Virginia Bioinformatics Institute, Virginia Tech, Washington Street, MC 0477, Blacksburg, VA 24061, USA

Department of Mathematics, Virginia Tech, 460 McBryde, Blacksburg, VA 24061-0123, USA

Department of Mathematics, The University of Tennessee, 227 Ayres Hall, 1403 Circle Drive, Knoxville, TN 37996-1320, USA

Department of Mathematics, Harvey Mudd College, 301 Platt Boulevard, Claremont, CA 91711-5901, USA

Department of Mathematics and Statistics, University of North Carolina - Greensboro, 116 Petty Building, Greensboro, NC 27402-6170, USA

Abstract

Background

Many biological systems are modeled qualitatively with discrete models, such as probabilistic Boolean networks, logical models, Petri nets, and agent-based models, to gain a better understanding of them. The computational complexity to analyze the complete dynamics of these models grows exponentially in the number of variables, which impedes working with complex models. There exist software tools to analyze discrete models, but they either lack the algorithmic functionality to analyze complex models deterministically or they are inaccessible to many users as they require understanding the underlying algorithm and implementation, do not have a graphical user interface, or are hard to install. Efficient analysis methods that are accessible to modelers and easy to use are needed.

Results

We propose a method for efficiently identifying attractors and introduce the web-based tool Analysis of Dynamic Algebraic Models (

Conclusions

Discrete modeling techniques are a useful tool for analyzing complex biological systems and there is a need in the biological community for accessible efficient analysis tools.

Background

Mathematical modeling is a crucial tool in understanding the dynamic behavior of complex biological systems. In addition to the popular ordinary differential equations (ODE) models, discrete models are now increasingly used for this purpose

The software tool introduced in this paper,

Results and Discussion

In this manuscript, we present the web-based tool

As the underlying computational approach, we propose a novel method to identify attractors of a discrete model. This method relies on the fact that many discrete models can be translated into the algebraic framework of polynomial dynamical systems. Using these polynomials, one can construct a system of polynomial equations, such that its solutions correspond to fixed points or limit cycles. Thus, the problem of identifying attractors becomes equivalent to solving a system of polynomial equations over a finite field. This is a long-studied problem in computer algebra, and can usually be solved efficiently by using Gröbner basis methods

In addition to providing access to mathematical theory for efficient analysis, algebraic models are a unifying framework and systematic approach for several model types. This allows for an effective comparison of heterogeneous models, such as a Boolean network model and an agent-based model. For community integration in the biological sciences,

General Features of ADAM

• Logical models generated with GINsim

• polynomial dynamical systems

• Boolean networks

• probabilistic polynomial dynamical systems, probabilistic Boolean networks (PBN)

The temporal evolution of the model can be visualized by the

With

All of these features can be computed assuming synchronous updates or sequential updates according to an update-schedule specified by the user. Note that the steady states are the same independent of the update schedule. This is due to the fact that updating any variable at a steady state does not change its value. It is irrelevant for a steady state analysis whether updates are considered to happen sequentially or simultaneously.

For probabilistic networks, i.e., models in which each variable has several choices of local update rules,

For Boolean networks,

In summary,

• wiring diagram

• phase space for small models

• steady states (for deterministic and probabilistic systems)

• limit cycles of specified length

• trajectories originating from a given initial state until a stable attractor is found

• dynamics for synchronous or sequential updates

• functional circuits for Boolean networks

• a complete description of the phase space for conjunctive/disjunctive networks.

Applications

We show how to use ^{18 }states. They analyze the model for steady states by manually solving a system of Boolean equations. They also analyze the temporal evolution of a specific initial state corresponding to the wild type expression pattern by repeatedly applying the Boolean update rules until a steady state is found. The update schedule of the model is synchronous with the exception of activation of SMO and the binding of PTC to HH (activation of PH), which are assumed to happen instantaneously. This can be accounted for by substituting the equations for SMO and PH into the update rules for other genes and proteins, rather than using SMO and PH themselves.

To analyze the model, we first rename the variables in the Boolean rules given in _{
i
}or SLP_{
i
}to _{1 }... _{60}, to standardize their format. The variables _{i }

Correspondence of genes and variable names

Cell 1

SLP

x1

wg

x2

WG

x3

en

x4

EN

x5

hh

x6

HH

x7

ptc

x8

PTC

x9

PH

x10

SMO

x11

ci

x12

CI

x13

CIA

x14

CIR

x15

Cell 2

SLP

x16

wg

x17

WG

x18

en

x19

EN

x20

hh

x21

HH

x22

ptc

x23

PTC

x24

PH

x25

SMO

x26

ci

x27

CI

x28

CIA

x29

CIR

x30

Cell 3

SLP

x31

wg

x32

WG

x33

en

x34

EN

x35

hh

x36

HH

x37

ptc

x38

PTC

x39

PH

x40

SMO

x41

ci

x42

CI

x43

CIA

x44

CIR

x45

Cell 4

SLP

x46

wg

x47

WG

x48

en

x49

EN

x50

hh

x51

HH

x52

ptc

x53

PTC

x54

PH

x55

SMO

x56

ci

x57

CI

x58

CIA

x59

CIR

x60

Genes and proteins in _{1},..., _{60}.

The rules in the model file are specified in

** ADAM: Analysis of steady states of Drosophila model**. Each row in the table corresponds to a stable attractor. Attractors are written as binary strings, where 0 represents non-expression of a gene (or low concentration of a protein), and 1 expression (or high concentration). Steady states of Drosophila Melanogaster as found with

Each row in the table in Figure

corresponds to the genes (and proteins) being expressed (or present in high concentration) in four cells from anterior to posterior compartments (compartment 1 to 4). The string can be translated back to a list of genes that are expressed in this stable attractor; see Table

Genes and proteins present in steady state

compartment 1

en, EN, hh, HH, SMO

compartment 2

ptc, PTC, PH, SMO, ci, CI, CIA

compartment 3

SLP, PTC, ci, CI, CIR

compartment 4

SLP, wg, WG, ptc, PTC, PH, SMO, ci, CI, CIA

Genes and proteins present in steady state corresponding to binary string (1).

Genes and proteins present in initial state

compartment 1

en, hh

compartment 2

ptc, ci

compartment 3

SLP, ptc, ci

compartment 4

SLP, wg, ptc, ci

Genes and proteins present in initial state corresponding to binary string (2).

By clicking

** ADAM: Trajectory of Drosophila model**. Temporal evolution of given initial state until steady state is reached.

Furthermore, we used

Benchmark Calculations

We analyzed logical models available in the GINsim model repository

Runtime of steady state calculations of several logical models from

**Runtime of steady state calculations of several logical models from **

In addition to the published models in ^{15 }- 10^{45 }states) and an average of average in-degrees of 1.6848. The steady state calculations took less than half a second for each network on a 2.7 GHz computer.

Comparison to Other Systems

In this section, we describe the functionality of several state-of-the-art software tools for the analysis of discrete models of biological systems. They are all capable of identifying steady states and limit cycles by exhaustive enumeration of the state space for small models (less than 32 variables)

Software Comparison

**Steady State**

**Analysis**

**Limit Cycle**

**Analysis**

**Input**

**Format**

**System**

**Requirements**

ADAM

Yes^{‡}

Yes^{◊}

Boolean (or polynomial) functions Logical Models (GINsim)

None, web based

GINsim

Yes^{‡}

For small models

Parameters (non-zero truth tables) Logical Model

Java virtual machine^{○}

BoolNet R package

For small ^{† }models

For small ^{† }models

Boolean functions

R statistics software

DDLab

For small models

For small models

Logical tables

BN/PBN Matlab Toolbox

For small models

For small models

Logical tables

Matlab

Comparison of different software tools regarding attractor analysis: ‡ less than 1 second on published gene regulatory networks with up to 72 variables; ◊ only for short limit cycles; † heuristic methods are available for larger networks; ○ installation necessary, available for common operating systems.

GINsim provides algorithms that use binary decision diagrams (BDD) for the determination of steady states

Non-heuristic analysis is limited to networks with 29 variables. For larger networks, steady states can be inferred heuristically, which does not guarantee that all steady states are identified.

Remarks about Logical Models

In this manuscript, we distinguish between three different update types: synchronous, sequential according to an update schedule, and asynchronous.

In models with asynchronous updates, as is common for logical models, one variable is updated at random at every time step, which results in a non-deterministic model. Models with sequential updates according to an update schedule produce dynamics that are different from that of models with asynchronous updates, i.e., logical models.

In GINsim, all models are

In multi-valued logical models, variables can have different maximum values. In an algebraic model, all variables are defined over the same algebraic field, i.e., have the same maximum value. When a multi-valued logical model is translated into an algebraic model, extraneous states might be introduced such that all variables are defined over the same field. An example of such an extension is given in Table

Multi-valued models

**next state of x**

**low x**

**medium x**

**high x**

_{1 }absent

low _{2}

medium _{2}

high _{2}

_{1 }present

medium _{2}

high _{2}

high _{2}

extension _{1 }present

medium _{2}

high _{2}

high _{2}

Updates for variable _{2 }in a logical model, where _{2 }depends on _{1 }and itself. The states 0 and 1 represent absent and present for the Boolean variable _{1}; 0, 1, and 2 represent low, medium, and high for the multi-valued variable _{2}. The last row is introduced in the polynomial dynamical system such that all variables are defined over

Architecture

All mathematical algorithms are programmed in Macaulay2

Output graphs are generated with Graphviz's

Model Repository

A model repository is part of the _{i }

New users can also use the repository to quickly familiarize themselves with the main functionalities of

Conclusions

Discrete modeling techniques are a useful tool for analyzing complex biological systems and there is a need in the biological community for easy to use analysis software.

After extensive experimentation with both discrete models arising in systems biology and randomly generated networks, we found that the algebraic algorithms presented in this manuscript are fast for sparse systems with few attractors, a structure maintained by most biological systems. All algorithms have been included in the software package

We hope to expand

Methods

Logical models, Petri nets, and Boolean networks are converted automatically into the corresponding polynomial dynamical system as described in

Gröbner basis calculation is for polynomial systems what Gauss-Jordan elimination is for linear systems: a structured way to transform the original system to triangular shape without changing its solution space. The triangular shape of the resulting systems allows for stepwise retrieval of the solutions of the system. For a more in depth discussion of Gröbner bases, see for example

In the worst case, computing Gröbner bases for a set of polynomials has complexity doubly exponential in the number of solutions to the system. However, in practice, Gröbner bases are computable in a reasonable time. It has been suggested, that in robust gene regulatory networks genes are regulated by only a handful of regulators

A Mathematical Background

A.1 Polynomial Dynamical Systems

To be self-contained, we briefly explain polynomial dynamical systems and their key features. A polynomial dynamical system**
**(PDS)

with coordinate functions _{i }
_{1},..., _{n}
_{i}
_{i }
_{i }
_{i }
_{1},..., _{n}

PDS have several dynamic features of biological relevance. These include the number of components, component sizes, steady states, limit cycles, and limit cycle lengths.

**Example **Let _{1}, _{2}, _{3}):

The wiring diagram of _{1}, _{2}, _{3}). The PDS described by

Wiring diagram: static relationship between variables

**Wiring diagram: static relationship between variables**.

Phase space: temporal evolution of the system

**Phase space: temporal evolution of the system**.

A probabilistic PDS**
**over a finite field

together with a probability distribution for every coordinate that assigns the probability that a specific function is chosen to update that coordinate. The coordinate functions _{i, j }
_{1},..., _{n}

A.2 Functional Edges

An edge in the wiring diagram from _{i }
_{j }
_{i}
_{i }
_{j}

B Algorithms

B.1 Analysis of stable attractors

Every attractor in a PDS is either a steady state or a limit cycle. For small models, ^{m}
_{i}
_{i }
^{m}

**Example **Fixed points of the system shown in the example in A.1 are solutions in

The only solution to this systems is the point (_{1}, _{2}, _{3}) = (0, 0, 0). This is in accordance with the state transition graph depicted in Figure ^{2}(

Again, (0, 0, 0) is the only solution, which means that there are no limit cycles of length two. Investigating ^{3}(

results in the solutions (0, 0, 0), (0, 1, 0), (0, 1, 1), (1, 1, 1). (0, 0, 0) is a steady state, and (0, 1, 0), (0, 1, 1), (1, 1, 1) are elements of a limit cycle of length 3. For all ^{m}

B.2 Conjunctive/Disjunctive Networks

Some classes of networks have a certain structure that can be exploited to achieve faster calculations. Jarrah et al. show that for conjunctive (disjunctive) networks key dynamic features can be found with almost no computational effort

Authors' contributions

FH led the algorithm and software development. BG, MB, and RM implemented the user interface and attractor analysis, executed benchmarking calculations, and drafted the initial manuscript under FH's direction. AV implemented the translation for logical algorithms to PDS used by

Acknowledgements

Dimitrova, Clemson University; J. Adeyeye, Winston-Salem State University; B. Stigler, Southern Methodist University; R. Isokpehi, Jackson State University are currently expanding