Systems/Computational Biology (I)Introduction

Kevin Burrage and Andre Leier

Advanced Computational Modelling CentreThe University of Queensland

leier@acmc.uq.edu.au

kb@maths.uq.edu.au

Introduction to Systems Biology

The cell

Wish to understand different types of dynamical processes and transports within a cell through experimentation (Single Particle Tracking, microarrays etc), mathematical models and simulations.

Three D EM image of a insulin mammalian

secreting cell (Marsh)

I. Cell Membranes and Lipid Rafts

Magnify a cell a million timesWater molecule: full stop; Protein: ping pong ball; Ribosome: soccer ball; Mitochondion: person; Nucleus: car; Cell membrane: <1 cm thick!

Singer and Nicholson, 1972:“protein icebergs floating in a sea of lipids”

II. Segmental oscillating gene expression

Chick embryo

III. Tubulation

Untreated cells have little tubulation

Massive tubulation event following EGF treatment

Modelling and Simulation

In silico humans-spatial & temporal scales

• 1 m person• 1 mm electrical length scale of cardiac tissue• 1 m cardiac sarcomere spacing• 1 nm pore diameter in a membrane proteinRange = 109

Requires a hierarchy of inter-related modelspathway models

ODEsstochasticmodels

PDEs (continuum models)gene reg.networks

• 109 s (70 yrs) human lifetime• 106 s (10 days) protein turnover• 103 s (1 hour) digest food• 1 s heart beat• 1 ms ion channel HH gating• 1 s Brownian motionRange = 1015

Michaelis – Menten ReactionS1+S 2

k1 S 3 a1 X =k 1S 1S2

S3k2S1+S 2 a 2 X =k 2S3

S3k3P+S1 a3 X =k 3S3

ν1=[−1−1

1 ] , ν 2=[ 11

−1] , ν3=[ 10

−1]The stoichiometric vectors are and the Law of Mass Action gives

X' t =∑ υj aj X t

An example: a system with three species of molecules and four reactions

X 1c10 a1 X =c1 X 1

X 1+X 1c2 X 2 a2 X =12

c2 X 12

X 2c3 X 1+X 1 a3 X =c3 X 1

X 2c4 X 3 a 4 X =c4 X 2

ν1=[−100 ] , ν 2=[−2

10 ] , ν3=[ 2

−10 ] , ν4=[ 0

−11 ] ,

If the current state is (x1,x2,x3)=(100,100,100),

After reaction 2, (x1,x2,x3)=(98, 101,100)

4, (x1,x2,x3)=(98, 100,101)

The four state transfer vectors are

(CalBioChem)

MAP kinase pathways in the cell: components, regulation, translocation and crosstalk between different signalling pathways

MAP kinase pathway activated by EGF receptor (230 reactions and 94 species) (Black: plasma membrane and cytosol, Green: internalized subsystem)

(Schoeberl, et al., Nature Biotech., 2002)

Basics of Molecular Cell Biology- A Review

All Living Things Are Made of Cells

Cells

have a variety of different shapes, sizes and functions,

grow and divide (reproduce themselves),

convert energy / digest, sense and respond to their environment,

are able to swim and to cooperate to form complex organisms,

…

HOW DOES THIS WORK?

All Living Things Are Made of Cells (cont.)

Alberts et al., Molecular Biology of the Cell, 4th Edition (2002).

The Cell – A Chemical Factory

Basic constituents inside cells are sugars (monosaccharides) fatty acids nucleotides amino acidsplus some ions, water (~70%), other organic molecules.

These are linked into macromolecules

All cells are governed by the same chemical machinery. Alberts et al., Molecular Biology of the Cell, 4th Edition (2002).

Genetic Regulation and cascading reactions

Elements of Genetic Regulatory Networks

• Transcription factors bind to DNA sequences in regulatory regions of genes.

• Binding regulates the rate at which transcripts (polymerase) of the gene are initiated.

• Protein is made off mature mRNA transcripts by translation (ribososomes).

Central Dogma: DNA RNA Proteintranscription translation

Feedback: DNA RNA Protein

DNA and its Building Blocks

Genetic information is encoded in the nucleotide sequence.

Prokaryotes: DNA in the cytoplasm; Eukaryotes: DNA in the nucleus (packaged into chromosomes).

Genome: all cell DNA.

Alberts et al., Molecular Biology of the Cell, 4th Edition (2002).

Gene Control Region

Promoter: DNA sequence to which RNAP binds to begin transcription. Transcription factor (TF): protein required to initiate or regulate

transcription.

Alberts et al., Molecular Biology of the Cell, 4th Edition (2002).

From DNA to Proteins: Transcription and Translation (cont.)

Differences in eukaryotic and prokaryotic cells:

Alberts et al., Molecular Biology of the Cell, 4th Edition (2002).

From DNA to Proteins:Transcription and Translation (cont.)

A ribosometranslates mRNA (using tRNA).

Transcription initiationby RNA polymerase (RNAP) II in an eukaryotic cell.

Alberts et al., Molecular Biology of the Cell, 4th Edition (2002).

Regulation Of Gene Expression

Alberts et al., Molecular Biology of the Cell, 4th Edition (2002).

Controlled by environmental signals(ligands bind to receptors)

Biology and Noise

Biological Evidence of noise

“Stochasticity is evident in all biological processes … the proliferation of both noise and noise reduction systems is a hallmark of organismal evolution” – Federoff et al.(2002).

“Transcription in higher eukaryotes occurs with a relatively low frequency in biologic time and is regulated in a probabilistic manner” – Hume (2000).

“Gene regulation is a noisy business” – Mcadams et al. (1999).

Stochastic mechanisms

Physiological activity and cell differentiation within a mammalian cell is controlled by more than 10,000 protein coding genes.

Many genes are expressed at low level copy numbers, which gene profiling methods cannot reliably detect.

“Initiation of gene transcription is a discrete process in which individual protein-coding genes in an off state can be stochastically switched on, resulting in sporadic pulses of mRNA production” – Sano 2001.

Internal Noise: stochastic fluctuations in number of proteins. Discrete interacting processes.CTRW describes position x(t)

Waiting time and jump y sampled from Sampled from Gaussian Sampled from Exponential Leads to Bownian motion and diffusive limit is Diffusion Eqn.

External Noise: fluctuations in environment/control parameters.Continuous Wiener processes.

Phenotypic noise: leading to qualitative differences in a cell phenotype (example: lysis-lysogen pathway).

Stablizable noise: leading to fluctuations in protein concentrations (robustness properties of biological systems).

x t+τ =x t +yt

τ ψ τ ϕ y

ψ

ϕ

Brownian Motion

Brown – 1827, Scottish Botanist Einstein – 1905, Einstein relation

Mesh example

PDE, Diffusion

PDF

CLT {Xj} iid, X is Ndist

wi t+Δt = 12

[ wi t +wi+ 1 t ]i-1 i i+1i-1 i i+1

∂ w∂ t

=D∂2 w∂ x 2

w t,x =c0

2π1

2 Dte

− x2

4 Dt

D= lim Δx 2

2Δt

X=∑ x i

ME [ X 2 t ]=2 Dt

D = Diffusion Constant

N(0,2Dt),

Observation:

In a stochastic simulation there is no one right answer. We may compute a single simulation for insight;compute many ensembles and compute statistics;solve a Fokker-Planck Eqn for the pdf.

Modelling Regimes

Discrete and stochastic – Small numbers of molecules. Exact description via Stochastic Simulation Algorithm (SSA) - Gillespie. Large computational time.

Continuous and stochastic - A bridge connecting discrete and continuous models. Described by SDEs – The Chemical Langevin Equation.

Continuous and deterministic – Law of Mass Action. The Reaction Rate equations. Described by ordinary differential equations. Not valid if molecular populations of some critical reactant species are small.

Noise in genetic regulatory networks

The Central Dogma

Genetransription mRNAtraslation protein |____________regulation___________|

For example, biological reactions in gene network

(X: transcriptional factor, D: binding site)

1. Binding reaction

2. Transcription

3. Translation

4. Degradation

X+Dk1 DXDX k2 D+X+mRNAmRNAk3 PPk4

A regulatory model

10 reactions, 6 unknowns. m (monomer protein)

D (dimer transcription factor)

RNA (mRNA produced by transcription)

DNA (free of dimers)

DNA1 (bound by D at binding site R1)

DNA2 (bound at sites R1 and R2). X = (m, D, RNA, DNA, DNA1, DNA2). t_end = 600; X_initial=[2 6 0 2 0 0]';

Transcription to mRNA when D occupies R1; mRNA translated into proteins (both can decay); protein dimerises to transcription factor, D; binding of D at R1 activates transcription of m; binding of D at R2 excludes RNA polymerase from

binding and transcription repressed.

DNA RNA+DNA 1RNA RNA+mm 0RNA 0

D

m+m

DNA+D

DNA1

DNA 1+D

DNA 2

The Hes1 gene

Hes1 oscillates in cultured mouse cells

Hirata et al., Science 298, 840–843 (2002).

Half–lives:

mRNA: ~24 min

protein: ~22 min

… but a non-delayed (ODE) model cannot oscillate…

Hirata et al. predicted extra components in the feedback loop.

hes1 Hes1 mRNA

Hes1 protein–

Eukaryotic transcription and time delays

• There is an irreducible delay of ~15–20 min from initiation of a transcript to appearance of

functional mRNA in the cytoplasm

• The delay can be much longer (>16 hrs for human

dystrophin)

• Delay equations should be used to model transcription

Delay estimates

RNA polymerase moves along DNA at 20 nucleotides/sec; Genome size; 8 minutes for introns to be spliced out; 4 minutes after splicing before mRNA in cytosol; mRNA translated by ribosomes at 6 nucleotides/sec.

Monk model for the Hes1 feedback loop

= 18.5 min

The transcriptional delay has now been observed directly for Hes7Bessho et al. Genes & Dev. 17, 1451 (2003).

dx / dt =- mx t +pg [ y t−t ]dy / dt =- νy t +qx t

xhes1 Hes1 mRNA

Hes1 protein y

g is a hill function with

n 4 – co-operativity

g y =1 / 1 y t−τ / y0n

Thank you