areejit samal regulation

System level dynamics and robustness of the genetic network regulating E. coli metabolism

Areejit SamalDepartment of Physics and Astrophysics

University of DelhiDelhi 110007 India

June 15, 2009 Areejit Samal

Outline

• Background

• System: E. coli transcriptional regulatory network controlling metabolism (iMC1010v1)

• Simulation results

• Design features of the regulatory network

• Conclusions

Cell can be viewed as a ‘network of networks’

Metabolic Pathway

Promoter

5’ 3’

Coding region

Gene A

Promoter

5’ 3’

Coding region

Gene B

Promoter

5’ 3’

Coding region

Gene CmRNA

mRNA

mRNA

Protein A Protein B Protein C

A DCB

DNA

Protein

Metabolite

TranscriptionalRegulatoryNetwork

ProteinInteractionNetwork

MetabolicNetwork

Cell

Cell can be viewed as a ‘network of networks’

Metabolic Pathway

Promoter

5’ 3’

Coding region

Gene A

Promoter

5’ 3’

Coding region

Gene B

Promoter

5’ 3’

Coding region

Gene CmRNA

mRNA

mRNA


A DCB

DNA

Protein

Metabolite


ProteinInteractionNetwork

MetabolicNetwork

Environment Cell

Boolean network approach to model Gene Regulatory Networks

• Boolean networks were introduced by Stuart Kauffman as a framework tostudy dynamics of Genetic networks.

• In this approach, gene expression is quantized to two levels:– on or active (represented by 1) and– off or inactive (represented by 0).

• Each gene at any point of time is in one of the two states (i.e. active orinactive).

• In this approach, time is taken as discrete.

• Also, the expression state of each gene at any time instant is determined bythe state of its input genes at the previous time instant via a logical rule orupdate function.


Simplified Diagram of the Transcriptional Regulatory Network controlling metabolism

Metabolic reaction

• An input may activate or repress the expression of the gene.For example:Gene B [t+1] = NOT Gene A [t]

• When there are more than one input to a gene, the expression state of the gene will be determined by the state of the inputs based on a logical rule.

• This logical rule may be expressed in terms of Boolean operators (AND, OR, NOT).

• For example:Gene C [t+1] = Gene A [t] AND NOT Gene B [t]

• The state of Gene C determines if the metabolic reaction can occur inside the cell.


Modelling Gene Regulatory Networks as Random Boolean Networks

In the absence of data on real genetic networks, Boolean networks have beenused primarily to study the dynamics of the genetic networks that were

– either members of ensemble of random networks or

– networks generated using the knowledge of the connectivity of genes andTF in an organism along with random Boolean rules at each node as inputfunction governing the output state of the gene


E. coli transcriptional regulatory network controlling metabolism (iMC1010v1)

In this work, we have studied the database iMC1010v1 containing thetranscriptional regulatory network (TRN) controlling E. coli metabolism hasbecome available. The network contained in the database was reconstructed fromprimary literature sources.

The database iMC1010v1 contains the following types of information:

– the connections between genes and transcription factors (TF)

– dependence of genes and TF activity based on presence or absence ofexternal metabolites or nutrients in the environment

– the Boolean rule describing the regulation of each gene as a function of thestate of the input nodes

Available at: Bernhard Palsson’s Group Webpage

(http://gcrg.ucsd.edu/)

June 15, 2009 Areejit SamalMetabolic reaction

Promoter

5’ 3’

Coding region

Gene A

Promoter

5’ 3’

Coding region

Gene B

Promoter

5’ 3’

Coding region

Gene CmRNA

mRNA

mRNA


DC

DNA

Protein

Schematic of Transcriptional Regulatory Network controlling metabolism


MetabolicNetwork


Description of the E. coli TRN controlling metabolism (iMC1010v1)

• There are 583 genes in this network which can be further subdividedinto– 479 genes that code for metabolic enzymes– 104 genes that code for TF

• The state of these 583 genes is dependent upon– the state of 103 TF and– presence or absence of 96 external metabolites

• The database provides a Boolean rule for each of the 583 genescontained in the network.


The pink nodes represent genes coding for TF, brown nodes represent genes that code for metabolic enzymes and the green nodes represent external metabolites.

The complete network can be subdivided into a large connected component and few small disconnected components.


Example of an input function in form of a Boolean rule controlling the output state of a gene

b2720

o2(e)b3202b2731

A CB

OUTPUT

b2720[t+1] = IF ( b2731[t] AND b3202[t] AND NOT o2(e)[t])

A B C OUTPUT

0 0 0 0

0 0 1 0

0 1 0 0

0 1 1 0

1 0 0 0

1 0 1 0

1 1 0 1

1 1 1 0

Truth Table

June 15, 2009Areejit Samal

The Dynamical System

We have used the information in the database to construct the following discrete dynamical system:

i

i

Gm

tg

tgi

)(

)1(583...1

denotes the state of ith gene at time t+1 that is either 1 or 0.

is vector that collectively denotes the state of all genes at time t

is a vector of 96 elements (each 0 or 1) determining the state of the environment contains all the information regarding the internal wiring of the network as well as the regulatory logic

( 1) ( ( ), )i ig t G g t m


State of the genetic network

The state of the 583 genes at any given time instant gives the state of the network.

1

2

3

583

( )( )( )

.

.

.( )

g tg tg t

g t

where gi(t) = 0 or 1; i = 1 …. 583Since each gene at any given time instant can be in one of the two states (0 or 1), the size of the state space is 2583.

g(t)


State of the environment

The presence or absence of the 96 external metabolites decide the state of the environment.

where mi = 0 or 1; i = 1 …. 96If an external metabolite or nutrient is present in the external environment, then we set the mi corresponding to it equal to 1 or else 0. In general, the concentration of external metabolites change with time. In the present study, we have considered buffered minimal media (i.e., vector m constant in time).

1

2

96

.

.

mm

m

m

E. coli TRN controlling metabolism as a Boolean dynamical system

Stuart Kauffman (1969,1993) studied dynamical systems of the form:

( 1) ( ( ))i ig t G g t

E. coli TRN controlling metabolism as a Boolean dynamical system

Stuart Kauffman (1969,1993) studied dynamical systems of the form:

( 1) ( ( ))i ig t G g t

( 1) ( ( ), )i ig t G g t m

The present database allowed us to systematically account for the effect of presence or absence of nutrients in the environment on the dynamics of the regulatory network.


Attractors of the E. coli TRN

• In the Boolean approach, the configuration space of the system is finite. Thediscrete deterministic dynamics ensures that the system eventually returns to aconfiguration which it had at a previous time instant. The sequence of statesthat repeat themselves periodically is called an attractor of the system.

• Starting from any one of the 2583 vectors as the initial configuration of genesand a fixed environment, the system can flow to different attractors for differentinitial configuration of genes.


The Network exhibits stability against perturbations of gene configurations for a fixed environment

( 1) ( ( ), )i ig t G g t m

Fix m to some buffered minimal media e.g. Glucose aerobic condition

Start with different g(t) as initial configuration of genes, and determine the attractor for the system for each initial configuration of genes.

Question 1: How many attractors of the system do we obtain starting from different initialconfiguration of genes and for a fixed environment?Answer 1: We found that the attractors of the genetic network were typically fixed points ortwo cycles. For a given environment, the number of different attractors were up to 8 fixedpoints and 28 two cycles. However, the maximum hamming distance between any twoattractor states for a given environment was 21. Hence, the states of most genes (≥562)was same in all attractor states for a given environment.

We found that the network exhibits homeostasis or stability against perturbations of initialgene configurations for a fixed environment.


Cellular Homeostasis

The graph shows that starting from even a initial configuration of genes that is inverse of the attractor for the glucose aerobic minimal media the system reaches the attractor in four time steps. Thus, any perturbation of gene configurations will be washed out in few time steps and the system is robust to such perturbations.

Time

0 1 2 3 4

Ham

min

g di

stan

ce w

.r.t.

gluc

ose

aero

bic

cond

ition

attr

acto

r

0

100

200

300

400

500

600

Random initial conditionHamming inverse of the attractorAttractor for glutamate aerobic mediumAttractor for acetate aerobic medium


E. coli TRN exhibits flexibility of response under changing environmental conditions

Question 2: How different are the attractors from each other for various environmentalconditions?Answer 2: We obtained the attractors of the system starting with 15,427 environmentalconditions. The largest hamming distance obtained between two attractors corresponding todifferent environmental conditions was 145.The system shows flexibility of response to changing environmental conditions.

We found that the system is insensitive to fluctuations in gene configurations for a given fixedexternal environment while it can shift to a different attractor when it encounters a change inthe environment. These properties ensure a robust dynamics of the underlying network.

( 1) ( ( ), )i ig t G g t m

Vary m across a set of 15427 buffered minimal media

Determine the attractors of the genetic system for different environments m


Flexibility of response

The graph shows that the largest hamming distance between two attractors from a set of attractors for 15,427 environmental conditions was 145.

Hamming distance

0 20 40 60 80 100 120 140

Freq

uenc

y

0

500x103

1x106

2x106

2x106

3x106

3x106

136 138 140 142 144 146


Flexibility of response

Each gene takes a value 0 or 1 in the 15427 attractors for the different environmental conditions. The standard deviation of a gene’s value across 15427 attractors is a measure of the gene’s variability across environmental conditions.

Standard deviation

0 0 - 0.1 0.1 - 0.2 0.2 - 0.3 0.3 - 0.4 0.4 - 0.5

Num

ber o

f Gen

es

0

50

100

150

200

250


Functional significance of attractors of TRN controlling metabolism

1010...1

Attractor for a given environment

Gene 1 is active: The enzyme is present to carry out a reaction in the metabolic network

Gene 2 is inactive: The enzyme is absent and a reaction cannot happen in the network

Met

abol

ic e

nzym

esTF

The attractor of the genetic network for a given environment constrains the set of active enzymes that catalyze various reactions in the metabolic network


Flux Balance Analysis (FBA)

List of metabolic reactions with stoichiometric coefficients

Biomass composition

Medium of growth or environment

Flux Balance Analysis

(FBA)

Growth rate for the given medium

Fluxes of all reactions

Reference: Varma and Palsson, Biotechnology (1994)

INPUT OUTPUT


Incorporating regulatory constraints within FBA

Biomass composition



(FBA)

Growth rate (pure)


INPUT OUTPUT

List of metabolic reactions



Biomass composition



(FBA)

Growth rate (pure)


INPUT OUTPUT


State of theenvironment

m



Biomass composition



(FBA)

Growth rate (pure)


INPUT OUTPUT


1010...1


Attractor of the genetic network

m



Biomass composition



(FBA)

Growth rate (pure)


INPUT OUTPUT


1010...1



Subset

m



Biomass composition



(FBA)

Growth rate (pure)


INPUT OUTPUT


1010...1



Subset

Growth rate (constrained)

The ratio of constrained FBA growth rate to pure FBA growth rate is ≤ 1.m


Answer 3(a): Histogram of the ratio of constrained FBA growth rate in the attractor of each of 15427 minimal media to the pure FBA growth rate in that medium. This is peaked at the bin with the largest ratio ≥ 0.9.

Ratio of constrained FBA growth rate topure FBA growth rate

0 - 0.1 0.1 - 0.2 0.2 - 0.3 0.3 - 0.4 0.4 -0.5 0.5 - 0.6 0.6 - 0.7 0.7 - 0.8 0.8 - 0.9 0.9 -1.0

Num

ber o

f med

ia

0

1000

2000

3000

4000

5000

6000

7000

Adaptability

Question 3(a): What is the ratio of the constrained FBA growth rate to pure FBA growth rate for various environmental conditions? In other words, is the regulatory network reaching an attractor that can make optimal use of the underlying metabolic network?


Adaptability

1010...1

1100...1

1101...0

.

.

.

.

.

.

.

.

t=0 t=1 t=∞

FBABiomasscomposition

GR(t=0) GR(t=1) GR(t=∞)

m


Adaptability

1010...1

1100...1

1101...0

.

.

.

.

.

.

.

.

t=0 t=1 t=∞

FBABiomasscomposition

GR(t=0) GR(t=1) GR(t=∞)

Question 3 (b): How well is the attractor of any particular medium “adapted” to that medium? Does the movement to the attractor “improve” the cell’s “metabolic functioning” in the medium?

Time

0 1 2 3 4 5

Gro

wth

rate

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Glutamine aerobic mediumLactate aerobic mediumFucose aerobic mediumAcetate aerobic medium

Answer 3(b):Growth rate increases by a factor of 3.5, averaged over pairs of minimal mediaFrom one minimal medium to another the average time taken to reach the attractor is only 2.6steps

Thus the regulatory dynamics enables the cell to adapt to its environment to improve its metabolic efficiency very substantially, fairly quickly.

m


The graph shows the genetic network controlling E. colimetabolism.


Design Features of the network explain Homeostasis and Flexibility

External Metabolites

Transcription factors

Metabolic Genes



This is an acyclic graph with maximal depth 4. Fixing the environment leads to fixing of TF states and also the leaf nodes leading to homeostasis. But when we change the environment, then the attractor state changes endowing system with the property of flexible response.



Metabolic Genes



The very few feedbacks from metabolism on to transcription factors are through the concentration of internal metabolites.



Metabolic Genes

Internal Metabolites


Modularity, Flexibility and Evolvability

This is a highly disconnected structure.

The disconnected components are dynamically independent and hence can be regarded as modules.

Such a structure can facilitate duringevolution to new environmental niches.


Almost all input functions in the E. coli TRN are canalyzing functions

• When a gene has K inputs, then in general there can be 2 to the power of 2K input Boolean functions that can exist. – As K increases the number of possible Boolean functions also

increases.• A Canalyzing Boolean function has at least one input such that for at

least one input value for that input the output value is fixed. • Stuart Kauffman proposed that Canalyzing Boolean functions are likely

to be over-represented in the real networks.• We found that all except four Boolean functions in the E. coli TRN were

canalyzing.


Design Features of the network

• The genetic network regulating E. coli metabolism is– Largely acyclic– Hierarchical– Root control with environmental variables– Disconnected and modular structure at the level of transcription factors– Preponderance of canalyzing Boolean functions

• There are some small cycles that exist due of presence of control byfluxes or internal metabolites but these cycles are very localized.

• Note that cycles are expected in developmental systems such ascell cycle which is a temporal phenomena.

• In metabolism, lack of cycles at the genetic level can be anadvantage as this is a slow process.

• Most cycles in metabolism exist at the level of enzymes and internalmetabolites such a process is faster.


Dynamics of the E. coli TRN controlling metabolism is highly ordered in contrast to that of Random Boolean Networks

Kauffman found that Random Boolean Networks (RBN) with K=2 are at the edge of chaos using Derrida Plot. Derrida plot is the discrete analog of the Lyapunovcoefficient. Derrida plot for RBNs with K>2 are found to be above the diagonal and their dynamics is quite chaotic.

Reference: S.A. Kauffman (1993)


Derrida Plot

100101

110111

000111

100111

t=0 t=1

H(0) = 2 H(1) = 1

H(0)

H(1

)

Derrida plot is a discrete analogue of the Lyapunov coefficient for continuous systems.

Ordered regime

Chaotic regime


Dynamics of the E. coli TRN controlling metabolism is highly ordered in contrast to that of Random Boolean Networks

Kauffman found that Random Boolean Networks (RBN) with K=2 are at the edge of chaos using Derrida Plot. Derrida plot is the discrete analog of the Lyapunovcoefficient. Derrida plot for RBNs with K>2 are found to be above the diagonal and their dynamics is quite chaotic.

Reference: S.A. Kauffman (1993)

H(0)

0 100 200 300 400 500

H(1

)

0

100

200

300

400

500

Reference: A. Samal and S. Jain (2008)

K can be as large as 8

The E. coli TRN controlling metabolism has input functions with K=8 also. However, the dynamics of the E. coli TRN is highly ordered .


System is far from edge of chaos

• The simple architecture of the genetic network controlling E. colimetabolism endows the system with the property of – Homeostasis– Flexibility of response

• Note that the dynamics is highly ordered and the system is far from the edge of chaos. It has been argued that the advantage of a system staying close to the edge of chaos lies in its ability to evolvable and be flexible.

• We have shown that the real system has an architecture with root control by environmental variables which is highly flexible, evolvable and far from the edge of chaos.

• Such an architecture of the regulatory network can also be useful for organisms with different cell types.


Acknowledgement

Collaboration

Sanjay JainUniversity of Delhi, India

Reference

areejit samal regulation

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