56:198:582 Biological Networks Lecture 8

Transcription

1 56:198:582 Biological Networks Lecture 8

2 Course organization Two complementary approaches to modeling and understanding biological networks Constraint-based modeling (Palsson) System-wide Metabolism Steady-state Network motifs (Alon) Small number of components at a time Gene regulation Dynamic

3 Transcriptional Networks: Basic Concepts

4 Transcriptional regulation... Signal 1 Signal 2 Signal 3 Signal 4 Signal N Environment Transcription factors X1 X2 X3... Xm genes gene 1 gene 2 gene 3 gene 4 gene 5 gene 6... gene k Fig 2.1 The mapping between environmental signals, transcription factors inside the cell and the genes that they regulate. The environmental signals activate specific transcription factor proteins. The transcription factors, when active, bind DNA to change the transcription rate of specific target genes, the rate at which mrna is produced. The mrna is then translated into protein. Hence, transcription factors regulate the rate at which the proteins encoded by the genes are produced. These proteins affect the environment (internal and external). Some proteins are themselves transcription factors, that can activate or repress other genes, etc.

5 Transcriptional regulation promoter Fig 2.2a DNA gene Y protein Y RNA polymerase mrna TRANSLATION TRANSCRIPTION gene Y Fig 2.2a GENE TRANSCRIPTIONAL REGULATION, THE BASIC PICTURE. (a) Each gene is usually preceded by a regulatory DNA region called the promoter. The promoter contains a specific site (DNA sequence) that can bind RNA polymerase (RNAp), a complex of several proteins that forms an enzyme That can synthesize mrna that is complementary to the genes coding sequence. The process of forming the mrna is called transcription. The mrna is then translated into protein.

6 Transcriptional regulation: Activators X Activator X Y Sx X binding site gene Y Y Y Y Y X X* X* INCREASED TRANSCRIPTION Bound activator Fig 2.2 (b) An activator X, is a transcription- factor protein that increases the rate of mrna transcription when it binds the promoter. The activator transits rapidly between active and inactive forms. In its active form, it has a high affinity to a specific site (or sites) on the promoter. The signal Sx increases the probability that X is in its active form X*. Thus, X* binds the promoter of gene Y to increase transcription and production of protein Y. The timescales are typically sub-second for transitions between X and X*, seconds for binding/ unbinding of X to the promoter, minutes for transcription and translation of the protein product, and tens of minutes for the accumulation of the protein,

7 Transcriptional regulation: Repressors Bound repressor X Y Sx X X* NO TRANSCRIPTION X* Unbound repressor X Bound repressor Y Y Y Y Fig 2.2c A repressor X, is a transcription- factor protein that decreases mrna transcription when it binds the promoter. The signal Sx increases the probability that X is in its active form X*. X* binds a specific site in the promoter of gene Y to decrease transcription and production of protein Y. Many genes show a weak (basal) transcription when repressor is bound.

8 Recall the E. coli lac operon The operon (a set of genes transcribed together) consists of three genes: lacz, lacy, and laca, which are used by E. coli to break down lactose Two DNA-binding regulatory proteins: LacI repressor binds to DNA only in the absence of lactose CAP (camp Activator Protein) binds to DNA only in the absence of glucose

9 Elements of transcription networks In this network, nodes are genes and edges represent transcriptional regulation of one gene by the protein product of another gene X Y means that the product of gene X is a transcription factor (TF) protein that binds to the promoter of gene Y The inputs to the network are signals that carry information from the environment. Each signal is a small molecule, protein modification, or molecular partner that directly affects the activity of one of the TFs. Signal S X can cause X to rapidly shift to its active state X The network thus represents a dynamic system: after an input signal arrives, TF activities change, leading to changes in the production rate of proteins

10 Transcription network of E. coli Fig 2.3. A transcription network that represents about 20% of the transcription interactions in the bacterium E. coli. Nodes are operons (groups of genes coded on the same mrna). Some operons encode for transcription factors. Edges directed from node X to node Y indicate that the transcription factor encoded in X regulates operon Y. This network contains data on verified direct transcriptional interactions from many labs, compiled in databases such as regulondb and Ecocyc. Source: Shen-Orr SS, Milo R, Mangan S, Alon U. Network motifs in the transcriptional regulation network of Escherichia coli. Nat Genet May;31(1):64-8.

11 The input function Suppose protein Y is regulated by transcription factor X (X Y ) and that X requires activation to an active form. We have rate of production of Y = f (X ), where X is the concentration of the active form of X The function f is the input function that describes the effect of the transcription factor X on its target gene Y

12 Separation of time scales Transcription networks are designed with a strong separation of timescales: the input signals usually change transcription factor activities on a sub-second timescale Binding of an active TF to its DNA reaches equilibrium in seconds Transcription and translate takes minutes Accumulation of the protein takes many minutes to hours Timescales for E. coli: 1. Binding of signaling molecule to a TF: 1 msec 2. Binding of active TF to its DNA site: 1 sec 3. Transcription and translation of the gene: 5 min 4. 50% change in concentration of translated protein: 1 h Hence 1 and 2 can be considered instantaneous when studying transcription networks

13 Modularity of transcription networks A remarkable property of transcription networks is the modularity of their components. One can take the DNA of a gene from one organism and express it in a different organism For example, one can take the gene for green fluorescent protein (GFP) from jellyfish and introduce it to bacteria. As a result, the bacteria produce GFP, causing the bacteria to turn green Regulation can also be added by adding a promoter region For example, control of GFP in bacteria can be achieved by pasting in front of the gene a DNA fragment from the promoter of another gene, say, one that is controlled by a sugar-inducible transcription factor. This causes E. coli to express GFP and turn green only in the presence of the sugar

14 Modularity of transcription networks Modular components make transcription networks very plastic during evolution and able to readily incorporate new genes and new regulation Transcription networks can evolve rapidly: the edges appear to evolve on a faster timescale than the nodes (coding regions) Related animals, such as mice and humans, have very similar genes, but the transcription regulation of these genes is evidently quite different. Many of the differences between animal species appear to lie in the differences in the edges of the transcription networks, rather than in the differences in the nodes (their genes)

15 The signs on the edges: Activators and repressors Each edge in a transcription network corresponds to an interaction in which a TF directly controls the transcription rate of a gene These interactions can be of two types: 1. Activation or positive control: TF increases the rate of transcription when it binds the promoter 2. Repression or negative control: TF reduces the rate of transcription when it binds the promoter Transcription networks often show comparable numbers of plus and minus edges, with more positive (activation) interactions than negative interactions, e.g., E. coli or yeast have 60 80% activation interactions

16 The signs on the edges: Activators and repressors Typically, TFs act primarily as either activators or repressors (i.e. the signs on the interaction edges out of a node are highly correlated) TFs tend to employ one mode of regulation for most of their target genes In contrast, the signs on the edges that go into a node are less correlated In short, the signs on the outgoing edges are rather correlated, but the signs on the incoming edges are not

17 The input function Typically, the Hill function is used for the input function Hill function for an activator: f (X ) = Hill function for a repressor: f (X ) = βx n K n + X n β 1 + (X /K) n

18 The input function The Hill function has three parameters, K, β, and n K is the activation (repression) coefficient. It has units of concentration. It defines the concentration of active X needed to significantly activate expression. Half-maximal expression is reached when X = K β is the maximal expression level of the promoter n is the Hill coefficient, which determines the steepness of the input function. The larger n is, the more the input function is like a step response. Typical input functions have n in the range of 1 to 4 Many genes have a non-zero minimal expression level called its basal expression level. This is achieved by adding a constant β 0 to the input function

19 Logic input functions It is often useful to approximate the Hill function with a step function Logic approximation for an activator: f (X ) = βθ(x > K), where θ = 0 if X K and θ = 1 if X > K. This approximation is achieved when the Hill coefficient n Logic approximation for a repressor: f (X ) = βθ(x < K)

20 Logic input functions β1 Y promoter activity 0.8 X Y Step function β θ (x>k) n=3 n=2 0.6 n=1 β/2 0.4 Hill function β x n /(K n + x n ) Activator concentration X*/K Figure 2.4a: Input functions for activator X described by Hill functions with Hill coefficient n=1,2 and 4. Black line: step function, also called logical input function. The maximal promoter activity is β, and K is the threshold for activation of a target gene (the concentration of X* needed for 50% maximal activation)..

21 Multi-dimensional input functions Many genes are regulated by multiple TFs multi-dimensional input function Often, these multi-dimensional input functions can be approximated by logic functions Many genes require simultaneous binding of two (or more) TFs in order to show significant expression. This is similar to an AND gate: f (X, Y ) = βθ(x > K)θ(Y > K) For other genes, binding of either activator is sufficient. This resembles an OR Gate: f (X, Y ) = βθ(x > K OR Y > K) Not all genes have Boolean-like input functions. For example, some genes display a SUM input function: f (X, Y ) = β X X + β Y Y

22 Multi-dimensional input functions Fig 2.5: Two-dimensional input functions. (a) Input function measured in the lac-promoter of E. coli, as a function of two inducers camp and IPTG. (b) an AND-gate like input function, which shows high promoter activity Only if both inputs are present. (c) an OR-gate like input function that shows high promoter activity if either input is present. Source: Setty, Y. et al. (2003) Proc. Natl. Acad. Sci. USA 100,

23 Dynamics and response time of simple gene regulation Consider the transcription interaction X Y When the signal S X appears, X rapidly transits to its active form X and binds the promoter of gene Y. Gene Y begins to be transcribed, and the mrna is translated, resulting in accumulation of protein Y. The cell produces protein Y at a constant rate β (units of concentration per unit time) The production of Y is balanced by two processes 1. Protein degradation: its specific destruction by specialized proteins in the cell 2. Protein dilution: the reduction in concentration due to the increase of cell volume during growth Total degradation/dilution rate (units of 1/time): α = α dil + α deg, where α dil is the dilution rate, and α deg is the degradation rate

24 Dynamics and response time of simple gene regulation Change in concentration of Y is described by the dynamic equation dy dt = β αy At steady state, Y reaches a constant concentration Y st. At steady state, 0 = dy dt = β αy st Y st = β α

25 Dynamics and response time of simple gene regulation Suppose we take away the input signal, so β = 0. Hence Y (t) = Y st e αt How fast does Y decay? We measure the speed at which Y levels change using the response time, T 1/2, the time to reach halfway between the initial and final levels Hence Y st 2 = Y ste αt 1/2 T 1/2 = log 2/α

26 Dynamics and response time of simple gene regulation Now consider the opposite case, in which an unstimulated cell with Y = 0 is provided with a signal, so that protein Y begins to accumulate. We now have Y (t) = Y st (1 e αt ) If αt 1, then hence e αt 1 αt, Y βt using Y st = β/α This makes sense: At early times the degradation term is negligible and Y accumulates at its production rate The response time is found using Y st 2 = Y st(1 e αt 1/2 ) T 1/2 = log 2/α

27 time [log(2) t/, cell cycles] Dynamics and response time of simple gene regulation Y/Yst Y/Yst 0.8 R(t) / Rst 0.6 R(t) / Rst time [cell cycles, log(2)/alpha] t / T 1/2 t / T 1/2 Fig 2.6a: Decay in protein concentration following a sudden drop in production rate. Note the response-time, the time it takes the concentration to reach half of its variation, is T 1/2 =log(2)/ α =1 in these units (dashed lines). Fig 2.6b: Rise in protein concentration following a sudden increase in production rate. Note the response time, the time it takes the dynamics to reach half of its variation, is T 1/2 =log(2)/ α =1 in these units (dashed lines). At early times, the protein accumulation is approximately linear with time, Y=β t (dotted line).

28 The response time of stable proteins Many proteins are not actively degraded in growing cells (α deg = 0). These are termed stable proteins The production of stable proteins is balanced by dilution due to the increasing volume of the growing cell, α = α dil. For such stable proteins, the response time is equal to one cell generation time Suppose a cell produces one protein then stops (β = 0) The cell grows and, when it doubles its volume, splits into two cells After one cell generation time τ, the protein concentration has decreased by 50%, hence T 1/2 = log 2/α dil = τ Bacterial cell generation times are on the order of 30 minutes to a few hours, and eukaryotic generation times are even longer. Thus, response time can be a limiting factor that poses a constraint for designing efficient gene circuits

29 The Autoregulation Network Motif

30 Transcription network of E. coli Fig 2.3. A transcription network that represents about 20% of the transcription interactions in the bacterium E. coli. Nodes are operons (groups of genes coded on the same mrna). Some operons encode for transcription factors. Edges directed from node X to node Y indicate that the transcription factor encoded in X regulates operon Y. This network contains data on verified direct transcriptional interactions from many labs, compiled in databases such as regulondb and Ecocyc. Source: Shen-Orr SS, Milo R, Mangan S, Alon U. Network motifs in the transcriptional regulation network of Escherichia coli. Nat Genet May;31(1):64-8. How to understand complexity of this network?

31 Patterns, randomized networks, and network motifs Our approach will be to look for meaningful patterns on the basis of statistical significance. To define statistical significance, we compare the network to an ensemble of randomized networks The randomized networks are networks with the same characteristics as the real network, but where the connections between nodes and edges are made at random. Patterns that occur in the real network significantly more often than in randomized ones are called network motifs The basic idea is that patterns that occur in the real network much more often than in randomized networks must have been preserved over evolutionary timescales against mutations that randomly change edges

32 Patterns, randomized networks, and network motifs To detect network motifs, we need to compare the real network to an ensemble of randomized networks. We will initially consider the ensemble of randomized networks introduced by Erdos and Renyi Suppose the real transcription network has N nodes and E edges. To compare it to the Erdos-Renyi (ER) model, one builds a random network with the same number of nodes and edges In a random network defined by the ER model, directed edges are assigned (uniformly) at random between pairs of nodes There are N 2 possible directed edges (including self-edges) In the ER model, E edges are placed at random in the N 2 possible positions each possible edge position is occupied with probability p = E/N 2

33 Autoregulation: A network motif Now we can begin to compare features of the real E. coli transcription network with the randomized networks. Let us start with self-edges, edges that originate and end at the same node The E. coli network that we use as an example has 40 self-edges. These self-edges correspond to transcription factors that regulate the transcription of their own genes Regulation of a gene by its own gene product is known as autogenous control, or autoregulation. Thirty-four of the autoregulatory proteins in the network are repressors that repress their own transcription: negative autoregulation Is autoregulation significantly more frequent in the real network than at random?

34 Autoregulation: A network motif What is the probability of having k self-edges in an ER random network with N nodes and E edges? The probability a randomly-placed edge is a self-edge is p self = N/N 2 = 1/N If E edges are placed at random, the probability of having k self-edges is approximately binomial: ( ) E P(k) = pself k k (1 p self) E k The expected number of self-edges and its standard deviation is N self rand = E/N, σ rand = E/N

35 Autoregulation: A network motif In the E. coli transcription network, the number of nodes and edges are N = 424 and E = 519 Thus, a corresponding random network with the same N and E would be expected to have only about one self-edge, plus minus one: N self rand 1.2, σ rand In contrast, the real network has 40 self-edges, which exceeds the random networks by many standard deviations The number of standard deviations difference can be calculated by Z = N self real N self rand σ rand 32, which marks a very high statistical significance Why is negative autoregulation a network motif?

36 Negative autoregulation speeds the response time Suppose transcription factor X represses its own transcription The dynamics of X are described by its production rate f (X ) and degradation/dilution rate: dx dt = f (X ) αx We use the decreasing Hill function for f : f (X ) = β 1 + (X /K) n

37 Negative autoregulation speeds the response time For simplicity, we use the logic approximation: f (X ) = βθ(x < K) To study response time, consider the case where X is initially absent and its production starts at t = 0. While X concentration is low, the promoter is not repressed and production is full steam at rate β: dx dt = β αx This results in an approach to a high steady-state value, as previously described. At early times, we can neglect degradation (αx β) to find linear accumulation of X with time: X (t) βt, when X < K and X β/α

38 Negative autoregulation speeds the response time Production stops when X levels reach the self-repression threshold, X = k Thus, X effectively locks into a steady-state level equal to the repression coefficient of its own promoter: X st = K

39 Negative autoregulation speeds the response time The response time T 1/2 can be found by asking when X reaches half steady-state so that X (T 1/2 ) = X st /2 For simplicity, we assume that there is strong autorepression, so X st = K β/α, hence we calculate the response time using linear accumulation of X, in which X = βt So βt (n.a.r.) 1/2 = K 2 T (n.a.r.) 1/2 = K 2β

40 Negative autoregulation speeds the response time In simple gene regulation, the steady state is a balance of production and degradation X st = β simple α simple In contrast, in the negative autoregulation case, X st = K We tune K so that both designs reach the same steady-state expression level: K = β simple α simple

41 Negative autoregulation speeds the response time The response time of simple regulation is So T simple 1/2 = log 2 α simple, while the response of the negative autoregulated circuit is T (n.a.r.) 1/2 T simple 1/2 T (n.a.r.) 1/2 = K 2β = Kα simple 2 log 2β = β simple 2 log 2β, which can be made arbitrarily small by making β large (making the response time of simple regulation fast requires increasing both β simple and α simple, which is wasteful)

42 Negative autoregulation speeds the response time 1.5 X/Xst T 1/2 (nar) T 1/2 (simple) time α t Fig 3.4 Dynamics of negatively auto-regulated gene product (full line) and simply regulated gene product (dashed line) which reach the same steady-state level and have equal degradation/dilution rates α. The response time is the time that the protein level reaches 50% of the steady state, denoted T 1/2 (nar) and T 1/2 (simple) for the negatively auto-regulated and simply regulated gene products. The parameters β=5, α =1, β simple =1 were used.

43 Negative autoregulation promotes robustness In addition to speeding response time, negative autoregulation confers the benefit of increased robustness of the steady-state expression level with respect to fluctuations in the production rate β The production rate of a given gene β fluctuates over time due to overall fluctuations in the metabolic capability of the cell and its regulatory systems and to stochastic effects in the production of the protein Twin cells usually show differences in the production rates β of most proteins, which are typically on the order of a few percent to tens of percents Parameters such as the repression threshold K vary much less from cell to cell Simple gene regulation is affected quite strongly by fluctuations in the production rate β (X st = β/α), but not negative autoregulation (X st = K)

44 What about positive autoregulation? X / Xst 1 Negative Auto-regulation 0.8 Simple regulation 0.6 Positive Auto-regulation cell-generations [log(2)/α ] Fig 3.5: Dynamics of negatively auto-regulated gene, a simply regulated gene and a positively auto-regulated gene. The negatively and positively auto-regulated genes have a Hill-input function with Hill coefficient n=1. Shown is protein concentration normalized by its steady-state value,x/xst, following an increase in production rate. Time is in cell-generations, or for actively degraded proteins,log(2)/α, where alpha is the protein degradation/dilution rate. Note that the response-time is T 1/2 =log(2) / α =1 for simple regulation, T 1/2 =0.21 for negative auto-regulation, and T 1/2 ~2 for positive auto-regulation with the present parameters. The response-time is constructed by the intersect of the dynamics with horizontal line at X/Xst=0.5. Positive autoregulation has an effect that is opposite to that of negative autoregulation. The former slows response times, whereas the latter speeds response times. The slow dynamics provided by positive autoregulation can be useful in processes that take a relatively long time, such as developmental processes

45 What about positive autoregulation? In addition, a positive autoregulation system can become bi-stable. Once the gene is activated, it is locked into a state of high expression and keeps itself ON, even after the original activation input has vanished memory