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---> Material for the lecture (slides, papers, software) <---

1. What is Systems Biology any why do we need it

  • Material:
    • cf. Chapter 1 of Skript (Very Draft Lecture Notes) (These lecture notes are created for a full systems biology lectures; nevertheless you can use the respective chapters for this lecture, too.)

  • What is Systems Biology?
  • What is a System?
  • Motivation - Why a systems approach? - Emergent Phenomena
  • Major Innovations of Systems Biology
    • Standards
    • Tight integration of experiment and theory
    • Automation of the scientific cycles
  • Example: SBML - Systems Biology Markup Language (very briefly)
  • Basic Approaches to Describe a System
  • Dynamical System and State Space
  • Model - System with Purpose to Abstract System Phenomena
  • Types of Systems

2. Reaction networks and modeling with differential equations.

  • Material:
    • Explained in Chapter 2 of Skript (Very Draft Lecture Notes) if your notes from the lecture should not be sufficient. The chapter contains also further examples for exercise.

  • How to translate biochemical reactions to differential equations?
  • Reaction network = set of species + set of reaction rules
  • Stoichiometric matrix
  • flux vector = kinetic laws
  • Mass action kinetics
  • Chemical differential equation
  • Dynamical interpretation / simulation
  • Example: Transcription factor activating a gene.

3. From reaction networks (RN) to gene regulatory networks (GRN)

  • Material:
  • With RNs we can create very detailed models of genes, their dynamics, and interactions.
  • The derived dynamical model is usually a differential equation (ODE), dx/dt = f(x), which can be used in three different ways:
    1. Find an explicit solution. (Only possible for very simple equations like dx/dt = k x, exponential growth)
    2. Simulation. E.g., by the simple scheme: x(t+dt) = x(t) + dt * f(x) with dt being the time-step.
    3. Qualitative analysis. E.g., by deriving the steady state, simply by solving f(x) = 0.
  • Overall picture, noting that there are different timescales, e.g., three time scales:
    1. very fast: activation of a protein, e.g., by phosphorylation [ignored in this lecture, assumed to be instantaneous]
    2. medium: binding of TF to promoter region, going into steady state; [focus of this lecture]
    3. slow: expression of a gene, creation of gene product (here, protein), [focus of next lectures]
  • As an exercise: "detailed" mode of gene Y being activated by transcription factor X forming a complex before binding to promotor region.
    • Time scale: medium
    • Approach: (2) Simulation (shown by using octave).
  • One dimensional model of gene activation:
    • n X + Yoff -> n X + Yon, Yon -> Yoff
    • Deviation of Hill-kinetics, beta(x).
    • Discussion of parameters: n and k.

4. Gene regulatory network motifs I (cf. U. Alon)

  • Basic model of a a factor X* activation a gene Y: dy/dt = beta(x) - alpha y
    • Steady state: y_st = beta(x) / alpha
    • Response time / half level activation time t1/2 = log 2 / alpha (independent of activation rate).
    • Graphical illustration of steady state and the dynamics towards steady state.
  • Negative feedback loop
    • Basic model dy/dt = beta(y) - alpha y * Use inhibitory Hill-kinetics for beta(y) similar to kinetics derived in previous lecture.
    • Deviation of core properties:
      • Reduces response time (gene switches quicker)
      • Increases robustness, with respect to decay rate alpha and max-level activation beta_m, but not according to k (binding kinetic constants).
    • Graphical illustration of steady state and the dynamics towards steady state.

  • Positive feedback loop
    • Basic model dy/dt = beta(y) - alpha y * Uuse Hill-kinetics for beta(y) as derived in previous lecture.
    • Deviation of core properties:
      • Can be bi-stable (if decay rate is not too high).
      • Decreases robustness,
      • Increases response time.
    • Graphical illustration of steady state and the dynamics towards steady state.

5. Gene regulatory network motifs II (cf. U. Alon)

  • Motifs with three genes
    • 13 possible non-trivial 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 "NK-Network" for short.
  • Global dynamics x(t+1) = F(x(t)) with x(t) an N-dimensional 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(n-1)} with x(i+1 mod n) = F(x(i)) for i = 0, ..., n-1.
    • 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 cell-cell 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

8. Practical Exercise (planned)

  • Simulation using COPASI/Matlab (practical exercise, Location: FRZ Linux Pool) OR
  • Practical exercises with pen and paper. Not the exciting but very useful.
Topic revision: r9 - 2019-11-25 - PeterDittrich
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