Transcriptional Regulatory Networks 01/12/12

Transcription

1 Transcriptional Regulatory Networks 01/12/12

2 Molecular interaction networks change constantly All the phenomena of life are the manifestation of the interactions of molecules that constitute the organism. Organism Cell All possible molecular interactions However, such molecular interactions are not fixed or static, rather, they are in constant change according to the environmental fluctuations, so that the organism s survivability is maximized.

3 Transcriptional regulatory networks are the information processing devices of the cell Changes in molecular interactions are mainly carried out through the alteration of the concentration of involved gene products---protein protein or RNA molecules; The most important factor that determine the concentration of a protein is the rate at which it is produced---the rates of its transcription and translation----in particular, the rate of transcription; Therefore, the cell has to constantly monitor its environment and determine the rate at which each gene product should be produced; Transcriptional regulatory networks are the information processing devices in the cell that determine the rate of production of each gene product according to the cell s needs.

4 The communication between transcriptional regulatory networks and environmental changes External stimuli Internal stimuli External stimuli: Temperature Osmotic pressure Signal from other cells Module 1 Module 2 Module i Module n Nutrients Toxic chemicals Internal stimuli: Levels of metabolites Energy supply DNA damage Membrane damage Activated TFs Response networks Proteins and RNAs Regulon 1 Regulon 2 Regulon i Regulon n Transcriptional regulatory networks

5 Environmental changes are reflected in the activities of transcription factors in the cell The active concentration of the transcription factors in the cell: R ([T* 1 ], [T* 2 ],, [T* i ],, [T* n ]), can be considered as the internal representation ti of the environmental factors. Environmental factors F 1 F 2 F 3 F N Signals S 1 S 2 S 3 S n Activated TFs T* 1 T* 2 T* m Genes g 1 g 2 g 3 g 4 g 5 g 6 g 7 g 8 g 9 g k

6 The elements of transcriptional regulatory networks 1. A transcription factor determines what environmental factor is reflected; 2. A target gene determines what cellular effect to be produced; 3. cis-regulatory elements (TF binding sites (TFBSs)) determine the efficiency and the type of regulation on the target gene. Transcription factor Activator + Gene cis-regulatory element Transcription factor Repressor - Gene cis-regulatory element

7 A TF can increase the rate of transcription through its interaction with RNAP: activator Promoter DNA TF X binding site RNAP binding site Gene S x activator X X* DNA S x X* RNAP Gene mrna

8 A TF can decrease the rate of transcription through its Interaction with RNAP: repressor S x repressor X X* S x X* No transcription DNA RNAP binding site TF X binding site Gene DNA RNAP Gene mrna

9 Graphic representation of transcriptional regulatory networks All the transcription regulation relationship in the cell forms a transcription network, which can be represented by a digraph; That the products of gene X is a TF that binds the promoter of gene to control the rate at which gene is transcribed is represented by X ; A sign or numbers can be added to the edges to reflect the type and the efficiency of such regulation: Activator X X Repressor Z Z K X + K Z - Gene Gene

10 A partial transcription network in E. coli in graph representation Global transcription factors Local transcription i factors Genes Curr Opin Microbiol Oct;6(5):482-9.

11 More about activator and repressor in E. coli The following conclusions can be drawn through studying this partial transcription network in E. coli K12: 1. It seems that there are more positive transcriptional regulations than negative regulations in a cell; 2. A TF can be an exclusive activator or an exclusive repressor; 3. A TF can also be a activator for some genes and a repressor for some other genes; but it usually dominates as either an activator or a repressor. Such a TF is called a duoregulator; 4. A gene can be regulated by multiple TFs, which can be either an activator or a repressor. On average, a gene is regulated by two TFs.

12 Transcriptional regulatory networks are designed by Nature with a strong separation of timescales The different steps from an environmental change to the accumulation of the protein have very different timescales. Environmental factor F Signal transduction sec~min pathways S x TF DNA ms X X* S x sec X* RNAP hr Gene min min mrna

13 Abstraction and simplification for modeling a transcriptional regulatory network This strong separation of timescales can be used to simplify the formulation of modeling of a transcriptional regulatory network; For example, when we study the dynamics of the production of a protein, the following factors can be considered to be in a steady state to simplify the problem: 1. The level of the signal S x that activate X; 2. The level of the TF X* that control the rate of s s transcription; 3. The levels of other resources for making. The details (time course) from the environmental change to the steady state activation of TF can also be neglected; With these simplifications the rate of protein production With these simplifications, the rate of protein production controlled by a TF can be unified in a rather simple mathematical description.

14 Input function of a transcriptional regulatory regulation The strength of the effect of a TF on the transcription rate of its target gene is described by an input function. Given the transcriptional regulation relationship X, the number of molecules of protein produced per unit time is called the rate of production of, and is given by d = f ( X * ) dt where X* is the concentration of X in its active form. This equation is called the dynamics equation of. f (X*) can be precisely described by the Hill equation, which we will derive later based on the mass action law, and equilibrium theory in chemistry. k on k nx + E EX n k off

15 The Hill equation for activator When X is an activator, then the Hill equation is in the form: f ( X *) = k βx n + * X n * n n β ( X * / k) = 1+ ( X * / k ) where, β is the maximal rate of the expression of, which is reached when X* >> k, thus, all the promoters are in full speed to produce mrna. k is a constant called activation coefficient, and has units of concentration. 1 X * = k, when f ( X *) = β. 2 n is called the Hill coefficient, which governs the steepness of the input function. The larger n is, the more step-like the input function. n typically takes 1~4. n,

16 Hill functions for activator X with n=1,2 and 4. β1 X prom moter act tivity β/ Step function n=3 β θ (x>k) n=2 Hill function β x n /(K n + x n ) n= Activator concentration X*/K Figure 2.4a from ur Alon:

17 The Hill equation for repressor When X is a repressor, then the Hill equation is in the form: f ( X *) = β * X 1+ ( K where, β is the maximal rate of the expression of, which is reached when X* = 0. k is the repression coefficient, 1 X * = k when f ( X *) = β. 2 n is called the Hill coefficient, which governs the steepness of the input function. The larger n is, the more step-like the input function. n typically takes 1-4. ) n,

18 Hill functions for repressor X with n=1,2 and 4. β 1 X 0.8 n=1 n=2 n=4 Step function β θ (x<k) prom moter act tivity β/ Hill function β K n /(K n + x n ) Repressor concentration X*/K Figure 2.4b in ur Alon:

19 The numbers on the edges of a transcriptional regulatory network graph If a Hill equation is used to describe the regulation of a gene by a TF, then such regulation can be described by a three-element vector (β,k,n). kn) Each edge in the transcriptional regulatory network digraph can be thought to carry such a vector plus a sign indicating the type of regulation. Activator X X Repressor Z Z K X =(β,k,n,+) K Z =(β,k,n,-) Gene Gene Each element of this vector is subject to evolutionary selection.

20 Logic approximation of single-input function Hill equations are useful for detailed modeling, e.g., the change in the production rate of with the change in X*; However, when such details are not necessary, e.g., we assume X* is constant, it is often useful to use even simpler functions that capture the essential behavior of these input functions; In the logic approximation, the input function can be given by f ( X *) = βθ ( X *), where β is the maximal production rate and θ is a step function: θ ( X * > k) θ ( X * < k) 1 = 0 0 = 1 X* > X* X* X* < k k k k if if X is an activator, X is a repressor.

21 Logic input function Thus, logic input functions are step-like approximation for the smoother Hill functions, which is equivalent to a very steep Hill functions with Hill coefficient n. prom moter activi ity X Step function β θ (x>k) n=3 n=2 n=1 Hill function β x n /(K n + x n ) n=2 n=4 prom moter activ vity Step function β θ (x<k) n=1 0.6 β/2 Hill function β K n /(K n + x n ) X Activator concentration X*/K Repressor concentration X*/K Using logic approximation, dynamics functions can be easily solved by graphic methods.

22 Multi-dimensional input functions When a gene s transcription is controlled by more than one TF, the rate of transcription of the gene is a multi-dimensional input function of different transcription factors, d = dt f ( X *, X * 2,..., X * 1 n ) f ( X *, X * 2,..., X * ) 1 n can be described by a multi-dimensional Hill equation, f ( X * 1, X * 2,..., X * n ) = i 1 β ( X i + i ( X * * i i / k i / k i ) ) n i m i.

23 Multi-dimensional logic input functions Often, multi-dimensional input function can be more conveniently approximately by logic functions just as in the case of the single factor input function. Let s consider a gene Z regulated by two TFs, X and : 1. If both X and are required for the transcription of Z, this is similar il to an AND gate: f ( X *, *) = βθ ( X * > k ) θ ( * k ) ~ X X > * AND *. 2. If either X or is required for the transcription of, this is similar to an OR gate: f ( X *, *) = βθ ( X * > k OR * k ) ~ X X > *OR *. 3. If X and have additive result, then a SUM function is used, f ( X *, *) = β Xθ ( X * > k X ) + βθ ( * > k ).

24 Two-dimensional input functions An input function measured in the lacpromoter of E. coli, as a function of two inducers camp and IPTG, has two plateaus---a SUM like function. An AND-gate like input function An OR-gate like input function Setty,. et al. (2003) PNAS. USA 100,