Random Boolean Networks

Random Boolean Networks

Boolean network definition The first Boolean networks were proposed by Stuart A. Kauffman in 1969, as random models of genetic regulatory networks (Kauffman 1969, 1993). A Random Boolean network (RBN) is a system of N binary-state nodes (genes) with K inputs to each node representing regulatory mechanisms. The two states (on/off) represent a gene being active or inactive. The variable K is typically held constant, but it can also be varied. In the simplest case each gene is assigned, at random, K regulatory inputs from among the N genes, and one of the possible Boolean functions of K inputs. Thus the state space of any such network is 2 N. Simulation of RBNs is done in discrete time steps. The state of a node at time t+1 is a function of the state of its input nodes and the Boolean function associated with it. The behavior of specific RBNs and generalized classes of them (ensembles) has been the subject of much of Kauffman's research. 2 CBS, Department of Systems Biology

Random Boolean Network (RBN) Representations Figure 2. A three-node RBN with the logical updating functions of its nodes (left) and the corresponding state space (right). Nodes in the state space with self-loops are the network s attractors; connected sets of nodes are basins of attraction. Geard & Willadsen, Birth Defects Research (Part C) 87:131 142 (2009) 3 CBS, Department of Systems Biology

Why RBNs? Simplicity of model Deterministic Can be explored comprehensively and exhaustively for small systems Rules can be associated with network dynamics 4 CBS, Department of Systems Biology

Circuit logic AND : all input must be on/high NAND : Not AND, one or more are off OR : one or more are on NOR : Not OR, all input must be off XOR/EX-OR : Exclusive OR, either but not both XNOR/EX-NOR : Exclusive NOR 5 CBS, Department of Systems Biology

Summary of logic functions Inputs Outputs X Y AND NAND OR NOR EX-OR EX-NOR 0 0 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 1 6 CBS, Department of Systems biology Presentation name 17/04/2008

Three input systems (non-canonical) Inputs Output A B C AND NAND OR NOR 0 0 0 0 1 0 1 0 0 1 0 1 1 0 0 1 0 0 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1 0 1 0 1 0 1 1 0 1 1 0 0 1 1 0 1 1 1 1 0 1 0 7 CBS, Department of Systems biology Presentation name 17/04/2008

RBN (discrete) dynamics Boolean network dynamics usually represented on a transition graph State space: the 2 N possible activity states, N-dimensional hyper-cube Attractor: subset of the state space where transition graph frequently leads to, Basin of attraction: set of states that lead to a particular attractor 8 CBS, Department of Systems Biology

Example, N=10, k=2, OR Regulatory Inputs [1] 1 0 0 0 0 0 0 1 0 0 [2] 0 0 0 0 1 1 0 0 0 0 [3] 1 0 1 0 0 0 0 0 0 0 [4] 1 0 0 0 1 0 0 0 0 0 [5] 0 0 1 1 0 0 0 0 0 0 [6] 1 0 0 0 0 1 0 0 0 0 [7] 0 0 0 0 1 0 1 0 0 0 [8] 0 0 0 0 0 0 1 0 0 1 [9] 1 0 0 0 0 0 0 0 1 0 [10] 0 0 1 0 0 0 0 0 1 0 [Time] States [0] 0 0 0 0 0 1 0 0 0 1 [1] 0 1 0 0 0 1 0 1 0 0 [2] 1 1 0 0 0 1 0 0 0 0 [3] 1 1 1 1 0 1 0 0 1 0 [4] 1 1 1 1 1 1 0 0 1 1 [5] 1 1 1 1 1 1 1 1 1 1 [6] 1 1 1 1 1 1 1 1 1 1 9 CBS, Department of Systems Biology

genes time steps RBN Dynamics genes [Time] Gene States [0] 0 0 0 0 0 1 0 0 0 1 [1] 0 1 0 0 0 1 0 1 0 0 [2] 1 1 0 0 0 1 0 0 0 0 [3] 1 1 1 1 0 1 0 0 1 0 [4] 1 1 1 1 1 1 0 0 1 1 [5] 1 1 1 1 1 1 1 1 1 1 [6] 1 1 1 1 1 1 1 1 1 1 time steps time steps time steps Genes genes genes genes 10 CBS, Department of Systems Biology time steps

Example synchronous updates genes Black indicates the gene is ON, white : OFF time steps genes N=100 Rows are time-steps Each column is a gene time steps K=3 Steps=30 90% OR fcn genes time steps 11 CBS, Department of Systems Biology

Example asynchronous updates genes time steps time steps genes N=100 K=3 Steps=100 90% OR fcn genes time steps 12 CBS, Department of Systems Biology

State Space Representation of an RBN Figure 2. A three-node RBN with the logical updating functions of its nodes (left) and the corresponding state space (right). Nodes in the state space with self-loops are the network s attractors; connected sets of nodes are basins of attraction. Geard & Willadsen, Birth Defects Research (Part C) 87:131 142 (2009) 13 CBS, Department of Systems Biology

Transition graphs and Attractors (discrete systems) Each node in a transition graph is a specific state, i.e. a defined expression state represented by 0 s or 1 s E.g. x=(0,1,1,1,0,1,0,0,0,1,0,0,1,0) Single state attractor: fixed point == point attractor == steady state Multiple state attractor: limit cycle, unstable state 14 CBS, Department of Systems Biology

Basin of Attraction for an RBN (a) (b) Fig. 6. Geometric representation of the structure of the configuration space of Boolean networks with scale-free topology. Each bold point corresponds to a dynamical configuration of the system. A link between any two points means that one point is the precursor of the other one. The arrows indicate the direction of the dynamical flow. The cases shown correspond to three different values of the scale-free exponent for networks with N = 10 elements (Ω = 1024 configurations). (a) Ordered phase: there is only one attractor consisting of two points. Note that the vast majority of points (99.22%) do not have precursors. (b) Critical phase: there are two attractors, each consisting of only one point. (c) Chaotic phase: there are five attractors of different lengths. In this case approximately 10% of the points have precursors. M. Aldana/Physica D 185 (2003) 45 66 15 CBS, Department of Systems Biology

Example Attractors 16 CBS, Department of Systems Biology DDLab-gallery of Andrew Wuensche

Chapter 4 Feed-Forward Loops

Network Motifs with 3 Nodes Fig. 1. (A) Examples of interactions represented by directed edges between nodes in some of the networks used for the present study. These networks go from the scale of biomolecules (transcription factor protein X binds regulatory DNA regions of a gene to regulate the production rate of protein Y), through cells (neuron Xissynapticallyconnected to neuron Y), to organisms (X feeds on Y). (B) All13typesofthree-nodeconnectedsubgraphs. Milo R, et al. Science 2002 18 CBS, Department of Systems Biology

Structures of FFL Circuit a Coherent FFL Coherent type 1 Coherent type 2 Coherent type 3 Coherent type 4 b S X c S X X X X X X X S Y S Y Y Y Y Y Y Y Z Z Z Z AND AND Incoherent FFL Incoherent type 1 X Y Z Incoherent type 2 X Y Z Incoherent type 3 X Y Z Incoherent type 4 X Y Z Z Figure 2 Feedforward loops (FFLs). a The eight types of feedforward loops (FFLs) are shown. In coherent FFLs, the sign of the direct path from transcription factor X to output Z is the same as the overall sign of the indirect path through transcription factor Y. Incoherent FFLs have opposite signs for the two paths. b The coherent type-1 FFL with an AND input function at the Z promoter. c The incoherent type-1 FFL with an AND input function at the Z promoter. S X and S Y are input signals for X and Y. Z Alon U, Nat. Rev. Genet. 2007 19 CBS, Department of Systems Biology

FFLs are over-represented network motifs Table 1 Statistics of occurrence of various structures in the real and randomized networks Appearances in real Appearances in network randomized network Structure (mean ± s.d.) P value Coherent feedforward loop 34 4.4 ± 3 P < 0.001 Incoherent feedforward loop 6 2.5 ± 2 P! 0.03 Operons controlled by SIM (>13 operons) 68 28 ± 7 P < 0.01 Pairs of operons regulated by same two transcription factors 203 57 ± 14 P < 0.001 Nodes that participate in cycles* 0 0.18 ± 0.6 P! 0.8 * Cycles include all loops greater than size 1 (autoregulation). P value for cycles is the probability of networks with no loops. Shen-Orr et al. Nat Genet. 2002 20 CBS, Department of Systems Biology

Abundance of 8 FFL Subtypes Figure 1. The eight FFL types and their relative abundance in the transcription networks of E. coli and S. cerevisiae. Relative abundance is the fraction of each type relative to the total number of FFLs in the network (138 in E. coli and 56 in S. cerevisiae in the networks presently studied). The coherent FFL types are denoted C1 through C4, and the incoherent types I1 through I4. Mangan et al., J. Mol. Biol. 2006 21 CBS, Department of Systems Biology

Protection against input fluctuations (C1-FFL) X Y AND Z S X Z Y Input S x 1 S X X 0.5 0 1 Y 0.5 0 1 Z 0.5 Delay 0 0 2 4 6 8 10 12 14 16 18 20 Time Alon U, Nat. Rev. Genet. 2007 22 CBS, Department of Systems Biology

On vs Off Dynamics of FFLs: C1-FFL vs simple, 2-input system b Arabinose system Lac system ON step of S X OFF step of S X CRP S X = camp S Y = arabinose CRP S X = camp S Y = allolactose 0.5 0.4 1 0.8 arabad laczya AraC LacI Z/Z st 0.3 Z/Z st 0.6 AND AND 0.2 0.4 Z = arabad, arafgh Z = laczya 0.1 0.2 0 10 20 30 40 Time (min) 0 10 20 30 40 Time (min) Alon U, Nat. Rev. Genet. 2007 23 CBS, Department of Systems Biology

On vs Off Dynamics of FFLs: C1-FFL with OR logic c ON step of S X OFF step of S X FlhDC FliA S X S Y Z/Z st 1 0.9 0.8 0.7 0.6 0.5 0.4 Z/Z st 1 0.9 0.8 0.7 0.6 FliA present FliA deleted OR flilmnopqr 0.3 0.2 0.1 0.5 0.4 0 40 60 80 100 120 140 160 Time (min) 0.3 0 20 40 60 80 100 120 140 160 Time (min) Figure 3 The coherent type-1 feedforward loop (C1-FFL) and its dynamics. a The C1-FFL with an AND input 24 CBS, Department of Systems Biology

Dynamics of Incoherent Feed-forward loop (FFL-I1) a 1 S X 0.5 0 1 c CRP camp galactose 2 1.5 gale-wt Y 0.5 GalS Z/Z st 1 Z b Z/Z st 0 1 0.5 0 1 0.5 0 0.5 1 1.5 2 2.5 Time I1-FFL 1.5 1 Simple regulation 0.5 galetk 0.5 gale-mut 0 0 1 2 3 Time (cell generations) Figure 4 The incoherent type-1 feeforward loop (I1-FFL) and its dynamics. a The I1-FFL can generate a pulse of Z expression in response to a step stimulus of S x. This occurs because once Y has passed its threshold (indicated by an orange circle) it starts to repress Z. b The I1-FFL shows faster response time for the concentration of protein Z than a simple-regulation circuit with the same steadystate expression level. c An experimental study of the dynamics of the I1-FFL in the galactose system of E. coli. Response acceleration in the wild-type system (marked gale-wt ) is found following steps of the input signal (glucose starvation). The acceleration is disrupted when the effect of the repressor GalS is abolished by mutating its binding site in the promoter of the output gene operon galetk (marked gale-mut ). T 1/2, response time; Z/Z st, Z concentration relative to the steady state. 0 0 0.5 1 1.5 2 2.5 3 3.5 4 T 1/2 I1-FFL T 1/2 (Simple regulation) Time 454 JUNE 2007 VOLUME 8!2007!Nature Publishing Group! Alon U, Nat. Rev. Genet. 2007 25 CBS, Department of Systems Biology

FFL Summary FFLs are over-represented in regulatory networks (only motif with n=3) FFLs are evolutionarily conserved Coherent Type-1 FFL (C1-FFL) Sign-sensitive delay Persistence detector (filter) Incoherent Type-1 FFL (I1-FFL) Pulse generator Response accelerator Of the 8 possible FFLs, only C1-FFL and I1-FFL are common Other FFLs have reduced functionality, insensitivity to inputs 26 CBS, Department of Systems Biology