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
Genomics/proteomics
Interactions between molecules
Intracellular networks
Tissue level processes
complexity
Whole organism
Y. Lazebnik, Cancer Cell, 2002
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
1
,…,
f
n
be Boolean functions in
variables
x
1
,…,
x
n
.
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
.
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
Boolean networks
The phase plane
Compound
y
Compound
x
dx /dt = f
(
x,y
)
dy /dt = g
(
x,y
)
(
x
o
,y
o
)
dx = f
(
x
o
,y
o
)
dt
dy = g
(
x
o
,y
o
)
dt
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
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
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
An Example
X = z
Y = x
Z = y
x
y
z
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
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