Discrete models of biological networks

Segunda Escuela Argentina de Matematica y BiologiaCordoba, Argentina

June 29, 2007

Reinhard LaubenbacherVirginia Bioinformatics Institute

andMathematics Department

Virginia Tech

Topics

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 f1,…,fn be Boolean functions in variables x1,…,xn. A Boolean network is a time-discrete dynamical system

f = (f1,…,fn) : {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.

f1 = : x2

f2 = x4 OR (x1 AND x3)

f3 = x4 AND x2

f4 = x2 OR x3

Boolean networks

The phase plane

Com

pou

nd

y

Compound x

dx /dt = f (x,y)dy /dt = g(x,y)

(xo ,yo)

dx = f (xo ,yo) dt

dy = g(xo ,yo) dt

Courtesy J. Tyson

Cellular automata

Definition. A 1-dimensional (binary) cellular automaton (CA) f is a Boolean network f in which fi only depends on some or all of

xi-1, xi, xi+1 (modulo n).

Example. fi = xi-1 XOR xi+1.

t =1:

t =2:

Initial State:

t =3:

t =4:

t =5:

t =6:

t =7:

t =8:

t =9:

Example

Rule 90 with 5 nodes

f(x1,x2,…,x5) = (x5 XOR x2, x1 XOR x3, … , x4 XOR x1)

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[x1,…,xn] / <xi2 – xi>,

and conversely.

Proof: x AND y = xyx OR y = x+y+xy

NOT x = x+1(x XOR y = x+y)

Polynomial dynamical systems

Let k be a finite field and f1, … , fn k[x1,…,xn]

f = (f1, … , fn) : kn → kn

is an n-dimensional polynomial dynamical system over k.

Natural generalization of Boolean networks.

Fact: Every function kn → k can be represented by a polynomial, so all finite dynamical systems kn → kn

are polynomial dynamical systems.

Example

k = F3 = {0, 1, 2}, n = 3

f1 = x1x22+x3,

f2 = x2+x3,

f3 = x12+x2

2.

Dependency graph(wiring diagram)

Sequential polynomial systems

k = F3 = {0, 1, 2}, n = 3

f1 = x1x22+x3

f2 = x2+x3

f3 = x12+x2

2

σ = (2 3 1) update schedule:

First update f2.

Then f3, using the new value of x2.

Then f1, using the new values of x2 and x3.

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 (PDSs) 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)

Open mathematical problems

• Determine the relationship between the structure of the fi and the dynamics of the system for special classes of models (see later lectures).

• Determine the effect of the update schedule on dynamics.

• Develop a categorical framework for (sequential/stochastic) PDSs.

• Determine and study a good class of “biologically meaningful” polynomial functions.

Example

A. Jarrah, B. Raposa, and R. Laubenbacher,

Nested canalyzing, unate cascade, and polynomial functions, Physica D, in press

Logical models

E. Snoussi and R. ThomasLogical identification of all steady states: the concept of feedback loop characteristic statesBull. 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 x1, … , xn; X1, … , Xn

(Xi = genes and xi = gene products, e.g., proteins. A gene product x regulates a gene Y, with a certain time delay.)

Each variable pair xi, Xi 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 xi 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

• Ability to relate structure (+ and - feedback loops) to dynamics (multistationarity, fixed pt vs. periods)

An Example

X = z

Y = x

Z = y

xy

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

Dynamic Bayesian networks

Definition. A Bayesian network (BN) is a representation of a joint probability distribution over a set X1, … , Xn of random variables. It consists of

• an acyclic graph with the Xi as vertices. A directed edge indicates a conditional dependence relation

• a family of conditional distributions for each variable, given its parents in the graph

An example

http://www.cs.ubc.ca/~murphyk/Bayes/bnintro.html#repr

Inference

Bayes’ rule: P(R=r | e) = P(e | R=r)P(R=r)/P(e)Cond. Prob.: P(A | B) = P(A∩B)/P(B)

BN models of gene regulatory networks

Can use BNs to model gene regulatory networks:

Random variables Xi ↔ 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 {Xi}, 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(Xi(t) | Pai(t-1)), where Pai = {Xj | (Xj, Xi) E}

(Doyer et al., BMC Bioinformatics (2006) 7)

Dynamic Bayesian networks

DBNs generalize Hidden Markov Models and linear dynamical systems.

Recently used for inference of gene regulatory networks from time courses of microarray data.

Summary

Modeling frameworks:

Boolean networks

Polynomial dynamical systems

Logical models

Dynamic Bayesian networks

(Petri nets)

Model inference from data

Goal: Given a set of experimental observations, infer the most likely model of the network that generated the data.

Model framework: polynomial dynamical systems over a finite field

Data discretization

Step 1: Discretize real-valued data into finitely many states.

This is a difficult problem.

E. Dimitrova, P. Vera-Licona, J. McGee, and R. Laubenbacher, Comparison of data discretization methods for inference of biochemical networks.

Model inference from data

Variables x1, … , xn with values in a finite field k.

(s1, t1), … , (sr, tr) state transition observations

with sj, tj kn.

Goal: Identify a collection of “best” dynamical systems

f=(f1, … ,fn): kn → kn

such that f(sj)=tj for all j.

Network inference

Problem: Given D={(sj, tj) kn×k}, find the “most likely” model f: kn → k such that

f(sj) = tj

Let M = {f: kn → k | f(sj) = tj } be the subset of k[x1, … , xn] of all possible models for a particular variable.

Network inference

Let f, g M. Then f(sj) = g(sj) for all j. So

(f-g)(sj) = 0 for all j. Let

I = {h k[x1, … , xn] | h(sj)=0 for all j}Let f 0 be any element of M. Then

M = f 0+I.

Note that I is an ideal, since it is closed under + and × by arbitrary polynomials.

Model selection

In the absence of additional network information, choose a “minimal” model f from M

(f only reflects relationships among variables that are inherent in the data)

If f = hg +f’, with g I and f’ is not divisible by any r I,

then f’ is preferable to f because hg vanishes on all sj.

Model selection

Strategy:

1. Compute f 0 M and the coset f 0+I.

2. Compute f f 0+I with the property that f is not divisible by any g I.

Could use other criteria for model selection:

f must contain certain variables and can’t contain others.

Could also require certain constraints on the dynamics.

Fundamental computational problem

Given I and f, decide whether f I. If not, compute the remainder of f under

“division by I.”

This is known as the “ideal membership problem.”

This problem can be solved by Gröbner basis theory.

Wiring diagrams

Goal: Compute all possible minimal wiring diagrams for a given data set.

Wiring diagram:

Vertices = variables

Edges: xi → xj if xi is involved in the regulation of xj, that is, if xi appears in fj.

Wiring diagrams

Problem: Given data (si, ti), i=1, … , r,

(a collection of state transitions for one node in the network), find all minimal (wrt inclusion) sets of variables y1, … , ym {x1, … , xn} such that

(f 0+I) ∩ k[y1, … , ym] ≠ Ø.

Each such minimal set corresponds to a minimal wiring diagram for the variable under consideration.

The “minimal sets” algorithm

For a k, let Xa = {si | ti = a}.

Let X = {Xa | a k}.

Then

f 0+I = M = {f k[x1, … xn] | f(p) = a for all p Xa}.

Want to find f M which involves a minimal number of variables, i.e., there is no g M whose support is properly contained in the supp(f).

Example

Let n = 5, k = F5. Let

(s1, t1) = [(3, 0, 0, 0, 0); 3](s2, t2) = [(0, 1, 2, 1, 4); 1](s3, t3) = [(0, 1, 2, 1, 0); 0](s4, t4) = [(0, 1, 2, 1, 1); 0](s5, t5) = [(1, 1, 1, 1, 3); 4]

ThenX0 = {s3, s4}, X1 = {s2}, X2 = Ø, X3 = {s1}, X4 = {s5}.

The algorithm

Definitions. • For F {1, … , n}, let

RF = k[xi | i F].• Let ΔX = {F | M ∩ RF ≠ Ø}.

• For p Xa, q Xb, a ≠ b k, let

m(p, q) = pi≠qi xi.

Let MX = monomial ideal in k[x1, … , xn] generated by all monomials m(p, q) for all a, b k.

(Note that ΔX is a simplicial complex, and MX is the face ideal of the Alexander dual of ΔX.)

The algorithm

Proposition. A subset F of {1, … , n} is in ΔX if and only if the ideal < xi | i F > contains the ideal MX.

Proof. Let F ΔX. Then Y ∩ RF ≠ Ø.

Let p Xa and q Xb, with a ≠ b. Then there is f k[xi | i F] such that

f(p) = a and f(q) = b.

So p and q differ in a coordinate j F. Hence m(p, q) contains xj as a factor, so is contained in

I = <xj | j F>.

Therefore, MX I.

The algorithm

Conversely, suppose MX <xi | i F>.

Then all generators m(p, q) are in terms of the xi, i F.

Therefore, p Xa and q Xb differ in coordinates i F.

For p Xa and for all a k, define f to be the polynomial function

f(p) = a for p Xa, for all a k;f(p) = 0 otherwise.

Then f M and depends only on variables xi, i F. Hence

f M ∩ RF.

This completes the proof.

The algorithm

Corollary. To find all possible minimal wiring diagrams, we need to find all minimal subsets of variables y1, … , ym such that MX is contained in <y1, … , ym>.

Example

Let MX = < x1x2, x2x3, x1x4 >. Then

MX = < x1, x2x3, x1x4 > ∩ < x2, x2x3, x1x4 >

= < x1, x2x3 > ∩ < x2, x1x4 >

= < x1, x2> ∩ < x1, x3 > ∩ < x2, x1> ∩ < x2, x4 >

= < x1, x2> ∩ < x1, x3 > ∩ < x2, x4 >.

(primary decomposition of MX)

Therefore, the collection of minimal wiring diagrams includes

{x1, x2}, {x1, x3}, {x2, x4}.(minimal primes in the primary decomp.)

Can be done algorithmically, implemented in computer algebra systems.

Model selection

How do we choose a “best” one from this list?

Example of a scoring method. (See alternative methods in (Jarrah, L., Stigler, Stillman)).

• First assign a score to each variable xi, i=1, … ,n.

• Then use these scores to assign a score to each minimal variable set.

• Choose the minimal set with the highest score.

Scoring method

Let F = {F1, … , Ft} be the output of the algorithm.

For s = 1, … , n, let Zs = # sets in F with s elements.

For i = 1, … , n, let Wi(s) = # sets of size s which contain xi.

S(xi) := ΣWi(s) / sZs

where the sum extends over all s such that Zs ≠ 0.

T(Fj) := ΠxiFj S(xi).

Normalization probability distribution on F of min. var. sets

This scoring method has a bias toward small sets.

Example

F1 = {x1, x2}, F2 = {x1, x3}, F3 = {x2, x4}

Z1=0, Z2=3, Z3=Z4=0;

W1(2) = 2, W2(2) = 2, W3(2) = 1, W4(2) = 1.

S(x1) = 2/2·3 = 1/3 = S(x2), S(x3) = 1/2·3 = 1/6 = S(x4).

T(F1) = 1/9, T(F2) = 1/18 = T(F3).

Example with data

1 0 0 2

1 2 2 1

0 2 1 1

1 2 1 2

2 2 0 2

0 1 1 2

x1: { {x1, x3}, {x1, x2, x4}, {x2, x3, x4}}x2: {{x1}, {x2, x3} }x3: { {x1, x3}, {x1, x2, x4}, {x2, x3, x4}}x4: { {x1, x3}, {x2, x3}, {x1, x2, x4} }

returned from min. sets algorithm

Example with data

Consider the variable sets for variable x1:F1 = {x1, x3}F2 = {x1, x2, x4}F3 = {x2, x3, x4}

Highest scoring set(s) for each variablex1: {x1, x3}x2: {x1}x3: {x1, x3}x4: {{x2, x3}, {x1, x3}}

S(x1) = 1/1 + 1/(2×3) = 7/6S(x2) = 2/(2×3) = 1/3S(x3) = 1/1 + 1/(2×3) = 7/6S(x4) = 2/(2×3) = 1/3

T(F1) = (7/6)(7/6) = 49/36T(F2) = (7/6)(1/3)(1/3) = 7/54T(F3) = (1/3)(7/6)(1/3) = 7/54

winner

Method Validation:Segment polarity network in the fruitfly

Network in cell: 21 genes, proteins

Albert model: 21 Bool. functions – 44 known interactions

Time series data– Generated wildtype, knockout– < 0.01% of 221 total states

Minimal Sets Algorithm– 89% interactions– 0 false positives, 5 false negatives– PDS: identified 19/21 functions

J. Theor. Biol. 2003

The topology of the regulatory

interactions predicts the expression pattern of the

segment polarity genes in Drosophila

melanogaster

Albert and Othmer

Pandapas network• 10 genes, 3 external biochemicals• 17 interactions

Time series data: 9 time points• Generated 8 time series for wildtype, knockouts G1, G2, G5 • 192 data points• G6, G9 constant

Data discretization• 5 states per node• 95 data points

– 49% reduction – < 0.00001% of 513 total states

Method Validation:Simulated gene network

Method Validation:Simulated gene network

Minimal Sets Algorithm

• 77% interactions• Identified targets of P2, P3 (x12, x13)• 11 false positives, 4 false negatives

Pandapas Reverse engineered

Summary

Algorithmic method to find all possible minimal wiring diagrams, given a data set:

Finds all possible minimal sets of variables for which there exists a PDS that is consistent with the data.

Provides a statistical measure to select most likely wiring diagram(s).

This algorithm can be used as a preprocessing step for the previous algorithm that actually finds dynamical models. It improves algorithm performance by reducing the variables to be considered.

Optimization

Goal: For a given data set, select a model from M which is optimal with respect to

• Model complexity

• Properties of wiring diagram

• Expected dynamic properties

dm2.chr2L CTGCGGGATTAGGGGTCATTAGAG---------TGCCGAAAAGCGAGT-TTATTCdp3.chr4_group3 CTGGAAGAGTTTTGATTAGTAGGGGATCCATGGGGGCGAGGAGAGGCCATCATCG

dm2.chr2L TATGGACTCACdp3.chr4_group3 TGT--ACTTAC

DroMel_4_ CTGCGGGATTAGGGGTCATTAGAGT---------GCCGAAAAGCGA---------GTTTDroPse_1_ CTGGAAGAGTTTTGATTAGTAGGGGATCCATGGGGGCGAGGAGAGGCCATCATCG---- DroMel_4_ ATTCTATGGACTCACDroPse_1_ ------TGTACTTAC

Each alignment can be summarized by counting the number of matches (#M), mismatches (#X), gaps (#G), and spaces (#S).

#M=31, #X=22, #G=3, #S=12

#M=27, #X=18, #G=3, #S=28

2(#M+#X)+#S=112 so #X,#G and #S suffice to specify a summary.

This notation follows Chapter 7 (Parametric Sequence Alignment) by Colin Dewey and Kevin Woods in the book Algebraic Statistics for Computational Biology.

Courtesy Lior Pachter

>melCTGCGGGATTAGGGGTCATTAGAGTGCCGAAAAGCGAGTTTATTCTATGGAC>pseCTGGAAGAGTTTTGATTAGTAGGGGATCCATGGGGGCGAGGAGAGGCCATCATCGTGTAC

For the sequences:

49 #x=24, #S=10, #G=2

There are eight alignments that have this summary.

the alignment polytope is:

Courtesy Lior Pachter

Parametric sequence alignment

Choose parameters a, b, c and minimize the linear functional

f(M, G, S) = aM+bG+cS

over the convex polytope spanned by the summaries of all possible alignments of the two sequences.

Theorem (Pachter, Sturmfels) This polytope can be described as the Minkowski sum of the Newton polytopes of a collection of polynomials.

The Dynamotope(joint work with A. Jarrah, B. Sturmfels, P. Vera-Licona)

Define the summary (S1, S2, S3, S4) of a polynomial model

g f + I:

S1 = w1·(u1, u2, u3, … ),

where ui is the number of limit cycles of length i, and w1 is a suitably chosen weight vector.

+ w2·(v1, v2, v3, … ),

where vi is the number of trees of height i, and w2 is a suitably chosen weight vector.

S2 = number of edges in the dependency graph of g.

S3 = “complexity” of g (including complexity of the polynomials gi and the “distance to being a normal form”).

Let w1 and w2 be (1, 1, …)S1 = (1,1,…)·(1, 1, 0, 1) = 3S2 = (1,1,…)·(0, 0, 1, 1, 1) = 3

Optimization

Choose parameters a, b, c, d. Minimize the linear functional

F = aS1 + bS2 + cS3 + dS4

on the convex polytope (dynamotope) spanned by all summaries (S1, S2, S3, S4) of models in

f + I.

Optimization

Problem: Don’t know how to describe this polytope.

Solution: Combinatorial optimization using an evolutionary algorithm.

Evolutionary algorithm

For f=(f1, … ,fn)

Gene = fi

Chromosome = f

Genotype = {f}

Evolution

Step 1: Choose an initial genotype

Step 2: Use mutation, cross-over of fittest models (with respect to linear functional F) to compute the next generation genotype

Step 3: Iterate many times

Step 4: Choose local/global minimum (if found)

Future work

• Optimal parameter choices for different biological problems

• Further validation of the algorithm with real and simulated data sets

• Characterize the dynamotope computationally

• Study optimal experimental design for this type of network inference