- 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

- 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.

- 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:
- Find an explicit solution. (Only possible for very simple equations like dx/dt = k x, exponential growth)
- Simulation. E.g., by the simple scheme: x(t+dt) = x(t) + dt * f(x) with dt being the time-step.
- 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:
- very fast: activation of a protein, e.g., by phosphorylation [ignored in this lecture, assumed to be instantaneous]
- medium: binding of TF to promoter region, going into steady state; [focus of this lecture]
- 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.

- Material:
- Look at the Cell Snapshot: http://www.sciencedirect.com/science/article/pii/S0092867410011360
- You can also watch Uri Alons video lectures (Lecture 2 - 4) avaiable at youtube. I basically took all the content from there. Except the relation between a reaction network and a regulatory network, which I showed at the beginning of the lecture. An analog deviation can be found in Alon's book "Systems Biology".
- Uri Alon's lectures at youtube

- 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.

- 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).

- 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.

- 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

- 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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