a cell constitutes the following
DESCRIPTION
cytoplasm. External signal. substrate. Cell wall + Membrane. Product. translation. A cell constitutes the following. Protein. Secondary messenger activation. nucleus. Translocation. mRNA. TF. Active TF. transcription. DNA. - PowerPoint PPT PresentationTRANSCRIPT
• 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.