COSC 348: Lab12
IMPORTANT: In the end of each lab submit your code and answers to the questions (worth 1% of the final mark).
Ising-like model of the dynamics of an abstract gene regulatory network.
The Ising model is defined on a discrete collection of variables called
spins, which can take on the value +1 or
−1 (up or down). The spins can be thought of as living on
a graph, where each node has exactly one spin, and each edge connects
two spins with a nonzero interaction value. The goal is to study phase
(state) changes in the Ising model as a simplified form of phase (state)
changes in real substances. The two-dimensional NxN square
lattice Ising model is one of the simplest models to study phase changes
and asymptotic behaviour of the modeled system.
Basic description of the model
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Let the gene expression be a two state variable.
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If the gene is over-expressed, then its state variable
S[i] = +1.
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If the gene is under-expressed, then its state variable
S[i] = −1.
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Gene interactions are expressed by interaction coefficients
w[i, j], which are real numbers from the (open) interval
(−1, +1).
They do not have to be symmetrical,
i.e w[i, j] ≠ w[j, i].
-
Let
b[i] be the basal expression of a gene,
a real number from the interval (−1, +1).
Visualising the model
Although we are dealing with linear data,
it is convenient to represent the data in a 2D lattice
(for instance we may have 16 genes, so there would be
16 (binary) spin values -- which can be displayed in
a 4x4 grid).
The simplest visualization of the NxN lattice can be to just
use text: where S[i] = +1 is represented by a black dot (or X);
and S[i] = −1 by a white dot (or .).
We will need it to observe how the state of the network evolves over time.
Example representation of 16 spin states:
XX.X
X..X
..XX
.XXX
The spin states need to be displayed at the start, and after each epoch
of change.
Initialisation of the model
In all experiments, initialize the network by assigning each gene a
state −1 or +1 with probability = 0.5
(flipping a virtual coin).
Updating the model
We can simulate time by iterating through the model, and discovering what
equilibrium behaviour develops within the model:
is there a Fixed point attractor; or a more dynamic
Limit cycle attractor (oscillation);
or Chaotic (strange) attractor?
These updates can be made as --
-
Asynchronous dynamics of the model:
At each iteration we randomly* pick one gene and update its
state according to this rule:
S[i] = +1
if ( ∑j
w[i, j] S[j] + b[i] ) ≥ 0
S[i] = −1
if ( ∑j
w[i, j] S[j] + b[i] ) < 0
Here we take into account the current states of the genes in the network,
S[j].
So if we were to perform an 'epoch update' (ie, each value is updated once),
then for some values of j, S[j] would be the
value generated in the previous 'epoch' iteration, and for other values
S[j] would be the value generated during this iteration.
* "randomly":
Ideally we want to train an entire epoch, updating gene-by-gene
-- so we
could simply update each gene 'in order' until the epoch is complete;
or we could decide a random permutation, and use that permutation of
gene orders for all epochs;
or we could use different random permutations for each epoch.
Obviously, it'll be easiest to process the genes "in order".
-
Synchronous dynamics of the model:
In this case we calculate all new values using the values for
S[j] that had been calculated in the previous iteration.
So, this is similar to epoch training, where we do all the calculations
before making any updates.
Experiments
Write a program to observe the evolution of the states over time, let us use:
N = 10,
so there will be 100 spin states;
and finish updating after twenty epochs
The first experiment -- point attractor dynamics
After certain number of epochs the system should stay in one state when
updating.
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Generate all interactions
w[i, j] as random real numbers
from the interval (−1, +1) in such a way that the
interaction matrix is symmetric, i.e. w[i, j] = w[j, i]
and there are no direct feedbacks, i.e. w[i, i] = 0
for all i.
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All basal expressions
b[i] = 0 for all i.
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The update rule is ASYNCHRONOUS.
Run the experiment several times, with different initial random configuration
of gene states BUT with the same interaction matrix W. Answer these questions:
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Does the system always converge to a stable state?
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Is this stable state always the same even when the initial distribution
of states is different?
The second experiment -- limit cycle attractor
After certain number of epochs the system should switch between just
few states when updating
-
Generate all interactions
w[i, j] as random real numbers
from the interval (−1, +1) in such a way that the
interaction matrix is symmetric, i.e. w[i, j] = w[j, i]
and there are no direct feedbacks, i.e. w[i, i] = 0
for all i.
-
Generate basal expressions
b[i] as random numbers from
the interval (−1, +1).
-
The update rule is SYNCHRONOUS.
Run the experiment several times, with different initial random configuration
of gene states BUT with the same interaction matrix W and the
same basal expression values b[i]. Answer these questions:
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Does the system always converge to the same sequence or cycle of states?
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Is this stable state sequence or cycle always the same even when the
initial distribution of states is different?
The third experiment -- chaotic attractor
After certain number of epochs the system should follow a sequence of
similar-but-never-the-same states.
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Generate all interactions
w[i, j] as random real numbers from
the interval (−1, +1) in such a way that the interaction
matrix is NOT symmetric, i.e. w[i, j] ≠ w[j, i] and
there are no direct feedbacks, i.e. w[i, i] = 0
for all i.
-
Generate basal expressions
b[i] as random numbers from the
interval (−1, +1).
-
The update rule can be either SYNCHRONOUS or ASYNCHRONOUS.
Run the experiment several times, with different initial random configuration
of gene states BUT with the same interaction matrix W and the
same basal expression values b[i]. Answer these questions:
-
Does the system always converge to the same sequence of similar states?
-
Is this state sequence always the same even when the initial distribution
of states is different?
If time permits, repeat the above experiments including feedback interaction,
i.e. generate w[i, i] as random numbers from the interval
(−1, +1).
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