56:198:582 Biological Networks Lecture 9

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1 56:198:582 Biological Networks Lecture 9

2 The Feed-Forward Loop Network Motif

3 Subgraphs in random networks We have discussed the simplest network motif, self-regulation, a pattern with one node We now consider larger patterns of nodes and edges, called subgraphs Two examples:

4 Subgraphs in random networks In total, there are 13 possible ways to connect three nodes with directed edges There are 199 possible four-node subgraphs, and 9364 five-node subgraphs

5 Subgraphs in random networks To find which subgraphs are significant, we need to compare the subgraphs in the real network to those in randomized networks We calculate the number of times a given subgraph G appears in a random Erdos-Renyi (ER) network Suppose G has n nodes and g edges For the feed-forward loop, for example, n = 3 and g = 3 There are N 2 possible directed edges, so the probability that a given edge is present is p = E/N 2

6 Subgraphs in random networks Most biological networks are sparse, i.e. p 1 For example, in the E. coli network, there are about 400 nodes and 500 edges, so p One reason that biological networks are sparse is that each interaction in the network is selected by evolution against mutations that would rapidly abolish the interaction. Thus, only useful interactions are maintained

7 Subgraphs in random networks The average number of occurrences of subgraph G in the network is approximately given by N G Nn p g, a where a is a combinatorial factor related to the structure and symmetry of each subgraph For the FFL, a = 1, and for the three-node feedback loop, a = 3

8 Subgraphs in random networks We recast this equation in terms of the mean connectivity of the network, give by λ = E/N Hence p = λ/n, so N G λg N n g a

9 Subgraphs in random networks Many patterns G are very rare in random networks, in the sense that they occur in vanishingly small numbers in large random networks. If any of these patterns occur in the real biological network, they are likely to be network motifs The dependence of N G on network size N is described by a scaling relation, which describes the way that the number of subgraphs depends on the size of the network ignoring prefactors: N G N n g The scaling of subgraph numbers in ER networks depends only on the difference between the number of nodes and edges in the subgraph, n g

10 Subgraphs in random networks For example, V-shaped patterns, such as subgraphs 1 and 2, have n = 3 nodes and g = 2 edges, and their number grows linearly with network size: N V shaped N n g = N If we double the number of nodes and edges in the random network, the number of V-shaped subgraphs will also double. These patterns are very common in random networks In contrast, the fully connected clique (subgraph 13) has g = 6 edges and n = 3 nodes, so the expected number of such subgraphs scales as N n g = N 3, and it therefore occurs very rarely in large random networks

11 Subgraphs in random networks Consider now the three-node feedback loop and feedforward loops The scaling of the expected number of these subgraphs with the network size goes as N n g = N 0. The numbers of these triangle patterns are constant in ER networks are do not increase with network size Triangles and more complex patterns occur rarely in random networks

12 The feed-forward loop is a network motif In the E. coli transcription network, there are 42 feed-forward loops and no feedback loops made of a cycle of three nodes In the corresponding randomized ER networks with the same mean connectivity λ = 500/ , the mean number of feed-forward loops is only about 2 and the mean number of feedback loops is less than 1: N FFL λ3 N 0 a N feedback λ3 N 0 a

13 The feed-forward loop is a network motif From randomly-generated networks, we obtain: Feed-Forward Loop Feedback Loop with 3 (FFL) nodes E. coli 42 0 ER random nets 1.7 ± 1.3 (Z = 31) 0.6 ± 0.8 Degree-preserving random nets 7 ± 5 (Z = 7) 0.2 ± 0.6 The feed-forward loop (FFL) is a strong network motif The three-node feedback loop is not a network motif In sensory transcription networks such as those of E. coli and yeast as well as higher organisms, the feed-forward loop is the only significant motif of the 13 possible three-node patterns

14 The feed-forward loop is a network motif Blue nodes= x y z FFL Fig 4.2 Feed-forward loops in the E. coli transcription network. Blue nodes participate in FFLs.

15 The feed-forward loop The feed-forward loop is composed of transcription factor X that regulates a second transcription factor, Y, and both X and Y regulate Z Two parallel regulation paths: 1. direct path from X to Z; single edge 2. indirect path through Y ; cascade of two edges Each of the three edges can correspond to activation or repression. Hence there are 2 3 = 8 possible types of FFLs

16 The feed-forward loop COHERENT FFL INCOHERENT FFL Fig 4.3 The 8 sign combinations (types) of feed-forward loops. Arrows denote activation and -- symbols denote repression.

17 The feed-forward loop The eight FFL types can be classified into two groups: coherent and incoherent The overall sign of a path is given by the multiplication of the sign of each arrow on the path In coherent FFLs, the indirect path has the same overall sign as the direct path In incoherent FFLs, the sign of the indirect path is opposite to the direct path

18 Frequency of FFL types The most abundant FFL is the type-1 coherent FFL (C1-FFL) The second most abundant type of FFL is the incoherent type-1 FFL (I1-FFL) The six other types appear much less frequently

19 Frequency of FFL types Fig 4.4 Relative number of the eight FFL types in the transcription network of yeast and E. coli. FFL types are marked C and I for coherent and incoherent. The E. coli network is based on the Ecocyc and Regulon DB databases (about twice as many edges as in the network of Fig 2.3).

20 The feed-forward loop We must also know how the inputs from the two regulators X and Y are integrated at the promoter of gene Z, i.e. the input function of gene Z We will consider two biological reasonable logic functions: AND logic and OR logic There are eight types of FFL sign combinations, each of which can appear with at least two types of input functions There are also two input signals to the circuit, S x and S y, which can be molecules that directly bind the transcription factors or modifications of the transcription factor caused by signal transduction pathways caused by external stimuli

21 The feed-forward loop Sx X Y Sy AND Z Fig 4.5a The coherent type-1 FFL with an AND input function: Transcription factor X activates the gene encoding transcription factor Y, and both X and Y jointly activate gene Z. The two input signals are Sx and Sy. An input-function integrates the effects of X and Y at the Z promoter ( an AND-gate in this figure).

22 The coherent type-1 FFL with AND logic S x X X* K xy X* X* Y* gene X gene Y gene Z Sy β y K xz Y Y* K yz β z Z Fig 4.6 The molecular interactions in the type-1 coherent FFL. The protein X is activated by signal Sx, which causes it to assume the active conformation X*. It then binds its sites in the promoters of genes Y and Z. As a result, protein Y accumulates, and, in the presence of its signal Sy, is active Y*. When Y* concentration crosses the activation threshold Kyz, Y* binds the promoter of gene Z. Protein Z is produced when both X* and Y* bind the promoter of gene Z (an AND input function).

23 The coherent type-1 FFL with AND logic Suppose that, at time t = 0, a strong signal S x triggers the activation of X The transcription factor X rapidly transits to active form X The active protein X binds the promoter of gene Y, initiating production of protein Y In parallel, X also binds the promoter of Z Since the input function at the promoter of Z is AND logic, X alone cannot active Z production

24 The coherent type-1 FFL with AND logic Production of Z requires binding of both X and Y The concentration of Y must build up to sufficient levels to cross the activation threshold for gene Z (K yz ) In addition, the second input signal S y must be present so that Y is in its active form Y Once S x appears, Y needs to accumulate in order to activate Z a delay in Z production

25 The coherent type-1 FFL with AND logic We describe the FFL using logic input functions The production rate of Y is given by production rate of Y = β y θ(x > K xy ) Thus dy dt = β y θ(x > K xy ) α y Y

26 The coherent type-1 FFL with AND logic The production of Z (governed by an AND gate input function) can be described by a product of two step functions: production of Z = β z θ(x > K xz )θ(y > K yz ) In the case of a strong step-like stimulation, X rapidly crosses the thresholds K xy and K xz The delay in the production of Z is due to the time it takes Y to accumulate and cross its threshold K yz Only after Y cross the threshold can Z production proceed at rate β z : dz dt = β zθ(x > K xz )θ(y > K yz ) α z Z

27 The C1-FFL is a sign-sensitive delay element We will consider ON and OFF steps in S x while assuming that S y is present, so the transcription factor Y is in its active form, i.e. Y = Y Following an ON step, Y begins to be produced at rate β y, so Y (t) = Y st (1 e αy t ) Recall that Y st = β y /α y

28 The C1-FFL is a sign-sensitive delay element Production of Z begins only after Y accumulates and crosses the activation threshold K yz T ON is the time needed for Y to reach its threshold. It can be found using Solving for T ON yields Y (T ON ) = Y st (1 e αy T ON ) = K yz T ON = 1 1 log α y 1 K yz /Y st

29 The C1-FFL is a sign-sensitive delay element X* Y* K yz Z T ON Fi g 4.7: Dynamics of the coherent type-1 FFL with AND logic following an ON step of Sx at time t=0. The activation threshold of Z by Y is K yz (dashed line). The production and degradation rates are α y =α z =1, β y =β z =1. The delay in Z production is T ON.

30 The C1-FFL is a sign-sensitive delay element α y T ON K yz / Y st Fig 4.8a: Delay in Z production the C1-FFL following an ON step of inducer S x as a function of the biochemical parameters of the transcription factor Y.. The delay T ON, made dimensionless by multiplying with the degradation/dilution rate of Y, α y, is shown as a function of the ratio of the activation threshold K yz and the maximal (steady-state) level of Y, denoted Y st.

31 The C1-FFL is a sign-sensitive delay element Recall that Y st is prone to cell-cell fluctuations owing to variations in protein production rates A robust design will have a threshold K yz significantly lower than Y st to avoid such fluctuations In bacteria, K yz is typically 3 to 10 times lower than Y st, and typical parameters give delays T ON that range from a few minutes to a few hours

32 The C1-FFL is a sign-sensitive delay element Following an OFF step, X rapidly becomes inactive and unbinds form the promoters of Y and Z Since Z is governed by an AND gate, Z production stops immediately no delay in Z dynamics after an OFF step

33 The C1-FFL is a sign-sensitive delay element X* Y* Z Fi g 4.8b: Dynamics of the C1-FFL following an OFF step of Sx at time t=0. All production and degradation rates are equal to 1. Time

34 The C1-FFL is a sign-sensitive delay element Sign-sensitive delay: the delay depends on the sign of the step, ON or OFF A sign-sensitive delay element can also be considered as a kind of asymmetric filter Only persistence pulses (longer than T ON ) result in Z expression This type of FFL is a persistence detector for ON pulses

35 The C1-FFL is a sign-sensitive delay element Sx Fig 4.8c: The coherent type-1 FFL with AND-logic as a persistence detector. A brief pulse of signal S x does not give Y enough time to accumulate and cross its activation threshold for Z. Hence Z is not expressed. A persistent pulse yields Z expression at a delay. Z expression stops with no delay when Sx is removed. Source: Shen-Orr 2002 Nature genetics.

36 The C1-FFL is a sign-sensitive delay element In engineering, a sign-sensitive delay is commonly used in situations where the cost of an error in not symmetric For example, the beam of light used to sense obstructions in an elevator door Only if you remove your hand for a persistent length of time do the doors close Put your hand in the doors, and they open immediately The cost of an error is asymmetric The environment of cells is often highly fluctuating, and sometimes stimuli can be present for brief periods that should not elicit a response

37 The arabinose system of E. coli The arabinose system is activated only in the arabinose AND NOT glucose environmental condition Absence of glucose is symbolized within the cell by the production of camp The regulators are connected in a C1-FFL with an AND input function, where S x = camp and S y = arabinose The observed delay in the arabinose FFL is on the same order of magnitude as the duration of spurious pulses in the input signal S x, which occur when E. coli transits between different growth conditions The circuit responds only to persistent stimuli, such as persistent periods of glucose starvation that require utilization of the sugar arabinose

38 The arabinose system of E. coli a) =camp =arabinose The arabinose system, FFL 1 =camp =allolactose The lactose system, simple regulation =arabad, =laczya b) arafgh ON step of S x OFF step pulse of S x Simple, FFL Simple FFL Fig 4.9 Experimental dynamics of the C1-FFL in the arabinose system of E. coli. The arabinose (ara) system encodes enzymes that utilize the sugar arabinose (arabad), and transport it into the cell (arafgh). It is activated upon glucose starvation by the activator X=CRP (signal S x=camp, a molecule produced upon glucose starvation), and in the presence of Sy=arabinose by the activator Y=araC. The input function is an AND-gate. As a control system with no FFL (simple regulation), the experiment used the lac operon, in which same activator X=CRP regulates the lactose operon, but X does not regulate Y 1=LacI. Dashed arrows: rapid, non-transcriptional positive and negative feedback loops in which the output gene products affect the signal (eg by transporting the sugar S y into the cell and degrading it). b) Dynamics of the promoter activity of the ara and the lac operons was monitored at high temporal resolution in growing cells by means of reporter-gfp fusions, in the presence of S y. The experiments followed The dynamics after ON steps and OFF steps of S x. Shown is GFP per cell normalized to its maximal level. Based on Mangan et al JMB 2003.

39 The C1-FFL with OR logic With an OR gate, Z is activated immediately upon an ON step in S x But Z is deactivated at a delay following an OFF step because both inputs need to go off for the OR gate to be inactivated

40 The incoherent type-1 FFL S x X Y S y K yz AND Z Fig 4.10a: The incoherent type-1 FFL with an AND gate at the Z promoter. The inputs are the inducers S x and S y. The repression threshold of gene Z by repressor Y is K yz.

41 The incoherent type-1 FFL Y* X* β Strong transcription X* Y* β Weak (basal) transcription also termed leakiness Fig 4.10b: The four binding states of a simple model for the promoter region of Z, regulated by activator X and repressor Y. Transcription occurs when the activator X* is bound, and to a much lesser extent when both activator and repressor Y* are bound.

42 The incoherent type-1 FFL Consider the response to a step addition of the signal S x in the presence of S y Since S y is present, Hence dy dt = β y α y Y Y (t) = Y st (1 e αy t ) In addition, Z begins to be produced at a rapid rate β z since its promoter is occupied by the activator X but there is not yet enough repressor Y, so and where Z m = β z /α z dz dt = β z α z Z Z(t) = Z m (1 e αz t ),

43 The incoherent type-1 FFL The fast production of Z lasts until Y reaches K yz, which is at time T rep = 1 1 log α y 1 K yz /Y st At times after T rep, the production rate of Z is reduced to β z, and the Z concentration decays exponentially to a new lower steady-state Z st = β z/α z : Z(t) = Z st + (Z 0 Z st )e αz (t Trep), where Z 0 is the level reached at time T rep by the original production Z 0 = Z m (1 e αtrep )

44 The incoherent type-1 FFL Let F be the repression factor defined as the ratio of the maximal and basal activity of the Z promoter: F = β z /β z = Z m /Z st When F 1, Z dynamics show a pulse-like shape The I1-FFL can act as a pulse generator

45 The incoherent type-1 FFL S x Y* K yz Z Z st Time Fig 4.11a: Dynamics of the I1-FFL with AND input function following an ON-step of S x. The step occurs at t=0, and X rapidly transits to its active form X*. As a Result, the repressor protein Y is produced, and represses Z production when it crosses the repression threshold K yz. In this figure, all production and decay rates are equal to 1.

46 The incoherent type-1 FFL T rep Fig 4.11b: Expression dynamics of Z in an incoherent type-1 FFL with repression coefficients F=2,5,20. The repression coefficient is the ratio of the maximal expression without active repressor to the steady-state expression with active repressor. T rep is the time when repression begins, and is the moment of maximal Z concentration.

47 The incoherent type-1 FFL The I1-FFL can accelerate response time Half steady-state is reached during the initial fast stage of Z production, before Y crosses its repression threshold The response time T 1/2 is given by solving which gives Z 1/2 = Z st /2 = Z m (1 e αz T 1/2 ), T 1/2 = 1 α z log 2F 2F 1 The response time becomes very small when F 1, approaching T 1/2 = 0 (within physical limits of translation and transcription delay and maximal production rate)

48 The incoherent type-1 FFL Z / Z st 1.5 I1-FFL Simple regulation o T 1/2 I1-FFL T 1/2 (simple reg.) Figure 4.12a: Response time of the I1-FFL is shorter than simple regulation that reaches same steady-state level. The normalized response time of simple regulation is log(2)~0.7. Simple regulation- dashed lines, I1-FFL- full lines)

49 The incoherent type-1 FFL The I1-FFL is a sign-sensitive response accelerator The response after the signal S x is removed occurs with the same dynamics as for a simply regulated gene

50 Three ways to speed response time So far encountered three ways to speed response time 1. Increased degradation rate: Production rate needs to be increased to maintain a given steady state; increased rate applies both to turn-on and turn-off gene expression 2. Negative autoregulation: Only turn-on is speeded, turn-off is not; the strategy works only for transcription factor proteins 3. Incoherent FFL: This can speed up ON responses and can be used for any target protein, not only transcription factors Designs 2 and 3 can work together with 1: a large degradation rate can further speed the response

51 Why are some FFL types rare? The I4-FFL has the same dynamic behavior as the I1-FFL, but has different steady-state logic Fig 4.14: The incoherent type-1 FFL and type-4 FFL I1-FFL X I4-FFL X Y Y AND AND Z Z

52 Why are some FFL types rare? S x S y Z st in I1-FFL Z st in I4-FFL (β z /α z ) 0 (β z/α z ) (β z/alpha z ) 0 (β z/α z ) S y does not affect the steady-state level of Z in I4-FFL This may explain why I4-FFL is selected less often than I1-FFL

53 Why are some FFL types rare? Fig 4.13: The effect of input signal S y on the dynamics of the I1-FFL. When Sy is absent, Y is not active as a repressor, and the concentration of protein Z shows an increase to a high unrepressed steady-state (dashed line) S x Y Z 1 S y absent 0.5 S y present