Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstrasse 1, 39106 Magdeburg, Germany

Abstract

Background

Mathematical modelling of cellular networks is an integral part of Systems Biology and requires appropriate software tools. An important class of methods in Systems Biology deals with structural or topological (parameter-free) analysis of cellular networks. So far, software tools providing such methods for both mass-flow (metabolic) as well as signal-flow (signalling and regulatory) networks are lacking.

Results

Herein we introduce

Conclusion

Background

Systems biology aims at a holistic analysis of biological networks. Mathematical modelling plays a pivotal role for this integrative approach. The arguably most common formalism for cellular networks is kinetic modelling, which has been successfully applied to the study of single pathways and networks of moderate size (e.g.

Implementation

The general set-up of CNA is shown in Figure ^{® }(Mathworks Inc.), a widely-used software for numerical computations and complex visualisations of numerical data. CNA has been programmed with the MATLAB language enabling to use built-in functions of MATLAB, in particular those for matrix operations. MATLAB also allows to call external C programs via the so-called MEX interface. CNA makes use of this interface for some computationally intensive calculations (see below).

General set-up of

**General set-up of CellNetAnalyzer**. For explanations see main text.

As CNA runs in the MATLAB environment and because MATLAB is available for many operating systems, CNA itself is platform-independent. Upon starting CNA in MATLAB's command window, CNA runs virtually autonomously as a graphical user interface.

Network projects

As a fundamental step, CNA facilitates the construction of

Example of an interactive network map of a simple signalling network

**Example of an interactive network map of a simple signalling network**. (Map and model were created with ProMoT [11] and exported to

A + B → C (1)

is interpreted as usual in MFNs, namely that the two reactants A and B are converted into C. A and B are consumed in this process representing the key characteristic of mass flows. MFNs are stored by the stoichiometric matrix and other variables such as the capacity and reversibility constraints of the reactions

In contrast, a reaction (or

Logical equations of the simple signal-flow network shown in Figure 2.

==> rec3

==> rec2

==> rec1

!rec2 + rec1 ==> a

rec2 ==> b

rec3 ==> b

a ==> c

c + b ==> e

d ==> c

b ==> f

!c ==> d

f ==> tf3

d ==> tf1

e ==> tf2

tf3 ==>

tf2 ==>

tf1 ==>

A "+" indicates an AND and a "!" a NOT operator.

3 A + !B → 2 C (2)

means that "C reaches level 2 if A is at level 3 AND B is inactive (level 0)". Using the formalism described above, CNA represents the logic of SFNs as

Specifically for SFNs, CNA also supports a

In principle, network analysis with CNA can be done without any graphical representation of the network. However, for visualisation purposes, each network project can be assigned one or several network maps (Figures

Example of an interactive network map of a mass-flow (metabolic) network in

**Example of an interactive network map of a mass-flow (metabolic) network in CellNetAnalyzer**. The network shows the central metabolism of

Example of an interactive network map in

**Example of an interactive network map in CellNetAnalyzer, here from a signal-flow network related to signalling pathways in T-cells**. The blue text boxes display a signalling path from the receptor CD4 to the transcription factor CRE. The menu of

An elegant solution for setting up logical network models is provided by a new feature of the modelling tool ProMoT: the model is created in a visual environment, and both the map and the underlying network can be exported to CNA and other formats

Results and Discussion

CNA provides a powerful battery of methods for

Metabolic networks

Regarding mass-flow networks, the majority of methods implemented in CNA belong to the constraint-based approach frequently used for metabolic network analysis

• general topological properties: (dead ends, blocked reactions, parallel reactions, enzyme subsets, etc)

• (elementary) conservation relations

• graph-theoretical features: shortest path lengths, connectivity analysis, network diameter etc.

• metabolic flux analysis: computing steady-state flux distributions from a set of given reaction rates (see example in Figure

• flux balance analysis: find optimal flux distributions for arbitrary linear objective functions

• metabolic pathway analysis with elementary modes

• minimal cut set analysis: intervention strategies for repressing a certain functionality in the network

Most of these functions were already part of

Apart from displaying EMs and MCSs directly in the interactive maps, CNA facilitates a detailed statistical assessment of large sets of MCSs and EMs. An important feature is the opportunity to select subsets of EMs or MCSs by specifying a set of criteria (e.g. "select all EMs involving reaction R1 but not R2"). Then, statistical properties can be calculated for the current selection and compared with other selections, useful e.g. to assess the importance of a reaction under different growth conditions. Such calculations include (relative) reaction participation, structural couplings, or optimal product yields.

Signalling and regulatory networks

CNA provides new algorithms designed for a functional analysis of signal-flow networks (most of the implemented methods were detailed in

Interaction graphs

The main features of CNA for studying interaction graphs comprises the computation and analysis of:

• general graph-theoretical network properties

• signalling paths and feedback loops (circuits)

• distance matrices capturing the lengths of the shortest negative/shortest positive path between all ordered pairs of species

• the

General graph-theoretical properties that can be computed include the number of components, the network diameter and others. A more sophisticated feature is the computation of signalling paths and of the network's feedback loops (circuits). For example, one can compute all (directed) signalling paths connecting a species

For computing paths and circuits, CNA utilises the same algorithmic approach as for EMs in metabolic networks

In very large networks, a full enumeration of paths and circuits becomes infeasible since the number of these objects depends exponentially on the network size. Instead of computing _{ik }and N_{ik}, respectively. Here we also allow _{kk }and N_{kk }represent the length of the shortest positive/negative circuit involving _{ik }and N_{ik }are usually different (sometimes only one of both is finite) and the minimum of both gives the (unsigned) shortest path length. For example, in Figure _{ik }= 2, N_{ik }= 3, N_{kk }= 4 and P_{kk }= ∞ (no positive circuit running over _{ik }= ∞ (a negative path from

Two examples of simple signed directed graphs (interaction graphs)

**Two examples of simple signed directed graphs (interaction graphs)**. Graph (b) is the same as graph (a) except that the negative arc from

The importance of shortest positive/negative paths and circuits has been emphasised in _{ik }and N_{ik }for all ordered pairs of nodes (_{ik }and N_{ik }of length ^{nd }iteration, a path length of 2 would be found for N_{ib}, P_{ik}, N_{ak}, P_{bd}, P_{cd}, P_{ke}, N_{df}, N_{ek}, P_{fd}. Hereby it is important to keep track of the predecessor node _{ik}, node _{ik}, node _{ib }and so on. This information can be used to reconstruct, at the end, the determined shortest path from _{ik }and N_{ik }it may here. For example, in Figure _{ik}= 2) and

With the algorithm described above, all the shortest positive/negative paths and circuits can be computed efficiently in large-scale networks (within seconds). They not only provide distance measures, but also show (i) whether _{ik}<∞ and N_{ik }= ∞ (there are only positive paths from _{ik }= ∞ and N_{ik}<∞ (all paths from _{ik }denotes the distance between _{ik }and N_{ik}):

Dependency matrix of the network in Figure 2 as displayed by

**Dependency matrix of the network in Figure 2 as displayed by CellNetAnalyzer**. The color of a matrix element M

• _{ik}= N_{ik }= ∞, i.e. D_{ik }= ∞ and there is no path from

• _{ik}<∞ and N_{ik }= ∞ and there is no node _{iz}<∞ and D_{zk}<∞ and N_{zz}<∞ (example:

• _{ik}<∞ and N_{ik }= ∞ and there is a node _{iz}<∞ and D_{zk}<∞ and N_{zz}<∞ (example:

• _{ik }= ∞ and N_{ik}<∞ and there is no node _{iz}<∞ and D_{zk}<∞ and N_{zz}<∞ (example:

• _{ik }= ∞ and N_{ik}<∞ and there is a node _{iz}<∞ and D_{zk}<∞ and N_{zz}<∞ (example:

• _{ik}<∞ and N_{ik}<∞ (example:

The global dependencies collected in the dependency matrix facilitate valuable qualitative predictions about the effects of perturbation or knock-out experiments

Note that this type of analysis is related to the theory on monotone systems and the notion of consistent graphs

In general, when computing (all or shortest) paths and feedback loops, CNA allows the user to exclude nodes or edges for testing knock-outs effects.

Logical networks

Boolean or logical networks have been extensively used for modelling small or medium-scale (gene) regulatory networks, typically characterised by having few (or no) inputs but many feedback circuits

(1) Logical steady states

CNA computes the logical steady state that follows from a user-defined scenario (consisting of a set of input stimuli, e.g. receptor X is activated and receptor Y not). This functionality enables to study how signals are propagated through the network and how a network responses to certain stimuli

Note that, sometimes, the logical steady state resulting from a given input pattern may be not unique for all nodes or a logical steady state does even not exist

(2) Minimal intervention sets

CNA provides a complex routine for computing (logical) minimal intervention sets (MISs;

To illustrate the concept of MISs consider Figure

For computing MISs, CNA uses an almost brute-force approach, since it checks systematically all

MISs can be displayed in the maps and assessed statistically. As outlined in

(i) searching for intervention strategies for repressing/provoking certain behaviours.

(ii) identifying fragile points in the network and estimating the importance of network elements for different functions (example: activated

(iii) identifying failure modes which might cause an observed abnormal (pathological) behaviour of the network (example (Figure

(iv) searching for candidates of missing links in the network by which experimental data not consistent with the current network model could be explained (for examples see

Additional features

A number of features, most of them available for both types of network projects, make work with CNA easier.

Integrating signal-flow and mass-flow networks

An important issue towards a holistic analysis of cellular networks is network integration, i.e. to facilitate the analysis of networks with mass

Though this approach is useful for a number of applications, it is unidirectional and is not able to close the loop, i.e. to account for the different kinds of interactions going from the metabolic network back to the regulatory or signalling part.

A quite different approach for connecting mass-flow with signal-flow networks, relying on interaction graphs, has been introduced recently

In our opinion, all of the above mentioned approaches enable the analysis of specific features of integrated mass-flow/signal flow networks but seem not yet general enough to consider all of the potential types of interactions that may occur. Accordingly, conceiving a more general conceptual framework for combining signal and mass flows and implementing it in CNA is a major aspect of our future work.

Conclusion

An increasing number of software tools is available for Systems Biology (see e.g.

Availability and Requirements

^{® }version 6.1 or higher. For a few calculations, the MATLAB Optimisation toolbox is required.

For academic purposes,

Commercial licenses are available for non-academic users.

List of abbreviations used

CNA CellNetAnalyzer

MFNs mass-flow networks

SFNs signal-flow networks

EMs elementary modes

MCSs minimal cut sets

MISs minimal intervention sets

Authors' contributions

SK developed

Acknowledgements

The authors thank support of the German Ministry of Research and Education (HepatoSys), the German Research Society (FOR521), and the Ministry of Education of Saxony-Anhalt (Research Focus Dynamical Systems). SK thanks Marcin Imielinski for drawing the attention to the Berge algorithm for computing minimal cut sets.