An Introduction to Systems Biology - SAMSI

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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