1. A cell constitutes the following
External signal
cytoplasm
substrate
Cell wall +
Membrane
Product
translation
Protein
Secondary messenger
activation
nucleus
Translocation TF
Active TF
mRNA
transcription
DNA
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)

2. transcription and translational processes
‘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.
Genetic Info.
DNA
mRNA
Protein
Regulation
By proteins
Physiology
Transcription
Translation
Polypeptide
Proteins (Inactive)
Phosphorylation-dephosphorylation(P-D)
acetylation
Active (changes the conformation of protein)

3. 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.

4. 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.

5. 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.
Models can help to identify different dynamically important regimes, which may be hard or inaccessible to experimentalists.
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.
Mathematical structure of the model helps to identify the similar regulatory processes that run through various biosystems

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

7. In many systems, both positive and negative feedback loops together play an
Important 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
Importance of feedback and inference from feedback circuit diagram and
the mathematical model based on feedback circuit diagram at a latter stage.

8. Classification of Feedback loops.
Positive feedback loop brings about
Instability (explosive growth)
Amplification of a weak signal
(c) Multistability (Multistationarity) and epigenetic modifications
Negative feedback loop brings about
Homeostasis
Control the unexplosive growth
Induces oscillations (not in all cases)
Robust again perturbations.

9. Terminologies and classification
Symbol Name
Induction +
DNA-Protein
interaction
- Repression
gene g1 g2
m-RNA (Processed)
Protein P1 P2 +
Positive regulation
Negative regulation
Inhibition
Protein-Protein
interaction

10. - - For present representation g1 p1 A B g2 P2 g1 g2 P1 A B
This circuit diagram is the
abstraction of the whole network.

11. + - + - + What are the other circuits with feedback loops + Positive
- negative
Autregulatory feedback loop + A B
Positive feedback loop A B + + - B C A + B C A -
Negative feedback loop
(daisy chain) +
Feedback loops are determined by the parity of the negative signs

12. Two and three element feedback loops (non-homogenous or mixed feedback loops)+ + A B A B + +
Bistability
Oscillations - + A B + + C
Negative feedback loop
Positive feedback loop
Oscillations
There is a common element ‘c’ that connects these feedback loops. It is called
‘Interlocked feedback loop’: Example is Circadian rhythms.

13. 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.

14. 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 x
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 y
Trajectory

15. Linear Stability Analysis Example of linear system This is the assumed
Eigen value
Example of linear system
Eigen vector
This is the
assumed
solution
coupled
Linear
Equations
Matrix form of
coupled
Linear
Equations
Characteristic
Equation
Characteristic
polynomial
Very important
Matrix!!!!!
Eigen Values
To determine
Steady state
Or Fixed point

16. Example: 1 Critical Point Called as STABLE NODE Equation Matrix form
Trajectories
Starting from
Different
Initial conditions
Equilibrium
Point
Phase potraint
Eigen values
REAL,SAME SIGN
AND UNEQUAL
IC-1
Eigen vectors
For different
Eigen values
IC-2
THE DYNAMICAL STATE IS
STABLE NODE
Time series

17. Example :2 Critical Point Called as SADDLE POINT Equation Matrix form
Equilibrium
Point
Phase potrait
Eigen values
REAL,
OPPOSITE SIGN
AND UNEQUAL
Eigen vectors
For different
Eigen values
THE DYNAMICAL STATE IS
SADDLE POINT
Time series

18. If the real part of eigen value is positive
Example :3
Critical
Point
Called as
Stable
Spiral
Equation
A =
Matrix form
Equilibrium
Point
Phase potrait
Eigen values
REAL, and
COMPLEX
THE DYNAMICAL STATE IS
STABLE SPIRAL/ FOCUS
If the real part of eigen value is positive
Then the critical point is unstable
Spiral / Focus.
Time series

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

21. This is the final form of the linearized
Nonlinear equation at the steadystate
Two things are to be determined
Jacobian
Eigen values of the characterestic
Equation.

22. Two steady states
Example 4: Lotka-volterra
Prey-predator model
Jacobian
Fixed or Equilibrium points
for
Is a saddle
Eigenvalues
for
Is a Center/oscillatory
Eigenvalues

23. Time series and phase portrait for α = 2, β =0. 002, γ = 0
Time series and phase portrait for α = 2, β =0.002, γ = and δ =2
In three dimension with time as the axis

24. - NATURE OF FEEDBACK LOOPS AND DYNAMICS What dynamics is expected when
Self regulatory feedback loop
mutual induction / mutual repression
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 made
These predictions are
What conditions will bring bistability ?
(b) What conditions will bring about oscillations ?
Use mathematical models to findout and use the guidelines to
Build the regulatory circuit.
So (a) first find a naturally occuring simple circuit.
(b) Modify the circuit to build the circuit that exhibit desired dynamics. + A A B B C A -

25. Nomenclature that are to be known for understanding Gene regulation
E.Coli, a prokaryotic system has
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

26. Some terms you need to Know

27. (Cluster or group of genes)

28. How gene expression is monitored
Gene expression is monitored through tagging the expressed gene with
GREEN FLOURESCENT PROTEIN (GFP)
Various variants are also available, but the principle is same.
For example,
Inducer P O
GENE
GFP
When Gene expresses, then GFP
also expresses with almost same
quantitative amount.
(Direction of expression)
Flourescent
Intensity
Inducer

29. PARALLEL BETWEEN ENGINEERING TERMINOLOGIES
Examples

30. CONSTRUCTION OF SYNTHETIC GENE CIRCUITS
GUIDED BY
MATHEMATICAL MODELS

31. Naturally ocurring system
Two genes cI and cro
Are under the control
Of promoter PR and
PRM.
Cro and λ-rep binds to the
Promoter and regulates
Each other production.
Cro λ
High Low State
Cro λ Lysis
λ Cro Lytic
Natural system
Exhibits bistability
Either can be in Lytic stage or Lysogenic state.

32. Classification of positive and negative feedback loops
One element autoregulatory feedback loop
+ Positive
- negative A B
(Exhibits stability)
(Exhibits bistability)
Natural system
Analogy:
Water flowing on all
directions from a tub
due to holes.
Control the flow to one direction by blocking all the holes except one.
Tetracycline responsive
transactivator
Modified system
High inducer concentration
Low inducer concentration
No regulation
Feedback regulation
regulated
unregulated

33. Two element feedback loop (homogeneous)
GENETIC TOGGLE SWITCH + ON A B
[A]
bistability A B
Off + P
+ x + = + (Positive feedback loop)
- x - = + (Positive feedback loop)
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. ON ON
[A]
Off
Off
Stimulus
SWITCH
GRADED RESPONSE
HYSTERESIS

34. Derivation of the box equation in Gardner paper. Circuit diagram
Kinetic equations
Reducing to
Dimensionless quantity
Final box equation
P M1 + P
M R1
R1γ + p2
R1γ p2
P = Promoter
R = repressor
Conservation equations
Questions
When bistability
Occurs?
What should be taken
Care when plasmids
Are constructed for
Toggle switch
Rate of synthesis

35. Nullcline of above equations i.e.,
Interpretations
Lumped parameters describes
RNA-binding, transcript and
polypeptide elongation,
etc.
Cooperativity from multimerization
Of repressor protein and and DNA and
Repressor binding
Nullcline of above equations i.e.,
BISTABILITY
Intersects at three points and isdue
To cooperative effect, i.e.,
Two SS and one USS
Bistability is possible only when the
Rate of synthesis of two repressors
are balanced.
If not, only mono stability is obtained
MONO STABILITY

36. Two parameter bifurcation diagram (Formation of Cusp)
Large region
CUSP
bifurcation
Rate of repressors if balanced properly, large region of bistability
Is 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.

37. Experimental observations of Synthetic toggle switchu v
Natural
What is observed in natural
bacteriophage lambda circuit
This is the constructed synthetic
toggle switch by R-DNA technology
taking into account of the theoretical
considerations from model
Toggle switch observed in experiments
and variations is seen by monitoring
Green flourescent protein (GFP).
Synthetic
Inducer LacI Clts GFP State
IPTG high low high ON
Heat Low high low OFF
(Isopropyl B-D-thiogalactopyroniside)

38. average cannot be predicted.
First synthetic construct from the model and good prediction is made about
the 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 inductiontime
And variability wwith IPTG
Cell to cell variation with IPTG
Fast induction time
With temperature

39. - + - + - + Three element feedback loops (homogenous) A B A B C C
negative feedback loop
Positive feedback loop
[A]
Oscillations
[A]
(USS)
SS-I (Off state) P P

40. - - B C A Numerical and Experimental observation of Repressilator
Leakiness
For standard
parameter
Decrease in
cooperativity
Decrease in
Repression
i.e., > 0

41. Prediction from the model
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

42. Design of Experimental system
Single cell isolated from colony
flouresence
Brigh field
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.

43. Since this oscillator is noisy and unstable
Since this oscillator is noisy and unstable. ‘Hysteretic’ based oscillator
Is proposed.
Hypothetical network is constructed and the where the degradation of the
Protein is controlled by a ‘slow’ subsytem.
This gives ‘Relaxation oscillation’
Still there is no experimental evidence.