formulated in terms of geometric properties of the bifurcation diagrams? Given desired geometric properties for bifurcation diagrams, how best can the problem be formulated mathematically to allow for tractable solution? Finally, we remark that the development of appropriate numerical methods, novel computational strategies and regularization techniques are necessary to allow for the upscaling of the current computational tool to high dimensional biological applications.