Systems biology
SAMSI Opening Workshop
Algebraic Methods in Systems Biology and Statistics
September 14, 2008
Reinhard Laubenbacher
Virginia Bioinformatics Institute and Mathematics Department
Virginia Tech

“Living systems, being nonlinear dynamical systems, have properties different from their constituents in isolation, properties which emerge from the interactions among the molecular constituents; accordingly, it is the organization of these intermolecular processes in organisms that underlies their characteristic living properties. A reductionist or antireductionist strategy alone does not do justice to this claim. A new strategy seems needed […]
F. C. Boogerd et al., 2007

Tissue level processes
Interactions between molecules
Intracellular networks
Genomics/proteomics complexity

Y. Lazebnik, Cancer Cell, 2002

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G. Koh et al., Bioinformatics, 2006

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Model Types
Ideker, Lauffenburger, Trends in Biotech 21 , 2003

Discrete models of molecular networks
“[The] transcriptional control of a gene can be described by a discrete-valued function of several discrete-valued variables.”
“A regulatory network, consisting of many interacting genes and transcription factors, can be described as a collection of interrelated discrete functions and depicted by a wiring diagram similar to the diagram of a digital logic circuit.”
R. Karp, 2002

Nature 406 2000

Discrete modeling frameworks
1. Boolean networks and cellular automata
(including probabilistic and sequential BNs)
2. Polynomial dynamical systems over finite fields
3. Logical models
4. Dynamic Bayesian networks

Boolean networks
Definition. Let f variables x
1
1
,…, x
,…, f n
. n be Boolean functions in
A Boolean network is a time-discrete dynamical system f = ( f
1
,…, f n
) : {0, 1} n →
{0, 1} n
The state space of f is the directed graph with the elements of {0,1} n as nodes.
There is a directed edge b → c iff f ( b ) = c .

Boolean networks f
1
= NOT x
2 f
2
= x
4
OR ( x
1
AND x
3
) f
3
= x
4
AND x
2 f
4
= x
2
OR x
3

The phase plane
Compound x dy = g ( x o
,y o
) dt
( x o
,y o
) dx = f ( x o
,y o
) dt dx /dt = f ( x,y ) dy /dt = g ( x,y )
Courtesy J. Tyson

Boolean network models in biology
Stuart A. Kauffman
Metabolic stability and epigenesis in randomly constructed genetic nets
J. Theor. Biol. 22 (1969) 437-467.
Boolean networks as models for genetic regulatory networks:
Nodes = genes, functions = gene regulation
Variable states: 1 = ON, 0 = OFF

Polynomial dynamical systems
Note: {0, 1} = k has a field structure (1+1=0).
Fact: Any Boolean function in n variables can be expressed uniquely as a polynomial function in k [ x
1
,…, x n
] / < x i
2 – x i
>, and conversely.
Proof : x AND y = xy x OR y = x+y+xy
NOT x = x +1
( x XOR y = x+y )

Polynomial dynamical systems
Let k be a finite field and f
1
, … , f n
 k [ x
1
,…, x n
] f = ( f
1
, … , f n
) : k n → k n is an n -dimensional polynomial dynamical system over k .
Natural generalization of Boolean networks.
Fact: Every function k n → k can be represented by a polynomial, so all finite dynamical systems k n → k n are polynomial dynamical systems.

Example k = F
3
= {0, 1, 2}, n = 3 f
1
= x
1 x
2
2 +x
3
, f
2
= x
2
+x
3
, f
3
= x
1
2 +x
2
2 .
Dependency graph
(wiring diagram)

Sequential polynomial systems f k = F
3
1
= {0, 1, 2},
= x
1 x
2
2 +x
3 n = 3 f
2 f
3
= x
2
+x
3
= x
1
2 +x
2
2
σ =
(2 3 1) update schedule:
First update f
2
.
Then f
3
, using the new value of x
2
.
Then f
1
, using the new values of x
2 and x
3
.

Sequential systems as biological models
• Different regulatory processes happen on different time scales
• Stochastic effects in the cell affect the “update order” of variables representing different chemical compounds at any given time
Therefore, sequential update in models of regulatory networks adds realistic feature.

Stochastic models
Polynomial dynamical systems can be modified:
• Choose random update order for each update
(see Sontag et al.
for Boolean case)
• Choose an update function at random from a collection at each update
(see Shmulevich et al.
for Boolean case)

Logical models
E. Snoussi and R. Thomas
Logical identification of all steady states: the concept of feedback loop characteristic states
Bull. Math. Biol. 55 (1993) 973-991
Key model features:
• Time delays of different lengths for different variables are important
• Positive and negative feedback loops are important

Model description
Basic structure of logical models:
1.
Sets of variables x
1
, … , x n
; X
1
, … ,
X n
( X i
= genes and x i
= gene products, e.g., proteins.
A gene product x regulates a gene Y , with a certain time delay.)
Each variable pair x i
, X i takes on a finite number of distinct states or thresholds (possibly different for different i ), corresponding to different modes of action of the variables for different concentration levels.

Model description (cont.)
2. A directed weighted graph with the x i as nodes and threshold levels, indicating regulatory relationships and at what levels they occur.
Each edge has a sign, indicating activation (+) or inhibition (-).
3. A collection of “logical parameters” which can be used to determine the state transition of a given node for a given configuration of inputs.

Features of logical models
• Sophisticated models that include many features of real networks
• Ability to construct continuous models based on the logical model specification
• Models encode intuitive network properties

X = z
Y = x
Z = y
An Example x z y

Features of logical models
• Include many features of real biological networks
• Intuitive but complicated formalism and model description
• Difficult to study as a mathematical object
• Difficult to study dynamics for larger models

Equivalence of models
Theorem. (A. Veliz-Cuba, A. Jarrah, L.)
A logical model can be encoded as a PDS, without loss of information.
(Boolean case: H. Siebert)
(Similarly, certain types of Petri nets can be encoded as
PDS.)
This aids model analysis.

Dynamic Bayesian networks
Definition. A Bayesian network (BN) is a representation of a joint probability distribution over a set X
1
, … ,
X n of random variables. It consists of
• an acyclic graph with the
X i as vertices. A directed edge indicates a conditional dependence relation
• a family of conditional distributions for each variable, given its parents in the graph

BN models of gene regulatory networks
Can use BNs to model gene regulatory networks:
Random variables X i
↔ genes
Directed edges ↔ regulatory relationships
Problem: BNs cannot have directed loops. Hence cannot model feedback loops.

Dynamic Bayesian networks
Definition. A dynamic Bayesian network ( DBN ) is a representation of the stochastic evolution of a set of random variables { X i
}, using discrete time .
It has two components:
• a directed graph (
V , E ) encoding conditional dependence conditions (as before);
• a family of conditional probability distributions
P ( X i
( t ) | Pa i
( t 1 )) , where Pa i
= { X j
| ( X j
, X i
)

E }
(Doyer et al ., BMC Bioinformatics 7 (2006) )

Dynamic Bayesian networks
DBNs generalize Hidden Markov Models.
Recently used for inference of gene regulatory networks from time courses of microarray data.

Open problems
• Find good model inference methods (system identification) using “omics” data
• Find experimental design strategies appropriate for systems biology
• Formalize systems biology along the lines of mathematical systems theory