Sharad Bhartiya Department of Chemical Engineering

IIT Bombay

Mathemight- 07 Jan18-20,2013

Department of Chemical Engineering IIT Bombay January 20, 20013

Department of Chemical Engineering

From Biology to Mathematics

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A Reductionist Approach:

Observe, Understand and

Characterize Components

HT Analytical Methods genomics

transcriptomics

proteomics

simple.wikipedia.org

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Need a systems approach

Schmula.com

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Integrative Approach:

Systems Biology

1. Components

2. Reconstruction

3. Modeling/simulation

4. Feedback analysis

1250 computers v/s massive analog computing

Hundreds of feedback loops v/s many many large

Millions of components v/s many many more

Historically, biology has been a descriptional science

Modern Biology has led to quantification at molecular level (sub-system)

Similar to engineering systems that are quantified to a level that they are designed, optimized and optimally operated.

Principles of system science can be applied to

component biology: System-wide Approach

1/19/2013 Process Control Group, ChE, IIT Bombay 7

Non-linear dynamics

Multiple feedback loops

Multiple interactions

Cascade structures

Feed forward loops

Interactions between modules

Timescale separation

Resulting in a Complex system

Experiment:

1. One maroon and One red Ball in a Box

2. Choose a ball randomly

3. Add and replace the same color ball into the box

4. What is the fraction of red balls after 50,000 steps

Picked maroon ball

Replaced the same

and added another

of same colour

P = 0.5

P = 0.5

P = 0.33

P = 0.66

P = 0.66

P = 0.33

Multiple outcomes possible depending on initial few steps

Fra

ctio

n o

f re

d b

alls

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Bottoms-up approach

Millions of components

Design manual available

Hundreds of feedback loops

Computer calculations

Robust

Top-down approach

No design principles available

No computation

Thousands of feedback loops

Molecular interactions

Robust

— Csete and Doyle, Science, 295, 2002

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DesignDesign

ControlControl

EvolveEvolve

Fault DiagnosisFault Diagnosis

OperationOperation

Economics

Model based

Smart systems/AI

Adaptive

Inherent

Robust performance/ stability

Molecular interaction

DNA repair/PCD

Evolution

Firefly Glows

Transgenic Plant made to Glow

Physiological state

Phosphorous release

using ATP

Metabolic network

Catalyzed by Luciferase Enzyme Protein network

Luciferase gene decoded RNA network

Luciferase Gene Genetic network

1/19/2013 Process Control Group, ChE, IIT Bombay 15

Genotype to Phenotype: An Integrated

Approach

• Genome

• Transcriptome

• Proteome

• Metabalome

Phenotype

(physiological

consequence)

Upsala Glacier, BBC website

Environment

Randomness

Genome Gene expression is triggered

Transcriptome mRNAs are synthesized

Proteome Necessary enzymes are made

Metabalome Enzymes catalyze substrates

Phenotype Metabolites react to trigger

Presence of genome does not ensure a phenotype

It requires a specific state in the hierarchical chain.

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Modeling

Logical/Boolean networks

CFD

Delayed ODE

PDE

Stochastic models

Multiscale

Data mining

Estimation theory

Nonlinear systems theory

Feedback control theory

Sensitivity

Optimization

Stability

Analysis

Key: Identify design principles

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Example 1: Central Dogma Illustrated:

Tryptophan System of E. coli

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Operon Activation

and Transcription

RNAP R

Chorismate L-tryptophan

trp L trp R P/O P/O trp D trp E trp C trp B trp A

T

T T

Attenuator

RNAP RNAP RNAP RNAP RNAP

EEDD

T

T

T

T

F/B Mechanism I:

Genetic Repression

1/19/2013 Process Control Group, ChE, IIT Bombay 20

Structural Enzymes

Anthranilate synthase

Phosphoribosyl

anthranilate

transferase

Indole glycerol

phosphate synthase

Tryptophan synthase

1 2

3 4

1 2

3

4 1 2

3

4

Translation

EEDD

BBAA

T

T

T T

F/B Mechanism II:

Attenuation

1 2

3 4

1/19/2013 Process Control Group, ChE, IIT Bombay 21

Chorismate AS

Anthranilate PRA PRT

IGPS

CdRP

IGPS

InGP TS

L-tryptophan

Tryptophan Synthesis

EEDD

T EEDD

T

T

T

Active

T

T EEDD Inactive

F/B Mechanism III:

Enzyme Inhibition

1/19/2013 Process Control Group, ChE, IIT Bombay 22

Example 1: Trp System: Model

Reduction

• Enables delineation of process and regulator

— Bhartiya, Rawool, Venkatesh, Eur. J. Biochem, 270, 2003

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Example 1: Tryptophan System in

Escherichia coli: Regulator and Process — Venkatesh, Bhartiya and Ruhela, FEBS Letters, 563, 2004

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Example 1: Multiple Loop v/s Single

Loop Design

CASE I: Multiple

CASE II:

Single

Are Multiple feedbacks loops a regulatory overkill?

(Freeman, Nature, 2003).

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G1 S M G2

Cyclin dependent kinese + cyclin •DNA repair

•Stress mediation

•Checkpoints

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Cig2Cdc2

Cdc13

Cdc2

Puc1

Cdc13

APC APC

P

P

P

P

P

Mik1

Mik1

Wee1

Wee1

Cdc13

Cdc2

P P

Rum1

P

Cdc13

Cdc2

Rum1

Puc1

Cdc2

Cig2

Cdc2

CAK

Cdc25

Cdc25

P

Rum1

PP1

P

A-DeP

Slp1

Slp1

APC

P

P

P

P

Ste9 P

SecSec

P

Separase

Separase

Ste9

APC

P

P

P

P

I

Cohsin Degradation for Sister

Chromatid Seperation

PP2I

PP1

PP1P

PP2

Ste9

Relative to previous works

• Synthesis of all proteins

• Role of multiple

phosphatases

• Detailed mitotic exit

regulation

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• Model could successfully simulate16 single, 14 double, 4 triple

deletion, 2 structural and 2 over expression mutants

• Math Question: Under what conditions will the limit cycle

become unstable? Identify sensitivity of period of oscillations to

parameters. Characterize DNA damage and repair.

Cycle time:

152 minutes

Synthetic Biology: Constructing Mathematical

Functions from biological components

Bistable circuits

Oscillators

Riboswitches

Logic Gates

Example 3:Bistable Circuits:

Natural

Bistable

ou

tpu

t

input

ON

OFF

Bistability in the lactose utilization network of E. coli

Nature 427, 737-740 (2004)

o Lactose utilization in E. coli.

o Lysis vs lysogeny in

bacteriophage λ.

o Sporulation in B. subtilis.

o Competence in B. subtilis.

A: TetR–VP16 (transactivator)

B: E-KRAB (transrepressor)

X: Erythromycin

PNAS July 5, 2005 vol. 102 no. 27 9517-9522

Circuit Design

Genetic Implementation

Example 3: Bistable Circuits:

Synthetic

Genes & Dev. 2007. 21: 2271-2276

A: Auto-feedback gene

B: Sensor gene

Natural Oscillatory Circuits

Example 4: Oscillatory Circuits: Natural

-

Nature Reviews Molecular Cell Biology 9, 981-991 (December 2008)

A: LacI

B: tetR

C: cI

Circuit Design

Genetic Implementation

Michael B. Elowitz and Stanislas Leibler; Nature. 2000

Repressilator

Tal Danino, Octavio Mondragón-Palomino, Lev Tsimring & Jeff Hasty; Nature.2010

Synchronized genetic clock

A: LuxI

B: aiiA

Frequency-modulated genetic arsenite biosensor

Nature 481,39–44 (05 January 2012)

Example 4: Oscillatory Circuits:

Synthetic

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Example 5: GAL system in yeast:

How to uptake galactose sugar

Yeast 1 (cerevisiae) and Yeast 2 (K. lactis) are evolutionary cousins

Both can metabolize galactose in absence of glucose

This is possible by the GAL system in the two yeasts

However the GAL system designs are different

Math Question: What is the evolutionary niche each have for their

respective designs

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Engineering

• Developed detailed growth related dynamic models for

both organisms (validated with experiments) • Pannala et al., Wiley Interdisciplinary Reviews: Systems biology and medicine, (2010)

• Pannala et al., FEBS Journal, (2010) [Steady state K. lactis]

• Pannala et al., IET Systems Biology, (In press) [comprehensive S. cerevisiae]

• Pannala et al., Systems and synthetic biology, (2011) [comparative study]

S. cerevisiae K. lactis

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Engineering

S. cerevisiae K. lactis

GAL80 mutant:

growth on glucose

Wildtype: growth

on galactose

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Evolutionary niche using optimization: MINLP

ConstraintConstraint--based Analysisbased Analysis

How often have I said to you that

when you have eliminated the

impossible, whatever remains,

however improbable, must be the

truth?

–Sherlock Holmes, A Study in

Scarlet

Flu

x C

Flux B

Flux A

Unbounded

Solution Space

Flu

x C

Flux B

Flux A

Unbounded

Solution Space

Flu

x C

Flux B

Bounded Convex Subset

Flu

x C

Flux B

Bounded Convex Subset

Constraints

(i) Stoichiometric (i)

(ii) Thermodynamic

(iii) Capacity

(iv) Rate

(v) Parameters

Based on the properties of the system.

Time constants for metabolic reactions are

very fast (sec - min) compared to cell growth

(hrs)

No net accumulation of metabolites in the cell

Thus, the steady-state approximation.

0 bvSX

dt

d

Venkatesh & Fell, Biotechnology and Bioengineering, 2004

Optimal

Biomass

Example 7: Elementary Modes

An elementary mode is a minimal subset of enzymes in

a network that can operate at steady state with all

irreversible reactions proceeding in the direction as

prescribed by thermodynamics

Elementary mode represents routes through substrate is

consumed to form products

Example 7: Methodology: Hypothetical

Network Elementary modes System chosen

Gayen and Venkatesh, BMC Bioinformatics. 2006; 7: 445

Example 7: Problem Formulation

Rates of external metabolites

In matrix form

Linear programming formulation

Experimentally

Determined

(known)

Gayen and Venkatesh, BMC Bioinformatics. 2006; 7: 445

Linear Algebra – metabolic flux analysis

Ordinary differential equations – lumped modeling

Partial differential equations – drug delivery, metastasis

Stochastic differential equations – heterogeneity, uncertain

Boolean algebra – large scale networks, drug discovery

Optimization – evolutionary biology, biotech

Statisitics – old companion of biologists

Artificial intelligence – bioinformatics

Lyapunov stability – perturbations before disease?

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Challenges: overcome jargon by reading biology

Interdisciplinary work requires interactive working

Be generous in sharing credit

Mathematics unifies seemingly different systems (a bacterial cell and a rocket) and this is her majestic might

Department of Chemical Engineering

Prof. K.V. Venkatesh

P. Anbumathi

Venkat Pannala

Nikhil Chaudhary

Department of Chemical Engineering