Logo of bmcbioiBioMed Centralsearchsubmit a manuscriptregisterthis articleBMC Bioinformatics
BMC Bioinformatics. 2009; 10(Suppl 12): S14.
Published online Oct 15, 2009. doi:  10.1186/1471-2105-10-S12-S14
PMCID: PMC2762063

A computational framework for qualitative simulation of nonlinear dynamical models of gene-regulatory networks

Abstract

Background

Due to the huge amount of information at genomic level made recently available by high-throughput experimental technologies, networks of regulatory interactions between genes and gene products, the so-called gene-regulatory networks, can be uncovered. Most networks of interest are quite intricate because of both the high dimension of interacting elements and the complexity of the kinds of interactions between them. Then, mathematical and computational modeling frameworks are a must to predict the network behavior in response to environmental stimuli. A specific class of Ordinary Differential Equations (ODE) has shown to be adequate to describe the essential features of the dynamics of gene-regulatory networks. But, deriving quantitative predictions of the network dynamics through the numerical simulation of such models is mostly impracticable as they are currently characterized by incomplete knowledge of biochemical reactions underlying regulatory interactions, and of numeric values of kinetic parameters.

Results

This paper presents a computational framework for qualitative simulation of a class of ODE models, based on the assumption that gene regulation is threshold-dependent, i.e. only effective above or below a certain threshold. The simulation algorithm we propose assumes that threshold-dependent regulation mechanisms are modeled by continuous steep sigmoid functions, unlike other simulation tools that considerably simplifies the problem by approximating threshold-regulated response functions by step functions discontinuous in the thresholds. The algorithm results from the interplay between methods to deal with incomplete knowledge and to study phenomena that occur at different time-scales.

Conclusion

The work herein presented establishes the computational groundwork for a sound and a complete algorithm capable to capture the dynamical properties that depend only on the network structure and are invariant for ranges of values of kinetic parameters. At the current state of knowledge, the exploitation of such a tool is rather appropriate and useful to understand how specific activity patterns derive from given network structures, and what different types of dynamical behaviors are possible.

Background

Although, up to now, there is no model that efficiently and accurately represents the gene interactions underlying regulatory mechanisms in their whole complexity, recent experimental evidence seems to confirm the adequacy of a specific class of ODEs to describe their essential dynamical features. These models assume that genes are controlled by transcriptor factors, and that the effect of a transcription factor on the transcription rate of a gene, the response function, is threshold-dependent. Such switch-like behaviors across variable thresholds are mathematically well represented by steep sigmoidal functions.

Although these models provide detailed description of gene regulatory molecular mechanisms [1-3], their predictive usefulness, at quantitative level, is rather limited even when the network at hand is very well studied. In fact, the exploitation of classical numerical approaches is mostly impracticable as precise and quantitative information on (i) the biochemical reaction mechanisms underlying regulatory interactions, and (ii) kinetic parameters and threshold concentrations are currently unknown and not identifiable from available data. However, at the current state of knowledge, qualitative predictions of the dynamical properties are not make-shift solutions but rather appropriate to get insight into the functioning of systems not completely understood as molecular interaction networks are.

The application of generic qualitative simulation approaches, as originally proposed in the Artificial Intelligence research framework [4] under the label QR, is not the proper solution. The mathematical tools they are grounded on are too simple to compensate for knowledge incompleteness. This results in a number of drawbacks, e.g. their inability to upscalability, the exponential growth of the generated behaviors, and the generation of spurious behaviors, that seriously limit the range of applicability of such methods to predict the nonlinear dynamics of regulatory networks. A qualitative study of GRNs dynamics could, in theory, be performed by analytical methods [5-8] based on the classical theory of qualitative analysis of dynamical systems, and properly adapted to the specific class of models. But, in practice, the network complexity makes quite hard or even unfeasible traditional analysis.

Pioneering work towards automated qualitative analysis and simulation of GRNs has resulted in a computational tool, called GNA[9]. Although GNA has been successfully applied to study real world networks, namely the initiation of sporulation in Bacillus subtilis [10], and the response to nutritional stress and carbon starvation in Escherichia coli [11,12], it is not flawless. GNA circumvents the hard problem of dealing with sigmoidal nonlinear response functions by approximating them with step functions, discontinuous in the threshold hyperplanes. Such an assumption considerably simplifies the analysis as the model results in piecewise-linear equations, but it raises the problem to find a proper continuous solution across the threshold hyperplanes, or, in other words, to seek for generalized solutions of ODEs with discontinuous right-side terms. The solution to this problem is not straightforward as (i) there exist in the literature several definitions of generalized solutions, (ii) it is not yet completely understood what are the relationships between different definitions, and then, (iii) it is not clear how to choose the "right" definition for a particular task [13]. GNA adopts the Filippov approach [14] that results particularly convenient to deal with control problems but it may present drawbacks when applied to approximate the limit solutions of a continuous ODE model: it might find "too many" solutions, and fail to reach all stable ones. Thus, GNA suffers from the same problems, that in addition to those raised by a further approximation introduced to deal with computational issues, might compromise its soundness and completeness (Dordan O, Ironi L, Panzeri L, Some critical remarks on GNA, in preparation).

The qualitative simulation algorithm we propose works under the assumptions that (i) threshold-dependent regulation mechanisms are modeled by continuous steep sigmoid functions, and (ii) any two genes are never regulated at the same threshold by a certain variable. The sigmoidal-nonlinearities make the problem quite hard to be tackled. But, the assumption that all sigmoids have very high steepness allows us to apply a systematic way of analysis. Let us observe that, due to the switch-like behavior of the response functions around the thresholds, the GRN dynamics occurs at different time-scales. To be able to deal with both slow and fast nonlinear dynamics we theoretically base our algorithm on a classical singular perturbation analysis method properly adapted to the assumed class of ODEs [8,15]. Such a method suitably combined with QR key concepts computationally drives, starting from initial conditions, the construction of all possible state transitions, and calculates the sets of symbolic inequalities on parameter values that hold when specific transitions occur.

A class of models of GRN dynamics

As phenomenological model of the complex dynamics of GRNs made up of n components, we consider the following generic equations:

equation image
(1)

where the dot denotes time derivative, xi is the concentration of the i-th gene product, γi > 0 is the decay rate of xi, Z is a vector with Zjk as components, and Zjk = S(xj, θjk, q) is a sigmoid function with threshold θjk. The response, or regulatory, function S : R+ → [0, 1] is a continuous monotonic S-shaped map depending on the parameter q (0 <q [double less-than sign] 1), that determines the steepness of S around the threshold value θjk, such that for q → 0 we have S(xj, θjk, q) = 0 (respectively 1) when the value of xj is smaller (greater) than θjk (Fig. (Fig.11).

Figure 1
Response function shape. The sigmoid response function S(x1, θ1, q) describes the relationship between the concentration of a gene product x1 and the relative production rate of the regulated gene. The parameter q determines its steepness around ...

Each xi, with domain Ωi [subset or is implied by] R+, is associated with ni thresholds ordered according to θij <θik if j <k. The state equations describe the balance between the production process fi(Z) and the degradation one, herein supposed to be linear. The functions fi are multilinear polynomials in the variables Zjk, and are frequently composed by algebraic equivalents of Boolean functions. More precisely,

equation image
(2)

where κil are real values that denote kinetic rate parameters, Li is the possibly empty number of interactions that synthesize xi, and, in accordance with the network structure, αjkl assumes value either equal to 1 when Zjk takes part in the l-th interaction or equal to 0 otherwise.

The validity of the biological assumptions underlying the model described by Eq. (1) has been confirmed by recent experimental evidence [16-28] and theoretical studies [29-32]. Thus, we can reasonably assume that such a model is suitable to give a phenomenological description of a wide range of regulatory systems in which the combined effects of a series of genetic processes, e.g. transcriptional and translational regulation, protein-protein interactions, metabolic processes, etc., can be properly described by threshold-dependent response functions.

Let us represent the dynamics of the n state variables xi, modeled by the Eq. (1) and associated with appropriate initial conditions, in the phase space. The ordered sets Θi of the ni threshold values θij associated with each xi naturally induce a partition of the phase space into qualitatively distinct domains. In the set Δ of all the domains identified by the partition, we can distinguish the set Δs [subset or is implied by] Δ of switching domains from the set Δr [subset or is implied by] Δ of regular domains, such that Δ = Δs [union or logical sum] Δr (Fig. (Fig.22).

Figure 2
An example of phase space partition. Partition of the phase space associated to a dynamical system with two state variables, where each variable xi [set membership] [0, An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i3.gif], i = 1, 2 is associated with two ordered thresholds θij, j = 1, 2. The white rectangles ...

A regular domain Dr (also called a box) is an open rectangular region between adjacent threshold hyperplanes in which the values of all response functions Zjk equal either 0 or 1. A switching domain Ds is a narrow region, whose width depends on the parameter q, surrounding either a section of a threshold hyperplane or an intersection of threshold hyperplanes. In a switching domain Ds we distinguish σ (Ds) switching variables, xs, from n - σ (Ds) regular ones xr. The values of a variables xs lies in the neighborhood of one of its thresholds and Zsk takes values in (0,1), while the values of a variable xr lies between two adjacent thresholds. The network dynamics in each domain D [set membership] Δ is described by different models. The motion in Dr [set membership] Δr is described by linear ODEs, and its analysis is straightforward. The motion in Ds [set membership] Δs, or equivalently around the thresholds, is described by nonlinear equations, and occurs at different time-scales. Therefore, to capture the salient features of the nonlinear dynamics in a switching domain, and to determine how the trajectories cross it to move towards other domains, a specific mathematical method is required.

Methods

Singular perturbation analysis in the switching domains

Singular Perturbation Analysis (SPA) is a classical approach to study phenomena that occur at different time-scales [33]. The dynamics of such phenomena are described by ODEs, associated with appropriate initial conditions, in which a small parameter (0 <q [double less-than sign] 1) multiplies either one of the derivatives or higher order derivative. Let us indicate this initial value problem by x2133q, and by x21330 the same problem where q = 0. As q → 0, the solution of x2133q identifies a "small" region, called boundary-layer region, of non-uniform convergence to the solution of the reduced system x21330. The region of uniform-convergence of x2133q to x21330 is called outer region. Taken together, the outer and boundary-layer solution approximate the solution of x2133q for small nonzero values of q.

Such a general approach is not directly applicable to our problem, but it must be tailored to it [8]. In outline, (i) the Eqs. (1) related to the xs variables are rewritten into the standard form of SPA through a change of coordinate system, (ii) the boundary-layer and outer solutions are calculated in the new coordinates, and (iii) they are converted back into the usual frame of reference.

Let Σ: Ω [mapsto] [0, 1]n be the coordinate transformation that converts the xs coordinates into the Zs ones. As under our assumptions, An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i4.gif, where ds is a continuous and limited function, we can rewrite Eq. (1) as in the following:

equation image
(3)

where ZS, ZR are the vectors of switching and regular variables Zs and Zr, respectively.

The fast dynamics in the boundary-layer is studied by suitably scaling the time variable, namely τ = t/q. In the limit, the fast dynamics is obtained by solving the system:

equation image
(4)

associated with appropriate initial conditions, where the prime denotes the derivative with respect to τ. As ZR is constant in any Ds [set membership] Δs, we focus on the switching variables Zs only. The system (4) has a manifold of stationary points, called slow-manifold, given by the solutions, for all s, of the stationary equations An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i7.gif = 0. We call exit point set (EP) the set of points in the slow-manifold that are stable and satisfy the Tikhonov-Levinson theorem, and we call Z-cube Ƶ(Ds) = An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i8.gif the frame of reference where we search for exit points.

The motion in the original time t along the exit points is described by the reduced system x21330. Thus, under the hypothesis that at least one exit point An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i9.gif exists, the motion equations of regular variables, or equivalently the outer solution, is represented by:

equation image
(5)

The problem (5) is linear, and then, given the initial conditions, the outer solution, that determines how the trajectories move along the xr directions, is easily calculated.

The calculation of the slow-manifold generally leads to an heavy, nonlinear computational problem as it consists in the solution of a set of polynomial equations. In the current implementation we adopt a further assumption that sounds realistic:

Assumption An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i11.gif. Every gene product only regulates one gene at each of its thresholds

Such an assumption considerably simplifies the calculation of the slow-manifold. Mathematically, it implies that each Zjk only occurs in one equation, and thus the terms fs(ZR, ZS) are linear.

Remark 1

The location of each exit point is crucial in our analysis as it indicates the next adjacent domains the trajectories are moving towards along the xs directions.

Let us observe that stationary points always exist on the vertices of Ƶ(Ds) as they are the roots of ds(Zs, θs) = 0. The computational cost of the search for all the other exit points could be quite high, but it can be considerably reduced by checking first a necessary condition for the existence of a stationary point on the other elements of Ƶ(Ds). Let F be a face or the interior of Ƶ(Ds). In [15], it has been proved that necessary condition for the existence of a stationary point in F is that the Jacobian matrix An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i12.gif restricted to the switching variables in F has a complete loop. This holds if and only if there is a non-zero loop involving all variables in JF, and can be checked by using concepts from graph theory.

Let An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i13.gif be a stationary point located on An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i14.gif [set membership] Ƶ(Ds), and An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i15.gif = {l : l [set membership] {l,..., σ (Ds)}, An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i16.gif [set membership] {0, 1}}. An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i13.gif is an exit point if (i) it is stable. If An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i14.gif is on the boundary of Ƶ(Ds), it is required, in addition to (i), that (ii) Zl, ∀ l [set membership] An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i15.gif, heads towards An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i14.gif. The stability of An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i13.gif is checked by analyzing the spectrum of the Jacobian matrix, and the condition (ii) is verified when, ∀ l [set membership] An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i15.gif, the sign of An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i17.gif, in a neighborhood of An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i13.gif is in agreement with the motion of Zl towards An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i14.gif. More precisely, in a neighborhood of An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i13.gif, An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i17.gif > 0 and An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i16.gif = 1 or An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i17.gif < 0 and An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i16.gif = 0.

Remark 2

SPA works out in the limit q → 0, but the calculated solution approximates the solution of Eq. (3) for sufficiently small q (0 <q [double less-than sign] 1). Moreover, it can be proved that the Jacobian matrix is stable for 0 <q <An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i18.gif [double less-than sign] 1. Thus, the exit points calculated in the limit are also valid for values of q <An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i18.gif.

Remark 3

Under the Assumption An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i11.gif, the reduced equations are always independent of the variables Zs in the Eq. (4), and the two sets of equations are mutually independent. Thus, the behavior of the variables xs in a Z-cube is completely independent of the values of variables xr, and the motion in a switching domain may be studied by first analyzing the former variables, and then the latter ones.

Qualitative reasoning concepts

Among the generic qualitative approaches proposed in the literature, QSIM provides both the most suitable formalism and algorithm to represent and simulate models qualitatively abstracted from ODEs [4]. Thus, we revise and ad hoc tailor its key concepts to our specific class of models.

Qualitative value

The real values of each state variable xi with domain Ωi = [0, An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i3.gif] are discretized into a finite ordered set of values qualitatively important from the biological point of view. In our context, the qualitative value space of each xi is defined by the ordered set Θi made up of both its ni threshold symbolic values and the endpoints of Ωi. The partition of the whole system domain, induced by the sets θi, i = 1,..., n identifies qualitatively distinct hyper-rectangles D that define all possible system qualitative values.

Qualitative state

Let A(D) be the set of domains adjacent to D [set membership] Δ. The qualitative state of D, QS(D), is defined by all of its adjacent domains Dk towards which a transition from it is possible:

equation image

Each transition from D identifies a domain next traversed by a system trajectory. More precisely, if we number by i the domain D traversed at time ti, each Dk [set membership] QS(D) will be traversed by different trajectories at time ti+1.

State transition

Qualitative simulation of network dynamics is achieved by iteratively applying local transition strategies from one domain to its adjacent domains. The possible transitions from any D are determined by different strategies according to whether D [set membership] Δr or D [set membership] Δs.

In the case D [set membership] Δr, like in traditional QR methods and in GNA[9], possible transitions are determined by the signs of An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i20.gif. As An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i20.gif are defined by linear expressions, such signs are easily determined by exploiting the inequalities that define the parameter space domain.

In the case D [set membership] Δs, a sign-based strategy is not practicable as the expressions for An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i20.gif are nonlinear. A convenient way to proceed is given by SPA: transitions from D towards adjacent Dk are determined by the locations of the exit points on the boundary elements of the associated Ƶ(D). Due to Assumption An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i11.gif, each boundary element of Ƶ(D) may contain at most one exit point. But, as, in a qualitative context, knowledge incompleteness on the parameter values is expressed by coarse constraints, the boundary element of Ƶ(D) where an exit point is located is not, in general, uniquely determined. Thus, unless the exit point is located in the interior of Ƶ(D) and a stable solution is reached, the successors of D are not uniquely determined. However, through symbolic computation procedures, it is possible to calculate the set of inequalities, An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i21.gif, on parameter values that hold when the transition from Di to Dk occurs. Then, the 3-tuple [left angle bracket]Di, Dk, An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i21.gif[right angle bracket] clearly and uniquely identifies each path from Di to Dk.

Qualitative behavior

A finite sequence of paths, where each path is clearly both linked and consistent with its predecessor and successor, defines a qualitative behavior: An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i22.gif, where D0 is the initial domain, and DF either contains a stable fixed point or identifies a cycle, i.e it is an already visited domain. I0 is the initial set of inequalities that defines the parameter space domain, and IF the set of inequalities on parameter values associated with DF.

Qualitative simulation

Starting from an initial domain D0 and a set, I0, of symbolic inequalities on parameter values, qualitative simulation generates all possible state transitions, and represents them by a directed tree rooted in D0, BT(D0), where the vertices correspond to Di, and the arcs, labeled by the inequalities An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i23.gif, to the transitions from Di to Dj. Each branch in BT(D0) defines a qualitative trajectory QB from D0, that traverses specific domains and occurs when the values of parameters satisfy its related inequalities. Such a trajectory abstracts all those numerical solution of the ODE model with initial conditions x0 varying in D0, and with kinetic parameter values fulfilling the inequalities.

Results

Given as input, (i) n symbolic state equations of the form (1); (ii) n qualitative value spaces Θi = {θij} of the state variables; (iii) D0 [set membership] Δ; (iv) a set of symbolic inequalities I0 on parameters defining a parameter space domain PSD0, the algorithm, through an iterative procedure initialized by [left angle bracket]D0, I0[right angle bracket], provides as output the entire range of possible network dynamics represented by BT(D0). Its main steps are outlined in the following:

1. Define the qualitative values by partitioning the phase space into regular and switching domains.

2. Calculate the qualitative state QS(Di) of the current domain Di.

(a) Calculate the symbolic state equations in Di, and the set A(Di);

(b) Determine constraints An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i21.gifon parameters that need to be fulfilled to have a transition from Di to Dk [set membership] A(Di);

(c) Check consistency of An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i21.gifwith I0, and build the path eik = Di Dk ;

3. Append [left angle bracket]Di, Dk, An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i21.gif[right angle bracket] to BT(D0), and mark Di as visited domain.

4. Repeat from step 2 for each not visited Dk.

Step 2, and in particular the calculation of the conditions on parameters An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i21.gif, is the core of the overall algorithm. A transition from Di to Dk actually occurs, and then Dk [set membership] QS(Di), only if the set An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i21.gif is consistent with the set I0, i.e. when it defines a not empty parameter space domain An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i24.gif such that An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i24.gif [subset, dbl equals] PSD0. The calculation of an inequality set An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i21.gif is performed by two distinct algorithms that implement a different transition strategy according to whether Di [set membership] Δ r or Di [set membership] Δs.

Moving from a regular domain

The algorithm in charge of the construction of the possible paths from regular domains is, in principle, similar to that one proposed by GNA, but it is more informative as it calculates the An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i21.gifs. In the limit, we indicate Di [set membership] Δr by the product An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i25.gif where (θji, θj(i+1)) denotes the interval of xj in Di.

(a) Symbolic state equations in Di [set membership] Δr. In each box Di, Zjk equals either 0 or 1 in the step function limit. This simplifies Eq. (1) as they reduces to linear equations:

equation image
(6)

where μj depends on Di, and is given by the sum of some κjls. From Eq. (6) we can easily find the focal point An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i27.gif the trajectories are heading towards. Herein, we assume that focal points do not belong to switching domains. If x* belongs to Di, there is a stable point in it. Such a stable point exists in Di, i.e. Di [set membership] QS(Di), if the set of inequalities An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i28.gifj [set membership] {1,..., n} exists and is consistent with I0. Otherwise, the trajectories move towards a switching domain Dk adjacent to Di.

(b) Constraints An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i21.gifon parameters. ∀ Dk [set membership] A(Di), the algorithm calculates the set of inequalities on parameters An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i21.gif that need to be fulfilled to have a transition from Di to Dk. As in Di all the equations (6) are linear, and all the trajectories head towards a focal point x* in a regular domain, such inequalities are calculated by imposing that the signs of state variable rates match the relative position of Dk with respect to Di. We define the relative position of Dk with respect to Di, indicated by An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i29.gif where vj [set membership] {-1, 0, 1}, through the comparison of the intervals defining Di and Dk. An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i21.gif, initialized to I0, is updated, ∀ j [set membership] {1,..., n}, with either the inequality An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i30.gif if vj = 1 or An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i31.gif if vj = -1.

(c) Possible transitions from Di to Dk. If the calculated inequality set defines a not empty parameter space domain An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i24.gif [subset, dbl equals] PSD0 then a transition towards Dk is possible and the qualitative state QS(Di) is updated accordingly.

To exemplify how the algorithm works we consider the ODE model:

equation image
(7)

where the parameters are positive, and all the response functions are expressed by the Hill function commonly used in the literature.

Let us consider the domain D11 in Fig. Fig.22 as current Di, and let us define I0 as follows:

equation image
(8)

The conditions on parameters in Table Table11 that are consistent with I0 identify the possible transitions from D11 towards adjacent domains in A(D11) = {D6, D7, D12, D16, D17}. Among the inequalities in Table Table1,1, An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i34.gif is the only one in agreement with I0. Thus, QS(D11) = {D12} and the only possible transition from D11 occurs when An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i34.gifI0 holds.

Table 1
Example parameter inequalities

Moving from a switching domain

In a domain Di [set membership] Δs, the nonlinear dynamics is characterized by fast and slow motions, respectively associated with xs and xr, that are independently calculated. Let us reindex the variables xj, Zj such that the switching variables come first, and proceed first with the construction of the fast motion by exploiting the SPA approach.

A – Fast motion

The study of the fast dynamics is performed in Ƶ(Di) in the scaled time τ = t/q, and aims at localizing the set of exit points in Ƶ(Di) rather than at detailing the dynamics within it. Such points clearly identify the next adjacent domains the trajectories are moving towards from Di along the xs directions.

To this end, the algorithm defines, in the limit, the mapping An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i46.gif: An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i47.gifƵ(Di), where An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i47.gif = Di [union or logical sum] A(Di), such that the interior of Ƶ(Di), and its boundary are the images of Di, and A(Di), respectively. More precisely, the domains Dk [set membership] A(Di) are mapped into the faces of Ƶ(Di) when Dk [set membership] Δs or into its vertices, otherwise. Let F be the set of both the faces and the interior of Ƶ(Di), and F its generic element.

To exemplify, let us consider the switching domain D7 in Fig. Fig.2.2. The set F has five elements, the four faces corresponding to D2, D6, D8, D12, and the interior of Ƶ(D7) that correspond to D7. Moreover, the vertices of Ƶ(D7) are the images, through the mapping Σ, of the adjacent regular domains D1, D11, D3, and D13 (Fig. (Fig.33).

Figure 3
Mapping between Di and Ƶ(Di). (a) Mapping of D7 and its adjacent domains in Fig. 2 into the elements of Ƶ(D7). The empty circles on the boundary of Ƶ(D7) denote the candidate exit points. (b) The filled circles denote the exit ...

(a) 1. Symbolic state equations in Di [set membership] Δs

The algorithm symbolically calculates the boundary-layer equations (4) in the Z variables associated with the current switching domain.

As an example, let us consider the model (7), and the domain D7, where both variables x1, x2 are switching in it, respectively, around the thresholds θ11, θ21. In D7, characterized by fast motion only, the boundary-layer system is given by:

equation image
(9)

2. Identification of the candidate next traversed domains Dk

Among all the domains Dk [set membership] A(Di) only those that correspond to an element on the boundary of Ƶ(Di) where a stationary point is located are possible domains next traversed by the trajectories. Then, the algorithm first builds the subset of elements of A(Di) that are candidate successors of Di.

Let us denote by EP the set of stationary points, initially made up of the vertices of Ƶ(Di). Under the Assumption A, each element F [set membership] F, may contain at most one stationary point. A necessary condition for the existence of a stationary point on F is the presence of a non-zero loop in JF, the Jacobian matrix calculated on F. Then, ∀ F [set membership] F, the algorithm calculates JF and searches for a non-zero loop involving all variables in JF. In case, it symbolically calculates the stationary point on F, and updates accordingly the set of candidate exit points EP. Let us observe that one internal stationary point exists if all variables in Ƶ(Di) are involved in a loop.

In the example, only An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i51.gif and An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i52.gif have a non-zero loop. Then, the algorithm looks for the stationary state on F2 and F12: An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i53.gif and An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i54.gif. Finally, the exit point candidate set is updated with the points An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i55.gif and An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i56.gif (Fig. 3(a)).

(b) Constraints An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i21.gif on parameters

The inequality set An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i21.gif, initialized to I0, is calculated for each candidate exit point An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i57.gif by requiring that each point is stable and fulfills other motion conditions on parameters. More precisely, the algorithm:

1. analyzes the spectrum of the Jacobian matrix JF, and imposes possible conditions An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i58.gif on parameters to ensure stability of An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i59.gif;

2. imposes conditions Ik, l on the sign of An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i17.gif, ∀ l [set membership] LF, in a neighborhood of An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i59.gif so that the motion of Zl heads towards An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i59.gif;

3. for those An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i59.gif located on elements of F, imposes the further condition, Ik,(0, 1), 0 <An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i60.gif < 1 for each An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i60.gif that does not take value 0 or 1.

Finally, An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i59.gif is an exit point if An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i61.gif holds.

Conditions 2. and 3. are easily checked, while the stability condition is checked by using concepts from graph theory, and the usual definition of stability based on the sign of the eigenvalues of JF. It follows from Assumption An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i11.gif that the matrix JF is block-structured, where each block is a permutation matrix associated with a sub-loop. It can be proved that An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i59.gif is stable if: (i) An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i62.gif has no blocks with dimension strictly greater than 2; (ii) in 1-dimensional blocks the elements are negative; (iii) in 2-dimensional blocks the loop products are positive.

Remark 4

Let us consider the general case when Di [set membership] Δs is characterized by both switching and regular variables. The motion from an exit point located in the interior of Di occurs in a sliding mode along a stable point in the slow-manifold of the boundary-layer system, and is described, in the normal time, by the reduced system. Then, a stable stationary point exists in Di, i.e. Di [set membership] QS(Di), if a stable point exists in the interior of Ƶ(Di), and the regular variable rates are zero in a point inside Di. The set of inequalities that checks the latter condition are defined by An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i63.gifj [set membership] {σ (Di) + 1,..., n}.

(c) Possible transition from Di to Dk

The exit points located on Ƶ(Di) clearly identify the set of all possible exit domains, i.e. those domains towards which a transition from Di is possible. Such domains are easily calculated by applying the map Σ-1 to each element of Ƶ(Di) that contains an exit point. Let us observe that the remaining unstable stationary points in EP are possible entrance points to Di.

In the example, both An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i51.gif and An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i52.gif are 1-dimensional block matrices with negative elements. Then, both exit points An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i55.gif and An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i56.gif fulfill the stability condition 1. without further constraints imposed on the parameters. The condition 2. on variable Zl, l = 2 imposes:

equation image
(10)
equation image
(11)

The inequality I12,2 defined by (11) is compatible with (8), but the inequality I2,2 is not. Then, An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i55.gif is removed from the exit point set. To be an exit point An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i56.gif must satisfy the condition 3.:

equation image

Finally, An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i56.gif is an exit point if An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i67.gif, defined by I0 I12,2 I12,(0,1), holds.

As for vertices in the example, the conditions 1. and 2. are fulfilled in the point An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i68.gif = (0, 1) that corresponds to the vertex defined as image of D11 by the map Σ.

Finally, the only exit domains are D12, D11, and then QS(D7) = {D12, D11} (Fig. 3(b)).

B – Slow motion

The slow motion of regular variables xr along the exit points is studied in the normal time t in the usual frame of reference, and it is reconstructed from the reduced system (5) through the same symbolic procedure given for regular domains.

Qualitative simulation outcomes

Through the described iterative procedure, the algorithm builds (i) a behavior tree, rooted in the initial domain, and (ii) sets of inequalities on parameter values that characterize specific state transitions. The behavior tree captures the entire spectrum of network dynamical behaviors that are invariant for ranges of parameter values defined by the calculated inequalities.

To illustrate the type of output the algorithm produces, let us consider the results of the simulation of the ODE model (7) starting from D1 with the parameter space defined by the inequalities I0 (8). Through the described iterative procedure, the algorithm builds the behavior tree, BT(D1), showed in Fig. Fig.4,4, and calculates the inequalities on the parameters, listed in Table Table2,2, that are associated with each path in BT(D1). As in the example n = 2, the trajectories described by the tree are also easily summarized in the phase plane (Fig. (Fig.5).5). Three reachable stable states, located in D11, D12 and D5, are identified by the final leaf of each branch in BT. As D12 [set membership] Δs, one of them is a singular stable point whereas the others are regular stable points. These stable states are reached by different predicted qualitative behaviors, each of them occurring under specific constraints on parameters. For example, the trajectory QB13 starting from D1, crossing D6, and reaching a stable point in D11 is allowed when the inequalities An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i69.gif and I11, consistent with I0, hold.

Table 2
Inequalities associated with BT(D1)
Figure 4
Behavior tree. Behavior tree rooted in D1, BT(D1), obtained by the simulation of the model Eqs.(7) with initial condition [left angle bracket]D1, I0[right angle bracket], where I0 is defined by Eq. (8). Each branch in the tree defines a qualitative trajectory that occurs when ...
Figure 5
Representation of the qualitative trajectories in the phase space. Phase space representation of trajectories defined by BT(D1) in Fig. 4, filtered of the spurious behaviors QB2 and QB11. Stable states are denoted by ·. The labels An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i21.gif denote the ...

In general, for n > 2, due to graphical difficulties, each branch in the tree is given a time representation. More precisely, a specific qualitative behavior, e.g. QB4, is described by the temporal evolution of each variable (Fig. (Fig.6).6). The behavior in Fig. Fig.66 abstracts all numerical trajectories of the ODE model (7) obtained with different initial values in D1, and different sets of parameter values that fulfill the same inequalities associated with QB4 (Fig. (Fig.77).

Figure 6
Temporal representation of a qualitative trajectory. Qualitative temporal representation of the behavior labeled QB4 in Fig. 4, namely of the variables x1 (a) and x2 (b). On the horizontal axis, the discrete sequence of symbolic instants at which the ...
Figure 7
Numerical simulation results. Time courses of the variables x1 (a) and x2 (b) obtained by numerical simulations of the model (7). The different plots result from simulations with different initial numeric values in D1, and different sets of parameter ...

Discussion

The algorithm is currently under implementation. As for symbolic calculus, the implementation requires to tackle complex tasks, such as: (i) update an inequality set with an another one; (ii) check the consistency of two sets of inequalities I1 and I2; (iii) solve systems of equations; (iv) find cycles in the Jacobian matrix. As for (iii), the original equations are multilinear in Zs, but due to Assumption An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i11.gif they can be straightforward solved and analyzed for stability. Also the solution of problems (i) and (ii) benefits from Assumption An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i11.gif as the inequalities are always linear. Then, thanks to the Assumption An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i11.gif, and to algorithms proposed both by the literature and common symbolic computation package, such as Mathematica [34], the tasks (i)-(iii) are simplified and feasible. As for the task (iv), it is performed by using cycle-detection algorithms and matrix graph theory tools [35].

Soundness and completeness

Soundness

The algorithm guarantees that the behavior tree captures all of the sound behaviors for values of q sufficiently small. A closed-form expression of a symbolic upper bound of q, An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i18.gif, in terms of model parameters guarantees the Jacobian matrix stability. Then, the generated behaviors are sound for values of q <An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i18.gif. However, for a complete formal proof of soundness we need to prove that assuming, in the limit, stability instead of asymptotic stability does not affect the results for values of q <An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i18.gif.

Completeness

At the current stage, the algorithm may generate spurious behaviors. There is a twofold explanation for that. First, we have not yet performed a thorough analysis with respect to entrance-exit transition, or in other words, we have not yet solved the problem of identifying the only admissible connections between entrance and exit points. Moreover, singular perturbation analysis is a local procedure that works quite well in a quantitative context but that needs, in a qualitative context, to be supported by a global criterion when local paths are connected to produce a specific trajectory. For example, the behavior QB2 is spurious as An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i67.gif is not consistent with An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i81.gif. Similarly, QB11 is spurious.

The characterization of the paths from one domain to the next ones by sets of inequalities constraining the model parameters is quite new in the field of qualitative simulation, as for both general-purpose and specifically tailored algorithms. Such a strategy may reveal quite useful in the definition of a "global criterion" that allows us to distinguish sound behaviors from spurious ones by requiring that the sets of inequalities that label the local paths in a specific trajectory are consistent with each other. Both the definition of such a criterion and its implementation are not a trivial task, especially from a computational point of view. As for algorithm completeness, another essential methodological and computational issue to be deepened deals with the definition of the transition map that properly connects the entrance points to the exit points associated with a switching domain.

Comparison between the algorithm and GNA

Until now, to the best of our knowledge, GNA is the only computational counterpart of the algorithm herein proposed. Unlike GNA, that simplifies the simulation problem by approximating the sigmoid response functions by step functions discontinuous in the thresholds, our qualitative simulation algorithm tackles the problem in all of its complexity, and works for models of GNAs with continuous sigmoid response functions. As a matter of fact, it has been designed to both provide a tool for a more realistic modeling framework and overcome the limits of GNA. Furthermore, in addition to more solid mathematical grounds for soundness and completeness, our algorithm differs from GNA as far as the required input information and the output outcomes are concerned. In GNA, the trajectories are constructed under the assumption that the locations of the focal points associated with the regular domains, symbolically expressed by a set of inequalities on parameters, are specified as input. But, as this precise knowledge is unlikely available, a thorough study of the dynamics of the system at hand might require several runs of the algorithm to simulate the GNA dynamics under possible different assumptions on focal point locations. Then, the study could result quite heavy due to the high number of possible scenarios, and the need to check, to avoid further causes of generation of spurious behaviors, the consistency of all the parameter inequalities that precisely localize a specific focal point.

Our algorithm does not need precise information on the location of focal points but, to be fired, it just requires the parameter space domain, that is inequalities on parameters that univocally determine the sign of direction of change of each state variable in the initial domain. Afterwards, it is up to the algorithm to calculate and refine the parameter inequalities that distinguish each simulated behavior. Due to more relaxed constraints, a single run of our algorithm leads to a richer and more informative set of behaviors than that one obtained by GNA as it is highlighted by the comparison of Fig. Fig.55 with Fig. Fig.8.8. Let us observe that any other possible definition of parameter inequalities would not have led to the set of behaviors in Fig. Fig.55 but to one of its subsets.

Figure 8
GNA simulation outcomes. Simulation outcomes of the example model obtained by the algorithm GNA given as input D1 and the parameter inequalities, compatible with I0, An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i82.gif; An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i83.gif; An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i84.gif; An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i85.gif (panel (a)), An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i86.gif; An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i87.gif; An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i88.gif; An external file that holds a picture, illustration, etc.
Object name is 1471-2105-10-S12-S14-i89.gif (panel (b)). The states generated by GNA, and labeled Sn, ...

A complete and fair performance evaluation of our approach when compared with GNA would not disregard its application to those real world networks successfully simulated within the GNA framework, namely the initiation of sporulation in Bacillus subtilis [10], and the response to nutritional stress and carbon starvation in Escherichia coli [11,12].

Application to Systems/Synthetic Biology

The qualitative simulation algorithm presented provides powerful computational grounds for dry experiments to be performed in both Systems and Synthetic Biology contexts. It is an economic tool for hypothesis testing and knowledge discovery as it allows us to understand how specific activation patterns are derived from models of fixed network structures, and which different dynamical behaviors are possible.

In a Systems Biology context, its exploitation is useful in the formulation phase of new hypotheses and theories that explain, in both physiological and pathological situations, experimental data either previously unobserved or not interpretable by the well-established biological knowledge. The result of the comparison of the simulated behaviors with the observed ones allows us either to confirm or to refuse the model that represents the new formulated biological hypotheses. Moreover, as qualitative simulation derives the entire range of network dynamics consistent with the initial conditions, the simulation outcomes might capture behaviors never observed. Such behaviors need to be experimentally tested by ad hoc designed experiments to possibly further confirm or suggest how to revise the model, i.e. the underlying biological hypotheses. The effective applicative potential of the proposed algorithm within a model-based reasoning cycle for the deep comprehension of complex real world biological systems is currently under study. More precisely, it will be applied (1) to explain and interpret the dynamics of the complex regulatory network that controls motility in Bacillus subtilis [36], and (2) to explore the molecular mechanisms that regulate the transition of a normal cell phenotype to an invasive-metastatic phenotype [37].

The model-based reasoning cycle results particularly convenient and useful also when applied to Synthetic Biology problems in the design phase. The construction of a synthetic biological network in the cell, able to perform specific tasks or to produce desired behaviors of a biological process in response to external stimuli, could benefit from a preliminary dry design of the network. To this end, models of different network structures can be simulated in correspondence to different inputs, i.e signals either exogenous or cellular, till the desired stable behaviors are obtained. Let us remind that, in our simulation framework, each specific simulated behavior is characterized by a chain of parameter inequalities. Thus, the actual construction of a synthetic network could benefit from such a piece of information as it provides the admissible parameter space domain for the target behaviors. The synthetic network IRMA[38] built in the yeast Saccharomyces cerevisiae is made up of a small number of components but rather complex in their interconnections as it includes multiple feedback loops generated by the combination of transcriptional activators and repressors. It offers a suitable benchmark for in vivo testing our computational framework.

Conclusion

The assumption of continuous response functions makes the simulation problem hard to be tackled but it is crucial in view of the realization of tools that can be gradually extended to tackle more and more realistic models. Our algorithm is grounded on a set of symbolic computation algorithms that carry out the integration of qualitative reasoning techniques with singular analysis perturbation methods: the former techniques allow us to cope with uncertain and incomplete knowledge whereas the latter ones lay the mathematical groundwork for a sound and complete algorithm capable to deal with regulation processes that occur at different time-scales.

The modeling framework and the simulation algorithm proposed can be applied to predict the possible qualitative behaviors of GRNs of any size and complexity, both natural and synthetic. The range of applicability of such a computational approach is quite large. First, it makes possible the validation of hypothesized models of GRNs by matching the simulated predictions against observed gene expression profiles, the derivation of the most plausible model, and the identification of parameter inequalities that account for the observations. Then, it greatly facilitates the investigation of the effects on the dynamics of a specific GRN in response to external stimuli. Finally, it allows us to build a total envisionment of the model through the generation of all possible state transitions, and to search, within them, for the parameter constraints and initial conditions that allow us to achieve a desired behavior, e.g. a stable solution. In term of tasks, besides its undoubted contribution to the comprehension of complex networks of genes and interactions, the proposed computational approach could be fruitfully used in the design of either new drugs or synthetic regulatory networks.

The code will be freely available for non-profit academic research by making a user licence request to ti.rnc.itami@inori.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

LI devised the work, and drafted the manuscript. Both authors participated in the design and development of the algorithms. LP implemented the software. All authors read and approved the final manuscript.

Acknowledgements

This research is carried out within the Interdepartmental CNR-BIOINFORMATICS Project. We gratefully acknowledge O. Dordan, E. Plahte, and V. Simoncini for the useful discussions on different methodological aspects and mathematical problems related to the presented work.

This article has been published as part of BMC Bioinformatics Volume 10 Supplement 12, 2009: Bioinformatics Methods for Biomedical Complex System Applications. The full contents of the supplement are available online at http://www.biomedcentral.com/1471-2105/10?issue=S12.

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