Introduction to the Analysis of Biochemical and Genetic Systems

Eberhard O. Voit* and Michael A. Savageau**

*Department of Biometry and EpidemiologyMedical University of South Carolina

VoitEO@MUSC.edu

**Department of Microbiology and ImmunologyThe University of Michigan

Savageau@UMich.edu

Three Ways to Understand Systems

• Bottom-up — molecular biology

• Top-down — global expression data

• Random systems — statistical regularities

Five-Part Presentation

• From reduction to integration with approximate models

• From maps to equations with power-laws

• Typical analyses

• Parameter estimation

• Introduction to PLAS

Module 1: Need for Models

• Scientific World View – What is of interest– What is important– What is legitimate– What will be rewarded

• Thomas Kuhn– Applied this analysis to science itself– Key role of paradigms

Paradigms

• Dominant Paradigms– Guides “normal science”– Exclude alternatives

• Paradigm Shifts– Unresolved paradoxes– Crises– Emergence of alternatives– Major shifts are called revolutions

Reductionist Paradigm

• Other themes no doubt exist

• Dominant in most established sciences– Physics - elementary particles– Genetics - genes– Biochemistry - proteins– Immunology - combining sites/idiotypes– Development - morphogens– Neurobiology - neurons/transmitters

Inherent Limitations

• Reductionist is also a "reconstructionist"

• Problem: reconstruction is seldom carried out

• Paradoxically, at height of success, weaknesses are becoming apparent

Indications of Weaknesses

• Complete parts catalog– 10,000 “parts” of E. coli

• But still we know relatively little about integrated system– Response to novel environments?– Response to specific changes in molecular

constitution?

X1 X3 X4X2t

012345678...

Dynamics

X1 X3 X4X2t

01234...

or ?

Critical Quantitative Relationships

X1

X3

X2

a

X1

X3

X2

b

c

X3

X1

Alternative Designs

Emergent Systems Paradigm

• Focuses on problems of complexity and organization

• Program unclear, few documented successes

• On the verge of paradigm shift

Definition of a System

• Collection of interacting parts, which constitutes a whole

• Subsystems imply natural hierarchies– Example: ... cells-tissues-organs-organism ...

• Two conflicting demands– Wholeness– Limits

Contrast Complex and Simple

Character Complex systems Simple systems

Numbers of variables Many Few

Interactions Strong Weak

Mode of coupling Nonlinear Linear

Processes Associative Additive

Quantitative Understanding of Integrated Behavior

• Focus is global, integrative behavior

• Based on underlying molecular determinants

• Understanding shall be relational

Mathematics

• For bookkeeping

• Uncovering critical quantitative relationships

• Adoption of methods from other fields

• Development of novel methods

• Need for an appropriate mathematical description of the components

Rate Law

• Mathematical function– Instantaneous rate– Explicit function of state variables that

influence the rate

• Problems

• The general case

Examples

• v = k1 X1

• v = k2 X1X2

• v = k3 X12.6

• v = VmX1/(Km+X1)

• v = VhX12/(Kh

2+X12)

Problems

• Networks of rate laws too complex

• Algebraic analysis difficult or impossible

• Computer-aided analyses problematic

• Parameter Estimation– Glutamate synthetase

• 8 Modulators

• 100 million assays required

Approximation

• Replace complicated functions with simpler functions

• Need generic representation for streamlined analysis of realistically big systems

• Need to accept inaccuracies

• “Laws” are approximations– e.g., gas laws, Newton’s laws

Criteria of a Good Approximation

• Capture essence of system under realistic conditions• Be qualitatively and quantitatively consistent with

key observations• In principle, allow arbitrary system size• Be generally applicable in area of interest• Be characterized by measurable quantities• Facilitate correspondence between model and reality• Have mathematically/computationally tractable form

Justification for Approximation

• Natural organization of organisms suggests simplifications– Spatial– Temporal– Functional

• Simplifications limit range of variables

• In this range, approximation often sufficient

Spatial Simplifications

• Abundant in natural systems

• Compartmentation is common in eukaryotes (e.g. mitochondria)

• Specificity of enzymes limits interactions

• Multi-enzyme complexes, channels, scaffolds, reactions on surfaces

• Implies ordinary rather than partial differential equations

Temporal Simplifications

• Vast differences in relaxation times– Evolutionary -- generations– Developmental -- lifetime– Biochemical -- minutes– Biomolecular -- milliseconds

• Simplifications– Fast processes in steady state– Slow processes essentially constant

Functional Simplifications

• Feedback control provides a good example– Some pools become effectively constants– Rate laws are simplified

• Best shown graphically

Rate Law Without Feedback

V

X

•

•

A

B

XBXA

Rate Law With Feedback

V

X

• •

•

A A'

C

XBXA

Consequence of Simplification

• Approximation needed and justified

• Engineering– Successful use of linear approximation

• Biology– Processes are not linear– Need nonlinear approximation

• Second-order Taylor approximation

• Power-law approximation

Module 2: Maps and Equations

• Transition from real world to mathematical model

• Decide which components are important

• Construct a map, showing how components relate to each other

• Translate map into equations

Model Design: MapsRibose 5-PATP

PP-Ribose-P Glutamine

P-Ribosyl-NH2

IMP

Amido-PRT

PP-Ribose-PSynthetase

ADP 2,3-DPG

Other Nucleotides

NADFAD

AMP, GMPATP, GTP

Example from Genetics

CRP

galS

galR

mgl

Galactose transport

gal

Galactose utilizat ion

Galactose transport

B A C

E T K (M)

P

gal

pp

p

p

p

--

+

+

-+

-

- ?

+

Components of Maps

• Variables (Xi, pools, nodes)

• Fluxes of material (heavy arrows)

• Signals (light or dashed arrows)

X4 X1 X2 X3

Rules

• Flux arrows point from node to node

• Signal arrows point from node to flux arrow

X1 X2

X3

X1 X2

X3

Correct Incorrect

Terminology

• Dependent Variable– Variable that is affected by the system;

typically changes in value over time

• Independent Variable– Variable that is not affected by the system;

typically is constant in value over time

• Parameter– constant system property; e.g., rate constant

Steps of Model Design1. Initial Sketch

Homoserine O-Homoserine-P Threonine-

Homoserinekinase

Threoninesynthetase

Threonyl-tRNAsynthetase

2. Conversion TableTable 2-1. Conversion Table for the Graph in Figures 2-11 and 2-12.

Variable Variable Variable

type name symbol

Dependent O-homoserine-P X1

Threonine X2Flux through O-homoserine-P V1Flux through threonine V2

Independent Homoserine concentration X3Homoserine kinase concentration X4Threonine synthetase concentration X5Threonyl-tRNA synthetase concentration X6

Aggregate None explicit --

Constrained None explicit --

Implicit ATP --

ADP --

Mg --

Inorganic phosphate --

threonyl-tRNA --

tRNAthr --

temperature --

pressure --

pH --

salt concentration --

geometry of reaction space --

3. Redraw Graph in Symbolic Terms

X 5X 4 X 6

-X 3 X 1 X 2

V1 V2

Examples of Ambiguity

• Failure to account for removal (dilution)

• Failure to distinguish types of reactants

• Failure to account for molecularity

• Confusion between material and information flow

• Confusion of states, processes, and logical implication

• Unknown variables and interactions

Failure to Account for Removal (Dilution)

Act. Xmas F.

Hageman F(XII) Act. Hageman F.

P.T.A. (XI)

Xmas F. (IX)

Act. P.T.A.

Ca++

Failure to Distinguish Types of Multireactants

X 1

X 2

X 3

X 1

X 2

X 3

X 2

X 3

X 1

X 2

X 3

X 1

Failure to Account for Molecularity (Stoichiometry)

X 1 X 2 X 1 X 2

X 1 X 22 X 1 X 22

Confusion Between Material and Information Flow

X 3X 1 X 2X 4

-X 1

Confusion of States, Processes, and Logical Implication

Hunger

Neuron Neuron

Pepsinogen

Pepsin

Food

Neutralizationof acid

Digestionproducts

Analyze and Refine Model

• There is lack of agreement in general

• Discrepancies suggest changes– Add or subtract arrows– Add or subtract Xs– Renumber variables

• Repeat the entire procedure– Cyclic procedure – Familiar scientific method made explicit

Open versus Closed Systems

X1X5 X4

X3

X2

X1X5 X4

X3

X2

Variables Outside the System

X 1

X 2

X 3

X 4

X 1

X 2

X 3

X 4

X 5

X 6

A

B

General System Description

• Variables Xi, i = 1, …, n

• Study change in variables over time

• Change = influxes – effluxes

• Change = dXi/dt

• Influxes, effluxes = functions of (X1, …, Xn)

• dXi/dt = Vi+(X1, …, Xn) – Vi

–(X1, …, Xn)

Translation of Maps into Equations

• Define a differential equation for each dependent variable:

dXi/dt = Vi+(X1, …, Xn) – Vi

–(X1, …, Xn)

• Include in Vi+ and Vi

– those and only those (dependent and independent) variables that directly affect influx or efflux, respectively

Example: Metabolic Pathway

dX1/dt = V1+(X3, X4) – V1

–(X1)

dX2/dt = V2+(X1) – V2

–(X1, X2)

dX3/dt = V3+(X1, X2) – V3

–(X3)

No equation for independent variable X4

X4 X1 X2 X3

Example: Gene Circuitry

A

/0/+ /0/+

Regulator gene /0/+

Effector gene

g45

g43

g15

g13

B

X1 mRNA

X2 Enzyme

X4 mRNA

X5 Regulator

X6 NA

X7 AA

X8 Substrate

X6 NA

X7 AA

X3 Inducer

g45g43 g13g15

/+

Power-Law Approximation

• Represent X1, …, Xn, Vi+ and Vi

– in logarithmic coordinates:

yn= ln Xn; Wi+ = ln Vi

+ ; Wi– = ln Vi

–

• Compute linear approximation of Wi+ and Wi

–

• Translate results back to Cartesian coordinates

Result• No matter what Vi

+ and Vi– , and even if Vi

+ and Vi–

are not known, the result in symbolic form is always

Vi+ i X1

gi1X2

gi2… Xn

gin

Vi– i X1

hi1X2

hi2… Xn

hin

“Power-Law Representation”

Parameters

gij: kinetic orders (positive, negative, or zero)

hij: kinetic orders (positive, negative, or zero)

i: rate constants (positive or zero)

i: rate constants (positive or zero)

Meaning of Kinetic Orders

0 < g, h < 1 -- Saturating functions

g, h > 1 -- Cooperative functions

– 1 < g, h < 0 -- Partial inhibition

g, h < – 1 -- Strong inhibition

– 2 < g, h < 2 -- Typical values (higher for fractal kinetics)

System Description

dXi/dt = Vi+(X1, …, Xn) – Vi

–(X1, …, Xn)

becomes “S-system”:

dXi/dt = i X1

gi1X2

gi2… Xn

gin

– i X1

hi1X2

hi2… Xn

hin

Summary of Power-Law Representation, S-systems

• Taylor series in logarithmic space

• Truncated to linear terms

• Interpretation of power-law function

• Estimation of parameter values

• Supporting evidence in biology

Components of a Typical Analysis

• Steady state– Numerical characterization– Stability– Signal propagation– Sensitivities

• Dynamics– Time plots– Bolus experiments– Persistent changes