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Introduction• Cells encounter different situations that require

different proteins.

• The cell continuously monitors its environment and calculates the amount at which each type of protein is needed.

• This information-processing function, which determines the rate of production of each protein, is largely carried out by transcription networks.

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Transcription factor

• A transcription factor (TF) is a protein that binds to the promoter region of its target gene and controls the expression (transcription) of the target gene.

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The cognitive problem of the cell

• To represent the different environmental states, the cell uses special proteins called transcription factors as symbols.

• Transcription factors are usually designed to transit rapidly between active and inactive molecular states, at a rate that is modulated by a specific environmental signal (input).

• The bacterium E. coli has an internal representation with about 300 degrees of freedom (transcription factors) and these regulate the rates of production of ~4000 proteins.

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E. coli – a model organism

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The mapping between environmental signals, transcription factors inside the cell, and the genes that they regulate

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Elements of transcription networks

Gene transcription regulation, the basic picture

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Transcription factors --- activators and repressors

activator

repressor

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Activators increase gene production

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Repressors decrease gene production

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Transcription networks

• In the network, nodes are genes and edges represent transcriptional regulation of one gene by the protein product of another gene.

• A directed edge X → Y means that the product of gene X is a transcription factor protein that binds the promoter of gene Y to control the rate at which gene Y is transcribed.

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Backtracking to find active sub-network

• Define differentially expressed genes

• Identify TFs that regulate these genes

• Identify further TFs that regulate these TFs

Active regulatory sub-network

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A transcription network that represents about 20% of the transcription interactions in the bacterium E. coli

Ref: Shen-Orr SS, Milo R, Mangan S, Alon U. Network motifs in the transcriptional regulation network of Escherichia coli. 2002. Nat Genet.2 31(1):64-8.

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Separation of timescales• The input signals usually change transcription factor

activities on a sub-second timescale. Binding of the active transcription factor to its DNA sites often reaches equilibrium in seconds.

• Transcription and translation of the target gene takes minutes, and the accumulation of the protein product can take minutes to hours

• The transcription factor activity levels can be considered to be at steady state within the equations that describe network dynamics on the slow time scale of changes in protein levels.

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Timescales for the reactions in the transcription network of bacterium E. coli

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Input function• The strength of the effect of a transcription factor on the

transcription rate of its target genes is described by an input function.

• When X regulates Y, represented in the network by X → Y, the number of molecules of protein Y produced per unit time is a function of the concentration of X in its active form X*

rate of production Y = f(X*)

• Typically, the input function is a monotonic, S-shaped function. It is a increasing function when X is an activator and a decreasing function when X is a repressor.

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Input function --- Hill functionactivatorfor function Hill )( *

**

nn

n

XKXXf

β: maximal expression level of the promoter; K: activation coefficient

n: Hill coefficient n governs the steepness of the input function

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Input function --- Hill functionrepressorfor function input Hill

)(1)( *

*

n

KX

Xf

K: repression coefficient

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Logic input function: a simple framework for understanding network dynamics

• For mathematical clarity, it is often useful to use even simpler functions that capture the essential behavior of these input function.

• Logic approximation: in this approximation, the gene is either OFF, f(X*) = 0, or maximally ON, f(X*) = β. The threshold for activation is K.

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Logic approximation for activatoractivatorfor ion approximat logic )()( ** KXXf

where θ (step-function) is equal to 0 or 1 according to the logic statement in the parentheses. The logic approximation is equivalent to a very steep Hill function with Hill coefficient n → ∞.

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Logic approximation for repressor

repressorfor ion approximat logic )()( ** KXXf

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**** ),( YXYXf yx

****** AND ~)()(),( YXKYKXYXf yx

****** OR ~) OR (),( YXKYKXYXf yx

Multi-dimensional input functions govern genes with several inputs

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Interim summary

• Input function: the rate of production of gene product Y is a function of the concentration of active transcription factor X*

• The input functions are often rather sharp and can be approximated by Hill functions or logic gates.

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Dynamics and response time of simple gene regulation• X → Y : transcription factor X regulates gene Y.

• The cell produces protein Y at a constant rate, which we will denote β (units of concentration per unit time).

• The degradation rate (αdeg , its specific destruction by specialized proteins in the cell) and the dilution (αdil , the reduction in concentration due to the increase of cell volume during growth), giving a total degradation/dilution rate (in units of 1/time) of αα=αdeg+ αdeg

• The change in the concentration of Y is due to the difference between its production and degradation/dilution, as described by a dynamic equation

dY/dt = β-αY (2.4.2)

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Response time

• The response time, T1/2 , is generally defined as the time to reach halfway between the initial and final levels in a dynamic process.

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Decay of protein concentration following a sudden drop in production rate

• At steady state, Y reaches a constant concentration Yst = β/α

• What happens if we take away the input signal, so that production of Y stops (β=0)? The solution of Equation (2.4.2) with β=0 is an exponential decay of Y concentration: Y(t) = Yst e-αt (2.4.4)

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Rise in protein concentration following a sudden increase in production rate

• dY/dt = β-αY and Y(0)=0 Y(t) = Yst (1 - e-αt)

• At early times, Y～βt when αt << 1

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References

• An Introduction to Systems Biology: Design Principles of Biological Circuits. 2006. Uri Alon.

• Transcription Factors. 2001.J. Locker.