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 Motifs with three genes
 13 possible nontrivial networks (directed graphs) with three genes
 Common motif: feed forward loop (FFL)
 When distinguishing activation and inhibition: 8 Feed forward loops (4 x coherent and 4 x incoheren); only one coherent and one incoherent is very common (Type 1).
 Coherent feed forward loop Type 1 with
 AND (delayed on, immediate off, filters transient on pulses)
 OR (immediate on, delayed off, filter transient off pulses)
 Incoherent feed forward loop Type 1 with
 AND (pulse generator, deviation of input signal, can detect an increase in the input, reduces response time)
 Motifs with more than three genes
 Single input multiple output (SIM): Primitive sequential "program" execution. Timing can be evolved by altering kon, koff
 More complex motifs; e.g. for a spiked program, which combines Incoherent Type 1 loops with Coherent AND FFLs (see Cell Snapshot).
 Notes:
 The motifs studied so far are common in all organisms, and in particular in bacteria (e.g., E. Coli)
 In particular in organisms with different cell types and complex morphogenesis other motifs can be found, in particular feedback loops involving more than one gene. (see lecture on Boolean networks).
6. Boolean networks for modeling gene regulatory networks
 Motivation: Study large networks with many complex feedback loops and many atractors.
 Rough history: since 200 years theory, since 100 years for computer engineering, since 50 years for biology (Stuart Kauffman)
 So, a lot of extremely powerful theory and tools available when modeling a GRN as a Boolean network.
 Boolean network  basic definition and dynamics
 N : number of genes = number of nodes / gates
 x_i(t) from {0, 1} : at time t state of gene i (can be either 0=false or 1=true), i = 1, ..., N
 x(t) = (x_1(t), ... x_n(t)) : global state of the network, just putting all gene states into one vector x.
 f_i : update function of gene i
 x_i(t+1) = f_i(x(t)) : Gene i is updated using the update function f_i
 Note that an update function f_i of gene i usually depends only on subset of other genes. K_i denotes the size of this set.
 K_i : number of inputs of gene i
 If all genes have the same number of inputs (i.e., K_i = K for all i = 1, ..., N), we call the network: "NK Boolean Network" or "NKNetwork" for short.
 Global dynamics x(t+1) = F(x(t)) with x(t) an Ndimensional Boolean vector denoting the state of the network at time t.
 Fixed point (stationary state): A state x is a stationary state, if F(x) = x.
 Atractor (or cycle) A: a set of states A = {x(0), x(1), ..., x(n1)} with x(i+1 mod n) = F(x(i)) for i = 0, ..., n1.
 n is called the period (or cycle length) of the atractor (cycle).
 A fixed point is an attractor with period (cycle length) 1.
 A basin of atraction B of an attractor A is the set of states that eventually lead to the attractor A, i.e., for all states x from B holds: F(2^N) is contained in A.
 An attractor (and a fixed point in particular) can be equated with a cell type.
 Random networks get easily chaotic (e.g. when there K>2 inputs per gene).
 Real networks are more ordered through canalizing functions.
 For modeling complex real networks (e.g., flower development) genes with three states are often used.
7. Stochastic dynamics in genetic systems
 Have a good feeling that we can do it this semester (19/20).
 But with an adapted, mathematically simplified, enhanced content.
 Background literature for the simulation algorithm:
 Paper: Gillespie1977 (just the first pages)
 Very nice, but a bit too complicated for this introductory lecture, ...
 Motivation:
 The problem: when molecular counts get low, stochasticity comes in.
 GRNs can become stochastic when transcription factor concentration or mRNA concentration is very low.
 This might explain cellcell variability and bursting gene expression.
 Can be useful: E.g. for decision making, switching dependent on stress (Yomo et al. model)
 Basic idea of stochastic simulation
 Gillespie Algorithm
 Draw one random number for deciding WHEN the next reaction happens.
 Draw another random number for deciding WHICH reaction then happens.
 Update current time and state accordingly.
 Note the difference of macro and micro rates: 2 S1 > S3 vs. S1 + S2 > S3
 Example simulation: Bursting gene expression
