Basic modeling approaches for biological systems

Mahesh Bule

The hierarchy of life from atoms to living organisms

Modeling biological processes often requires accounting for action and feedback involving

a wide range of spatial and temporal scale

Modeling and biology

• Life is one of the most complex phenomenon in the universe

• Biological systems are regulated at scales of many orders of magnitude in space and time, with space spanning from the molecular scale (10−10 m) to the living organism scale (1 m), and time from nanoseconds (10−9 s) to years (108 s)

• The systematic investigation of cells, organs, organisms and manly cellular processes such as communication, cell division, homeostasis and adaptation- is systems biology

• Systems biology offer chance to predict outcome of complex process e.g. cell growth, gene expression

Integrative systems biology involving the iterative cycle of wet and dry laboratory

research

Modeling approaches in biology

• Bottom up and top down

Approach for multi-scale model development in biology

Hierarchy of scale, related mechanisms and modeling approaches

Relation of modeling approach, scale and experimental procedure

Comparison of systemic and molecular views of the same metabolic system on the example of

the photosynthetic apparatus of purple bacteria

Systems Biology is Modeling

• It relies on the integration of experimentation, data processing and modeling

• Modelling biological process focusses on increasing the depth of understanding and prediction of reliable results

• Development of tools to aid modelling can aid in understanding of processes

• Development of multi-scale modelling can allow “dry experiments” or “in-silico experiments” to be used as a form of validation which can save time and resources

Systems Biology is Modeling

Properties of model

1. Model assignment is not unique

• Biological processes can be described in more than one way as follows: – A biological object can be investigated with different

experimental methods

– Each biological process can be described with different (mathematical) model

– The choice of a mathematical model or an algorithm to describe a biological object depends on problem

Systems Biology is Modeling

2. System state

• Different modeling approaches have different representations of state e.g. – In differential equation model for a metabolic

network, the state is a list of concentrations of each chemical species

– In stochastic model, its is a probability distribution and /or list of current number of molecules of species

– In a Boolean model of gene regulation, the state is string of bits indicating for of each gene whether it is expressed (“1”) or not expressed (“0”)

Systems Biology is Modeling

3. Steady state

• The concept of stationary states is important for the modeling of dynamical systems

• The asymptotic behavior of dynamic systems, i.e. the behavior after sufficiently long time, is often stationary

• Fast process often reach a quasi-steady state after short transition period

Systems Biology is Modeling

4. Variables, Parameters, and Constants

• Constant is fixed value- natural number

• Parameters are quantities that are assigned a value, such as the Km value of enzyme in a reaction

• Variables are quantities with a changeable value for which the model establishes relations

Systems Biology is Modeling

5. Modeling behavior

Two fundamental causes that determine the behavior of a system

• Influences from the environment (input)

• Processes within the system

Systems Biology is Modeling

6. Process classification For modeling, processes are classified with respect to criteria. • Reversibility – determines whether process can proceed in

a forward and backward direction • Irreversible- the process which can proceed only in one

direction • Periodicity- indicates that a series of state may be assumed

in the time interval (t, t+∆t) • Deterministic approach- when the motion through all

following states can be predicted with known conditions • Discrete model- where values taken from a discrete set • Continuous model- where values are taken from a

continuum

Typical aspects of biological systems and corresponding models

• Modularity

– interacting nodes w/ common function

– constrained pleiotropy

– feedback loops,

oscillators, amplifiers

• Network versus Elements

– A system consists of individual elements that interacts and thus form a network

• Robustness

– insensitivity to parameter variation

• Severe constraints on design

– robustness not present in most designs

Three basic approaches used for modeling biological process

Interactome (Tier 1)

Deterministic (Tier 2)

Stochastic (Tier 3)

Response measurment during model development

• Tier 1: Interactome – Which molecules talk to

each other in networks?

• Tier 2: Deterministic – What is the average case

behavior?

• Tier 3: Stochastic – What is the variance of

the system?

Out put of different tiers during model development

• Tier 1 – get parts list

• Tier 2 & 3 – enumerate biochemistry

Out put of different tiers during model development

• Tier 2 & 3 – enumerate biochemistry

– define network/mathematical relationships

– compute numerical solutions

• Tire 2 & 3 – Deterministic: Behavior of

system with respect to time is predicted with certainty given initial conditions

– Stochastic: Dynamics cannot be predicted with certainty given initial conditions

Introduction to different models used

• Deterministic – Ordinary differential equations

(ODE’s) • Concentration as a function of

time only

– Partial differential equations (PDE’s) • Concentration as a function of

space and time

• Stochastic – Stochastic update equations

• Molecule numbers as random variables

• functions of time

Tire 1: Static interactome analysis

• Protein-protein

• Signal

transduction

• Cell cycle

• Protein-DNA

• Gene regulation

• Metabolic

pathways

• Respiration

• cAMP

Tier 1: Static interactome analysis

• Goals – Determine network topology – Network statistics – Analyze modular structure

Tier 1: Static interactome analysis

• Limitations:

– Time, space, population average

– Crude interactions • strength

• types

– Global features • starting point for Tier 2 & 3

first time-varying yeast interactome (Bork 2005)

typical interactome

Tier 1: Static interactome analysis

• Analysis methods

– Functional Genomics

• expression analysis

• network integration

– Graph Theory

• scale free

• small world

Tier 2: Deterministic Models

• Goal – model mesoscale system

– average case behavior

• Three levels – ODE system

– ODE compartment system

– PDE (rare!)

• data limited…

lumped cell

cell compartments

continuous time & space (MinCDE oscillation)

Tier 2: Deterministic Modeling

• Results – Robust Chemotaxis

– MinCDE Oscillation

– Feedback in Signal Transduction

• Output – time series plots (ODE)

– condition on parameter values

Tier 2: Deterministic Modeling

• Example – Robustness in bacterial chemotaxis

• Bacterial chemotaxis robust to parameter fluctuations! – Chemotaxis: bacterial

migration towards/away from chemicals

– Parameters • concentrations

• binding affinities

Tier 3: Stochastic analysis

• Fluctuations in abundance of expressed molecules at the single-cell level

– Leads to non-genetic individuality of isogenic population

Tier 3: Stochastic Analysis

• When stochasticity is negligible, use deterministic modeling…

• Molecular “noise” is low:

– System is large • molar quantities

– Fast kinetics • reaction time negligible

– Large cell volume • infinite boundary conditions

Tier 3: Stochastic Analysis

• Molecular “noise” is high: – System is small

• finite molecule count matters

– Slow kinetics

• relative to movement time

– Large cell volume

• relative to molecule size

• Need explicit stochastic modeling!

Model development workflow in biology

Formulation of problem

Verification of available information

Selection of model structure

Establishing a simple model

Sensitivity analysis

Experimental test and model prediction

Iterative refinement of model

Major challenges and limitations

• Measurement of chemical kinetics parameters and molecular concentrations in vivo

– Differences between in vitro and in vivo data

• Compartmental specific reactions

• Data is the limit!!!

Major challenges and limitations

• Data is the limit!!! – Functional genomic data

(Interactomes)

– E. Coli chemotaxis (Leibler, deterministic/robustness)

• Important – parameter estimation

– feedback based estimation methods