CZ5225: Modeling and Simulation in Biology Lecture 11: Biological Pathways III: Pathway Simulation...

CZ5225: Modeling and Simulation in BiologyCZ5225: Modeling and Simulation in Biology

Lecture 11: Biological Pathways III: Lecture 11: Biological Pathways III: Pathway SimulationPathway Simulation

Prof. Chen Yu ZongProf. Chen Yu Zong

Tel: 6874-6877Tel: 6874-6877Email: Email: [email protected]@cz3.nus.edu.sg

http://xin.cz3.nus.edu.sghttp://xin.cz3.nus.edu.sgRoom 07-24, level 7, SOC1, NUSRoom 07-24, level 7, SOC1, NUS

22

““Genomes To Life” Computing RoadmapGenomes To Life” Computing Roadmap

Biological Complexity

ComparativeGenomics

Constraint-BasedFlexible Docking

Co

mp

uti

ng

an

d I

nfo

rmat

ion

In

fras

tru

ctu

re C

apab

ilit

ies

Constrained rigid

docking

Genome-scale protein threading

Community metabolic regulatory, signaling simulations

Molecular machine classical simulation

Protein machineInteractions

Cell, pathway, and network

simulation

Molecule-basedcell simulation

Current Computing

33

Characterising PathwaysCharacterising Pathwaysmetabolic networks as an examplemetabolic networks as an example

Barabasi & Oltvai, Nature Rev Gen 5, 101 (2004)

To study the network characteristics of the metabolism a graph theoretic description needs to

be established.

(a) illustrates the graph theoretic description for a simple pathway (catalysed by Mg2+-

dependant enzymes).

(b) In the most abstract approach all interacting metabolites are considered equally. The links

between nodes represent reactions that interconvert one substrate into another. For many

biological applications it is useful to ignore co-factors, such as the high-energy-phosphate

donor ATP, which results

(c) in a second type of mapping that connects only the main source metabolites to the main

products.

44

DegreeDegree

Barabasi & Oltvai, Nature Reviews Genetics 5, 101 (2004)

The most elementary characteristic of a node is its

degree (or connectivity), k, which tells us how

many links the node has to other nodes.

a In the undirected network, node A has k = 5.

b In networks in which each link has a selected

direction there is an incoming degree, kin, which

denotes the number of links that point to a node,

and an outgoing degree, kout, which denotes the

number of links that start from it.

E.g., node A in b has kin = 4 and kout = 1.

An undirected network with N nodes and L links is

characterized by an average degree <k> = 2L/N

(where <> denotes the average).

55

Degree distributionDegree distribution


The degree distribution, P(k), gives the probability

that a selected node has exactly k links.

P(k) is obtained by counting the number o f nodes

N(k) with k = 1,2... links and dividing by the total

number of nodes N.

The degree distribution allows us to distinguish

between different classes of networks.

For example, a peaked degree distribution, as

seen in a random network, indicates that the

system has a characteristic degree and that there

are no highly connected nodes (which are also

known as hubs).

By contrast, a power-law degree distribution

indicates that a few hubs hold together numerous

small nodes.

66Barabasi & Oltvai, Nature Rev Gen 5, 101 (2004)

Aa

The Erdös–Rényi (ER) model of a random network starts with N

nodes and connects each pair of nodes with probability p, which

creates a graph with approximately pN (N-1)/2 randomly placed links.

Ab

The node degrees follow a Poisson distribution, where most nodes

have approximately the same number of links (close to the average

degree <k>). The tail (high k region) of the degree distribution P(k )

decreases exponentially, which indicates that nodes that significantly

deviate from the average are extremely rare.

Ac

The clustering coefficient is independent of a node's degree, so C(k)

appears as a horizontal line if plotted as a function of k. The mean

path length is proportional to the logarithm of the network size, l log

N, which indicates that it is characterized by the small-world property.

Random networksRandom networks

77

Origin of scale-free topology and hubs in Origin of scale-free topology and hubs in biological networksbiological networks

Barabasi & Oltvai, Nature Rev Gen 5, 101 (2004)

The origin of the scale-free topology in complex networks

can be reduced to two basic mechanisms: growth and

preferential attachment. Growth means that the network

emerges through the subsequent addition of new nodes,

such as the new red node that is added to the network that

is shown in part a . Preferential attachment means that new

nodes prefer to link to more connected nodes. For

example, the probability that the red node will connect to

node 1 is twice as large as connecting to node 2, as the

degree of node 1 (k1=4) is twice the degree of node 2 (k2

=2). Growth and preferential attachment generate hubs

through a 'rich-gets-richer' mechanism: the more connected

a node is, the more likely it is that new nodes will link to it,

which allows the highly connected nodes to acquire new

links faster than their less connected peers.

88

Scale-free networksScale-free networks Scale-free networks are characterized by a power-law degree

distribution; the probability that a node has k links follows P(k) ~ k- -,

where is the degree exponent. The probability that a node is highly

connected is statistically more significant than in a random graph, the

network's properties often being determined by a relatively small number

of highly connected nodes („hubs“, see blue nodes in Ba).

In the Barabási–Albert model of a scale-free network, at each time point

a node with M links is added to the network, it connects to an already

existing node I with probability I = kI/JkJ, where kI is the degree of node

I and J is the index denoting the sum over network nodes. The network

that is generated by this growth process has a power-law degree

distribution with = 3.

Bb Such distributions are seen as a straight line on a log–log plot. The

network that is created by the Barabási–Albert model does not have an

inherent modularity, so C(k) is independent of k.

(Bc). Scale-free networks with degree exponents 2< <3, a range that is

observed in most biological and non-biological networks, are ultra-small,

with the average path length following ℓ ~ log log N, which is significantly

shorter than log N that characterizes random small-world networks.

99

Network measuresNetwork measures


Scale-free networks and the degree exponent

Most biological networks are scale-free, which means that their

degree distribution approximates a power law, P(k) k- , where

is the degree exponent and ~ indicates 'proportional to'. The

value of determines many properties of the system. The

smaller the value of , the more important the role of the hubs

is in the network. Whereas for >3 the hubs are not relevant, for

2> >3 there is a hierarchy of hubs, with the most connected

hub being in contact with a small fraction of all nodes, and for

= 2 a hub-and-spoke network emerges, with the largest hub

being in contact with a large fraction of all nodes. In general, the

unusual properties of scale-free networks are valid only for <

3, when the dispersion of the P(k) distribution, which is defined

as 2 = <k2> - <k>2, increases with the number of nodes (that

is, diverges), resulting in a series of unexpected features,

such as a high degree of robustness against accidental node

failures. For >3, however, most unusual features are absent,

and in many respects the scale-free network behaves like a

random one.

1010

Shortest path and mean path lengthShortest path and mean path length


Distance in networks is measured with the path length,

which tells us how many links we need to pass through to

travel between two nodes. As there are many alternative

paths between two nodes, the shortest path — the path

with the smallest number of links between the selected

nodes — has a special role.

In directed networks, the distance ℓAB from node A to

node B is often different from the distance ℓBA from B to

A. E.g. in b , ℓBA = 1, whereas ℓAB = 3. Often there is no

direct path between two nodes. As shown in b, although

there is a path from C to A, there is no path from A to C.

The mean path length, <ℓ>, represents the average over

the shortest paths between all pairs of nodes and offers

a measure of a network's overall navigability.

1111

Clustering coefficientClustering coefficient


In many networks, if node A is connected to B, and B is connected to C,

then it is highly probable that A also has a direct link to C. This

phenomenon can be quantified using the clustering coefficient CI =

2nI/k(k-1), where nI is the number of links connecting the kI neighbours of

node I to each other. In other words, CI gives the number of 'triangles'

that go through node I, whereas kI (kI -1)/2 is the total number of triangles

that could pass through node I, should all of node I's neighbours be

connected to each other. For example, only one pair of node A's five

neighbours in a are linked together (B and C), which gives nA = 1 and CA

= 2/20. By contrast, none of node F's neighbours link to each other,

giving CF = 0. The average clustering coefficient, <C >, characterizes the

overall tendency of nodes to form clusters or groups. An important

measure of the network's structure is the function C(k), which is defined

as the average clustering coefficient of all nodes with k links. For many

real networks C(k) k-1, which is an indication of a network's

hierarchical character.

The average degree <k>, average path length <ℓ> and average

clustering coefficient <C> depend on the number of nodes and links (N

and L) in the network. By contrast, the P(k) and C(k ) functions are

independent of the network's size and they therefore capture a network's

generic features, which allows them to be used to classify various

networks.

1212Barabasi & Oltvai, Nature Rev Gen 5, 101 (2004)

Hierarchical networksHierarchical networksTo account for the coexistence of modularity, local clustering and scale-

free topology in many real systems, it has to be assumed that clusters

combine in an iterative manner, generating a hierarchical network.

The starting point of this construction is a small cluster of 4 densely

linked nodes (4 central nodes in Ca).

Next, 3 replicas of this module are generated and the 3 external nodes of

the replicated clusters connected to the central node of the old cluster,

which produces a large 16-node module.

3 replicas of this 16-node module are then generated and the 16

peripheral nodes connected to the central node of the old module, which

produces a new module of 64 nodes. The hierarchical network model

seamlessly integrates a scale-free topology with an inherent modular

structure by generating a network that has a power-law degree

distribution with degree exponent = 1 + ln4/ln3 = 2.26 (Cb) and a

large, system-size independent average clustering coefficient <C> ~ 0.6.

The most important signature of hierarchical modularity is the scaling of

the clustering coefficient, which follows C(k) ~ k-1 a straight line of slope -

1 on a log–log plot (Cc). A hierarchical architecture implies that sparsely

connected nodes are part of highly clustered areas, with communication

between the different highly clustered neighbourhoods being maintained

by a few hubs (Ca).

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Constructing a pathway model:Constructing a pathway model:things to considerthings to consider

1. Dynamic nature of biological networks.Biological pathway is more than a topological linkage of molecular networks.

Pathway models can be based on network characteristics including those of invariant features.

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Constructing a pathway model:Constructing a pathway model:things to considerthings to consider

2. Abstraction Resolution:

• How much do we get into details?

• What building blocks do we use to describe the network?

High resolution

Low resolution

(A) Substrates and proteins

(B) Pathways

(C) “special pathways”

1515

Constructing a pathway modelConstructing a pathway modelStep I - DefinitionsStep I - Definitions

We begin with a very simple imaginary metabolic network represented as a directed graph:

Vertex – protein/substrate concentration.

Edge - flux (conversion mediated by proteins of one substrate into the other)

Internal flux edge

External flux edge

How do we define a

biologically significant

system boundary?

1616

dxS v

dt

Stoichiometry Matrix

Flux vectorConcentration vector

Constructing a pathway modelConstructing a pathway modelStep II - Dynamic mass balanceStep II - Dynamic mass balance

1717

A ‘simple’ ODE model of yeast glycolysis

1818

A model pathway system and its time-dependent behavior

Positive Feedback Loop

1919


2020


2121

dxS v

dt

Stoichiometry Matrix

Flux vectorConcentration vector Problem …

V=V(k1, k2,k3…) is actually a function of concentration as well as several kinetic parameters.

it is very difficult to determine kinetic parameters experimentally.

Consequently there is not enough kinetic information in the literature to construct the model.One Solution:

In order to identify invariant characteristics of the network, one can assume the network is at steady state.

Constructing a pathway modelConstructing a pathway modelStep II - Dynamic mass balanceStep II - Dynamic mass balance

2222

1. What does “steady state” mean?

2. Is it biologically justifiable to assume it?

3. Does it limit the predictive power of pathway model?

“ The steady state approximation is generally valid for some metabolic pathways because of fast equilibration of metabolite concentrations (seconds) with respect to the time scale of genetic regulation (minutes)” – Segre 2002

How about signaling pathways?

Constructing a pathway modelConstructing a pathway modelStep III - Dynamic mass balance at steady stateStep III - Dynamic mass balance at steady state

2323

dxS v

dt 0 S v

How can the steady state assumption solve pathway simulation problem?

Steady state assumption

0

0

0

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0,iv i Constraints on internal fluxes:

Constraints on external fluxes:

Source

Sink

Sink/source is unconstrained

In other words flux going into the system is considered negative while flux leaving the system is considered positive.

Remark: further constraints can be imposed both on the internal flux as well as the external flux…

0jb

jb

Constructing a pathway modelConstructing a pathway modelStep IV - Adding constraintsStep IV - Adding constraints

0jb

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Flux cone and metabolic capabilitiesFlux cone and metabolic capabilities

Observation: the number of reactions considerably exceeds the number of metabolites0 S v

0

0

0

The S matrix will have more columns than rows

The null space of viable solutions to our linear set of equations contains an infinite number of solutions.

“ The solution space for any system of linear homogeneous equations and inequalities is a convex polyhedral cone.” - Schilling 2000

C

Our flux cone contains all the points of the null space with non negative coordinates (besides exchange fluxes that are constrained to be negative or unconstrained)

What about the constraints?

2626

Flux cone and metabolic capabilitiesFlux cone and metabolic capabilities

What is the significance of the flux cone?

• It defines what the network can do and cannot do!

• Each point in this cone represents a flux distribution in which the system can operate at steady state.

• The answers to the following questions (and many more) are found within this cone:

• what are the building blocks that the network can manufacture?

• how efficient is energy conversion?• Where is the critical links in the

system?

2727

Navigating through the flux cone byNavigating through the flux cone byusing “Extreme pathways”using “Extreme pathways”

Next thing to do is develop a way to describe and interpret any location within this space.

• We will not use the traditional reaction/enzyme based perspective

• Instead we use a pathway perspective:

Extreme rays - “extreme rays correspond to edges of the cone. They are said to generate the cone and cannot be decomposed into non-trivial combinations of any other vector in the cone.”- schilling 2000

Differences

•Unlike a basis the set of, extreme pathways is typically unique

•Any flux in the cone can be described using a non negative combination of extreme rays.

We use the term Extreme Pathways when referring to Extreme rays of a convex polyhedral cone that represents metabolic fluxes

What is the analogy in

linear algebra?

2828

Navigating through the flux cone byNavigating through the flux cone byusing “Extreme pathways”using “Extreme pathways”

,1

{ | 0 }k

i i ii

C v v w EP w i

Pw v

• Extreme Pathways will be denoted by vector EPi (0≤ i ≤ k)

• Every point within the cone can be written as a non-negative linear combination of the extreme pathways.

C v

In biological context this means that :

any steady state flux distribution can be represented by a non-negative linear combination

of extreme pathways.

2929

The entire process from a bird’s eye viewThe entire process from a bird’s eye view

Compute steady state flux Convex

Identify Extreme pathways (using

algorithm presented in Schilling 2000)

+ constraints

3030

ExampleExample

Lets look at a specific vector v’ :

4

2

0

1

0

1

4

2

1

1

v

4

2

0

1

0

1

4

2

1

1

Is v inside the flux cone?Easy to check…1. Does v fulfill constraints?2. Is v in the null space of Sv=0 ?

3131

We can reformulate this concept using matrix notation:

,1

' 0 k

i i ii

v w EP w

can v be represented by using a non-negative linear combination of extreme pathways ?

The vector w gives us a pathway based perspective of the network functioning!

ExampleExample

=

1

2

3

4

5

6

7

w

w

w

w

w

w

w

1

2

3

4

5

6

7

1

2

3

4

v

v

v

v

v

v

v

b

b

b

b

w vP4

2

0

1

0

1

4

2

1

1

4

1

1

0

0

0

0

=

•v’=4*p1+1*p2+1*p3

1

2

3

4

5

6

7

w

w

w

w

w

w

w

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Flux Balance AnalysisFlux Balance Analysis• Simulates Flux or Flow through

Metabolic Pathways

• Kinetics of each step not required

• Predicts effects of gene knockouts

3333

The Multiscale Problem in Systems BiologyThe Multiscale Problem in Systems Biology

Multiple time scales (stiffness)

The presence of exceedingly large numbers of molecules that must be accounted for in SSA

In the heat-shock response in E. Coli, an estimated 20 - 30 sigma-32 molecules per cell play a key role in sensing the folding state of the cell and in regulating the production of heat shock proteins. The system cannot be simulated at the fully stochastic level due to:

3434

Computational Models Computational Models of Molecular Networksof Molecular Networks

Discrete and stochastic - Finest scale of representation for well stirred molecules. Exact description via Stochastic Simulation Algorithm (SSA) due to Gillespie.

This is the method described here

Continuous and stochastic - The Langevin regime. Valid under certain conditions. Described by Stochastic Differential Equations (SDE).

Continuous and deterministic - The chemical rate equations. Described by ordinary differential equations (ODE). Valid under further assumptions.

3535

Stochastic Simulation AlgorithmStochastic Simulation Algorithm

Well-stirred mixture

N molecular species

Constant temperature, fixed volume

M reaction channels

Dynamical state where

is the number of molecules in the system

S1,...SN

R1,...RM

X(t) (X1(t),..., XN (t))

X i(t)

Si

3636


Propensity function the probability, given ,

that one reaction will occur somewhere inside in the next

infinitesimal time interval

When that reaction occurs, it changes the state. The amount by

which changes is given by the change in the number of

molecules produced by one reaction

is a jump Markov process

a j (x)dt

X(t) X

R j

[t, t dt]

X i

i j

Si

R j

X(t)

3737


Draw two independent samples and from

and take

the smallest integer satisfying

Update X

1

a0(X)1n

1

r1

a j '(x) r2a0(x)j '1

j

X X j

r1

r2

U(0,1)

j

3838

Hybrid MethodsHybrid Methods

Slow reactions involving species present in small numbers are simulated by SSA

Reactions where all constituents present with large populations are simulated by reaction-rate equations

Haseltine and Rawlings, 2002Matteyses and Simmons, 2002

What to do about other scenarios???

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A Closer Look at the Multiscale Challenges A Closer Look at the Multiscale Challenges for Heat Shock Responsefor Heat Shock Response

The total “concentration” of 32 is 30-100 per cell

But the “concentration” of free 32 is .01-.05 per cell

DNAK

RNAP

RNAP

DnaK FtsH

4040

Stochastic Partial Equilibrium ApproximationStochastic Partial Equilibrium Approximation In deterministic simulation of chemical systems, the partial equilibrium

approximation assumes that the fast reactions are always in equilibrium. These fast reactions are thus treated as algebraic constraints.

In stochastic simulation, the states keep changing. The stochastic partial equilibrium approximation is based on the assumption that the distributions of the fast species remain unchanged by the fast reactions.

Partition and re-index reactions so that are fast reactions and are slow reactions. Partition the species so that are fast species (any species involved in at least one fast reaction), and are slow species

The virtual system with only fast reactions and fast species is assumed to be in stochastic partial equilibrium, on the time scale of the slow reactions

R1,...,RM '

RM '1,...,RM '

S1,...,SN '

SN '1,...,SN

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Foundations of SPEAFoundations of SPEA

Let the probability that, given , no slow

reaction fires during .

Define .

We will not simulate the fast reactions. Instead we take at

time as a random variable. The probability that one slow

reaction will occur in is given by

where mean over only.

P0( xs,x f ,t)

X(t) (x f ,xs)

[t, t )

as(X) a j (X)jM '1

M

x f

[, d)

E f (as(X f ,xs))d P(X f () x ' f )as(x ' f ,xs)dx ' f

E f

X f

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Let be so small that in the next at most one slow reaction could fire.

Then

(1)

Solving (1) we obtain

(2)

Define the next slow reaction density function as the probability

that, given , the next slow reaction will occur in

and will be an reaction. Then (2) leads to

(3)

Now apply the stochastic partial equilibrium approximation. Assume the virtual

system is at partial equilibrium and is small compared to .

Then can be taken as .

Thus

d

[, d)

00 0( , , ) ( , , )(1 ( ( ( ), )) )sf s f s f sP d x x t P x x t E a X x d

0 0( , , ) exp ( ( ( ), )t s

f s f stP x x t E a X s x ds

p'(, j x f ,xs,t)

X(t) (x f ,xs)

[t , t d)

R j

0'( , , , ) ( ( ( ), ) exp ( ( ( ), ))t s

f s j f s f stp j x x t E a X x E a X x

relax

X f ()

ˆ X f ()

0'( , , , ) ( ( ( ), ) exp( ( ( ( ), )))sf s j f s f sp j x x t E a X x E a X x

Foundations of SPEAFoundations of SPEA

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The Multi-scale Stochastic Simulation AlgorithmThe Multi-scale Stochastic Simulation Algorithm

Given initial time , initial state and final simulation time ,1. Compute the partial equilibrium state for the fast reaction channels.

This involves solving a nonlinear system arising from the equilibrium approximation and the local conservation laws for

2. For , calculate and . (This is the most

difficult task)

3. Generate two random numbers and from .

The time for the next slow reaction to fire is given by ,

where .

The index of the next slow reaction is the smallest integer

satisfying .

4. If , stop. Otherwise, update .

t

x0

T

E(X f (t))x

j M '1,..., M

a j (xs)

a s(xs)

r1

r2

U(0,1)

t

1

a s(X s)log

1

r1

j

a j '(xs) r2a s(xs)j 'M '1

j

t t

t t x x j

4444

Down-Shifting MethodDown-Shifting Method

The MSSA method captures the full distribution information for the slow species, but computes only the mean accurately for the fast species.

However, the distribution information of the fast species can be easily recovered by taking a few small time steps whenever this information is needed.

This ‘down-shifting’ works because the system is a Markov process!

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Heat Shock Response ModelHeat Shock Response Model

Stochastic Model involves 28 species and 61 chemical reactions. This is a moderate-sized model.

CPU time on 1.46 Ghz Pentium IV Linux workstation for one SSA simulation is 90 seconds.

CPU time for 10,000 simulations is more than 10 days. 12 fast reactions were chosen for the SPEA. The fast reactions

were identified from a single SSA simulation to be the ones that fired most frequently. These 12 reactions fire 99% of the total number of times for all reaction channels.

CPU time for the multiscale SSA Without down-shifting: 3 hours for 10,000 runs With down-shifting: 4 hours for 10,000 runs

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MSSA Heat Shock Response Model ResultsMSSA Heat Shock Response Model Results

Histogram (10,000 samples) of a slow species mRNA(DNAK) (left) and a fast species DNAK with (right) and without downshifting method

(middle) solved by the original SSA method (purple solid line with ‘o’) and the MSSA method (red dashed line with ‘+’), for the HSR model.

fast specieswith down shiftingone fast speciesone slow species

CZ5225: Modeling and Simulation in Biology Lecture 11: Biological Pathways III: Pathway Simulation...

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