• A cell constitutes the following

cytoplasm

nucleus

DNA

Cell wall + Membrane

Cell is dynamic as it involves in various processes;(a) it change its shape, i.e., it is elastic(b) it can communicate with other cells (cell-to-cell communication)(c) it can move (molecular motors)(d) It can process signals and involves in transport of molecules(e) Cell can also divide (Mitotic) and can make multiple copies (Replication)

mRNA

substrate

Product

Active TFTF

External signal

Secondary messenger

activationTranslocation

transcription

translation

Protein

• ‘GENETIC’ Information is also passed on from one generation to another through

transcription and translational processes

• ‘ENVIRONMENTAL’ Information flows in and out the cell through protein synthesis .

Simplified version of earlier cell diagram.

Transcription

Polypeptide

Translation

Proteins (Inactive)

DNA

mRNA

Protein

RegulationBy proteins

Physiologyacetylation Phosphorylation-dephosphorylation(P-D)

Active (changes the conformation of protein)

Genetic Info.

Biological regulatory network (BRN) consists of sets of DNA’s, proteins and enzymes that involves in mutually regulating each other.

• The regulation gives rise to (a) functioning of the cell at normal as well as in adverse conditions (b) set a stage for developmental by inducing phenotypic variations.

• Regulation occurs at both transcription and translation.

• Regulation involves multiple feedback loops

• Feedback loops are classified as positive and negative.

• Difficult to asess the complexity of feedback loops globally.

• So intution alone is not sufficient to understand. Requires mathematical modelling.

Various tools and methodologies are available for modelling.

Usefulness of Mathematical models in BRN’s

• a) To account for experimental observations and to determine the validity of experimental conclusions.

• (b) Clarification of hypothesis.

• (c) Difficult to rely only on intuition. Mathematical equations provides a strong foundation for validating concepts and analyze complex data that involves multiple coupled variables.

• (d) Models identify critical parameters for which certain phenomena can occur.

(e)Different regulatory mechanisms can be explored through models from which only plausible mechanisms can be identified. This cannot be carried out through experiments, which are expensive and time consuming.

(f) Models can help to identify different dynamically important regimes, which may be hard or inaccessible to experimentalists.

(g)Models can suggest experimentalists to perform new experiments to explore unknown, but biologically important and interesting regimes. Conversely, it can also validate the suitability of the models.

(h)Mathematical structure of the model helps to identify the similar regulatory processes that run through various biosystems

How to model BRN’s ?

• Differential equations

Ordinary (ODE) Partial (PDE) stochastic (SDE) Functional (FDE)

Further can be classified as Linear and non-linear differential equations.

To model GRN‘s the universal law of chemical mass action kinetics is used widely.

To use mass action kinetics, static biological circuit diagrams should beConstructed.

To construct biological circuits, interaction among various proteins and types of interactions are necessary. This is transfomed in to mathematical equations.

In many systems, both positive and negative feedback loops together play anImportant role.

Take different types of positive and negative feedback loops and analyze for different dynamical phenomena.

Visualization is important for to understand biological systems.

Certain conclusion can be made by visualizing simple circuit diagrams. These Circuit diagrams are the backbone and basis of large circuit diagrams.

After construction of biological circuit diagram, mathematical models are constructed. Models again can be visualized by two ways and are

Phase plane analysis (b) bifurcation diagrams.

We shall look at the first aspect now namely

(a) Importance of feedback and inference from feedback circuit diagram and (b) the mathematical model based on feedback circuit diagram at a latter stage.

Classification of Feedback loops.

Positive feedback loop brings about (a) Instability (explosive growth)(b) Amplification of a weak signal(c) Multistability (Multistationarity) and

epigenetic modifications

Negative feedback loop brings about(a) Homeostasis(b) Control the unexplosive growth(c) Induces oscillations (not in all cases)(d) Robust again perturbations.

Terminologies and classification

gene

Symbol Name

m-RNA (Processed)

Protein

Positive regulation+

Negative regulation

+Induction

- Repression

g1 g2

P1 P2

Inhibition Protein-Proteininteraction

DNA-Proteininteraction

g1

p1

A g2

P2

B

-

g1 g2

P1

-

A B

For present representation

This circuit diagram is theabstraction of the whole network.

What are the other circuits with feedback loops

+ Positive - negative Autregulatory feedback loop

A B

+

+

A BPositive feedback loop

+

+B

C

A

+

-

B

C

A

--

Negative feedback loop

Feedback loops are determined by the parity of the negative signs

(daisy chain)

Two and three element feedback loops (non-homogenous or mixed feedback loops)

A

+

B

-

A

+

B

-

+

B

C

A

+

++

+-

Negative feedback loop Positive feedback loop

There is a common element ‘c’ that connects these feedback loops. It is called ‘Interlocked feedback loop’: Example is Circadian rhythms.

Bistability Oscillations

Oscillations

Some definitions

Nonlinear dynamics: The rate of change of variables can be written as the linear function of the other related variables. Most nonlinear systems exhibit interesting dynamics like bistability and oscillations.

Fixed point: It is the point where the rate of change of all variables are exactly Zero. Small perturbations of the fixed point will bring the system back to the same point and perturbations are quenched. It is then called stable.

Multistability: Having more than one fixed point.

Bistability: A system that has two stable fixed points.

Phase plane analysis.

Its a graphical approach and can be only performed only for 2D systems.

Its only a qualitative analysis.

Its only for autonomous system. Where time does not occur explicitly in the differential equations. For example

Right hand side of this equation, ‘t’ is not involved.

The plot of x .vs. y gives the phase plane

The trajectories in the phase plane is called the phase portrait.

To find the critical points or steady state or Equilibrium points

x

y

Trajectory

Linear Stability Analysis

coupledLinearEquations

Matrix form ofcoupledLinearEquations

To determineSteady stateOr Fixed point

This is the assumedsolution

Eigen valueEigen vector

CharacteristicEquation

Characteristicpolynomial

Eigen Values

Example of linear system

Very importantMatrix!!!!!

Example: 1

Equation

Matrix form

EquilibriumPoint

Eigen valuesREAL,SAME SIGN AND UNEQUAL

Eigen vectors For differentEigen values

Phase potraint

Time series

THE DYNAMICAL STATE IS STABLE NODE

CriticalPointCalled asSTABLENODE

TrajectoriesStarting fromDifferentInitial conditions

IC-1

IC-2

Example :2

Equation

Matrix form

EquilibriumPoint

Eigen valuesREAL, OPPOSITE SIGN AND UNEQUAL

Eigen vectors For differentEigen values

THE DYNAMICAL STATE IS SADDLE POINT

CriticalPointCalled asSADDLEPOINT

Phase potrait

Time series

Example :3

Equation

Matrix form

EquilibriumPoint

Eigen valuesREAL, andCOMPLEX

THE DYNAMICAL STATE IS STABLE SPIRAL/ FOCUS

CriticalPointCalled asStableSpiral

Phase potrait

Time series

A = -0.5 1-1 -0.5

If the real part of eigen value is positiveThen the critical point is unstableSpiral / Focus.

Linear stability analysis around the steady states using Taylor expansion.What is Taylors expansion ?

Consider a function f(x,y) and expand around steady state (x*,y*).

Suppose if it is nonlinear equation, how to determine various dynamical states. It is Simple: Linearize nonlinear system around steady state.

This is the final form of the linearizedNonlinear equation at the steadystate

Two things are to be determined

(a) Jacobian

(b) Eigen values of the characteresticEquation.

Example 4: Lotka-volterraPrey-predator model

for

Eigenvalues Is a saddle

for

Eigenvalues Is a Center/oscillatory

Fixed or Equilibrium points

Two steady states

Jacobian

Time series and phase portrait for α = 2, β =0.002, γ = 0.0018 and δ =2

In three dimension with time as the axis

What dynamics is expected when

(a) Self regulatory feedback loop

(b) mutual induction / mutual repression

(c) Three elements repressing each other in a daisy chain or activationg each other in a daisy chain

Difficult to perform experiments unless certain predictions can be madeThese predictions are

(a) What conditions will bring bistability ?

(b) What conditions will bring about oscillations ?

Use mathematical models to findout and use the guidelines toBuild the regulatory circuit.

So (a) first find a naturally occuring simple circuit. (b) Modify the circuit to build the circuit that exhibit desired dynamics.

NATURE OF FEEDBACK LOOPS AND DYNAMICS

A+

A B

B

C

A

--

Nomenclature that are to be known for understanding Gene regulation

E.Coli, a prokaryotic system has

(a) A single circular chromosome

(b) PLASMIDS, an extrachromosomanal DNA(Usually this is manipulated to design new circuits)

Genes are referred in lower case italics, while proteins are referred in Uppercase.

For example,

Ecoli

Some terms you need to Know

(Cluster or group of genes)

How gene expression is monitored

Gene expression is monitored through tagging the expressed gene withGREEN FLOURESCENT PROTEIN (GFP)

Various variants are also available, but the principle is same.

For example,

GENE GFPOP

(Direction of expression)

InducerF

lou

resc

ent

Inte

nsi

ty

Inducer

When Gene expresses, then GFPalso expresses with almost samequantitative amount.

Examples

PARALLEL BETWEEN ENGINEERING TERMINOLOGIES

CONSTRUCTION OF SYNTHETIC GENE CIRCUITS

GUIDED BY

MATHEMATICAL MODELS

Natural systemExhibits bistabilityEither can be in Lytic stage or Lysogenic state.

Naturally ocurring system

High Low State

Cro λ Lysis

λ Cro Lytic

Two genes cI and croAre under the controlOf promoter PR andPRM.

Cro and λ-rep binds to thePromoter and regulatesEach other production.

Cro λ

Classification of positive and negative feedback loops

A+ Positive

One element autoregulatory feedback loop

B- negative

Natural system

Modified system

No regulation

Feedback regulation

(Exhibits bistability) (Exhibits stability)

Tetracycline responsivetransactivator

Low inducer concentration

High inducer concentration

unregulated

regulated

Analogy:

Water flowing on alldirections from a tubdue to holes.

Control the flow to one direction by blocking all the holes

except one.

Two element feedback loop (homogeneous) GENETIC TOGGLE SWITCH

A B

+

+

A B

- x - = + (Positive feedback loop)

[A]

P

bistability

Steady states (SS- I and II) can be seen in experiements, but unstable steady state (USS) cannot be seen in the experiments, but realized in mathematical model.

Off

ON

+ x + = + (Positive feedback loop)

Off

ON

Stimulus

Off

ON[A]

SWITCHGRADED RESPONSE HYSTERESIS

Derivation of the box equation in Gardner paper.

Kinetic equations

Conservation equations

Rate of synthesis

Reducing to Dimensionless quantity

Final box equation

P = PromoterR = repressor

Questions

When bistabilityOccurs?

What should be takenCare when plasmidsAre constructed forToggle switch

Circuit diagram

R2(R1) (P1)

(P2)

P M1 + PM1 R1R1γ + p2 R1γ p2

Interpretations

Lumped parameters describesRNA-binding, transcript and polypeptide elongation, etc.

Cooperativity from multimerizationOf repressor protein and and DNA andRepressor binding

Nullcline of above equations i.e.,

Intersects at three points and isdueTo cooperative effect, i.e.,

Two SS and one USS

Bistability is possible only when theRate of synthesis of two repressorsare balanced.

If not, only mono stability is obtained

BISTABILITY MONO STABILITY

Rate of repressors if balanced properly, large region of bistabilityIs possible

Slopes of the bifurcation lines are determined by the cooperativity Effect; High cooperativity effect, large bistable regime and vice versa

Contributes to the robustness of the system.

CUSPbifurcation

Two parameter bifurcation diagram (Formation of Cusp)

Large region

u v

Natural

Synthetic

Experimental observations of Synthetic toggle switch

What is observed in naturalbacteriophage lambda circuit

This is the constructed synthetictoggle switch by R-DNA technologytaking into account of the theoreticalconsiderations from model

Toggle switch observed in experimentsand variations is seen by monitoringGreen flourescent protein (GFP).

(Isopropyl B-D-thiogalactopyroniside)

Inducer LacI Clts GFP State

IPTG high low high ONHeat Low high low OFF

First synthetic construct from the model and good prediction is made aboutthe rate of synthesis and repressor stength for the occurrence of toggleswitch

But predicted only average behavior of the cell and variations about the average cannot be predicted.

For example, the time required for switching Toggle to high state for IPTG takes 3-6 hrs for different cells. Non-deterministic effects cannot be predicted.

The switching time is very slow and is not useful for practical purpose. So different toggle switch was constructed which was temperature sensitive. Theswitching time is fast and rapid.

Slow inductiontimeAnd variability wwith IPTG

Fast induction timeWith temperature

Cell to cell variation with IPTG

Three element feedback loops (homogenous)

B

C

A

+

++

B

C

A

-

--

Positive feedback loop negative feedback loop

[A]

P

Oscillations[A]

P

SS-I (Off state)

(USS)

B

C

A

--

- Numerical and Experimental observation of Repressilator

Decrease in cooperativity

Decrease in Repression

i.e., > 0

Leakiness For standard parameter

Prediction from the model

(a) Presence of strong promoters that binds the protein

(b) High cooperative binding of the repressors increases the range of oscillatory regime and robustness

(c) The life time of the mRNA and proteins should be similar for strong oscillations

Synthetic network that are to oscillate should be in accordance to the above condition

Design of Experimental system

But there is a large variation in the cell.

This is due to stochastic fluctuations of the molecules

When simulated with stochastic model, this variation is accounted for.

flouresence

Brigh field

Single cell isolated from colony

Since this oscillator is noisy and unstable. ‘Hysteretic’ based oscillatorIs proposed.

Hypothetical network is constructed and the where the degradation of theProtein is controlled by a ‘slow’ subsytem.

This gives ‘Relaxation oscillation’

Still there is no experimental evidence.