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TODO List
  • REF Malthus
  • REF Verhulst
  • Add model of ribosome binding in yeast (Shah et al, 2013)


Event-based models


Single level (autonomous) dynamical systems can be classified by the way they implement time and variables:

continuous time
discrete time
continuous variables
ODE
(continuous time requires a continuous variable)
MAP
(next state function with a defined interval)
discrete variables/
nominal entities
EVENT-BASED
(time continuous but not everything changing together only certain events)
FSMs
n-FSMs, CAs, Bool-nets (nominal states require discrete time)

So far we have taken FSMs as a center-stage and have shown how other model formalisms can be derived from them using short cuts on conditions defining FSMs.
At this stage we will introduce event-based models and compare them to ODEs. As a case example we use the logistic growth equation:
dN/dt = aN - bN^2.

Consider for a moment the history of the interpretation of the terms of this classic equation. For instance the N^2 term can be interpreted as representing competition. Alternatively in the case of dN/dt = rN(1-N/K), K is the carrying capacity, which is a concept which was derived from the population concept.
These interpretations are in fact later derived, while the original interpretation was much more arbitrary and stems from a conference called by king Leopold I of Belgium (19th century?) to discuss the implications of Malthus's (REF?) prediction about the human overpopulation (population doubling every 30 years!) as derived from British parish records. At the conference, Verhulst (REF?), a physicist, put forward a simple solution to the problem and was so able to calm the poor king's fears.
Verhulst used the argument that any function (including a growth function) can be approximated by a Taylor expansion , and with a little reasoning a reasonable human population growth function can be derived from dN/dt = a + bN + cN^2 + ... + nN^y. The first term can obviously be dropped because there is no external influx into the human population. Moreover there should be at least one additional term other than aN and it should be negative, otherwise the population would grow to infinity, which is obviously a physical impossiblity. And so, obviously there was no reason to fear for infinite overpopulation! Notwithstanding this reasoning, it still depends on the coefficients whether overpopulation is alarming or not! The main point here however, is that the interpretations of model terms can be very minimalistic indeed.

Gillespie algorithm for a birth/death process


We can model a stochastic birth/death process using a stochastic ODE:
dN/dt = aN - bN^2 + noise
In the ODE-formalism, the variables change continously, i.e. something is happening at every time. However, we can also assume that things only happen at certain times, as events. In this event-based formalism each event is modelled using probabilities which determine which event occurs when in continuous time. Since we use probabilities for the events, this automatically gives a stochastic description of the system. In this formalism each term needs to be explicitly interpreted:
  • aN = birth + death
  • bN^2 = extra death + reduced birth due to competition

Now, assume that a1 is the per capita birth rate, a2 the per capita death rate, b1 the reduction in births due to competition, and b2 the extra deaths due to competition (hence a = a1+a2 and b = b1+b2). There are two possible events that can take place: birth-events and death-events. Then, the frequency of all events (birth and death) is:
E0 = (a1+a2)N - b1N^2 + b2N^2,
and the time we have to wait to the next event is drawn as
tau = 1/E0 * ln(1/rand1),
where rand1 is a random number between 0 and 1. Hence, on average we have to wait 1/E0 time till the next event, but this can vary (stochasticity). Last, we have to determine whether the event at time T+tau is a birth of a death:
  • N -> N+1 (birth) if (a1N - b1N^2) < rand2*E0 where rand2 is again a random number between 0 and 1;
  • N -> N-1 (death) else.

The behaviour in this model allows for stochasticity which is comparable to the chaotic regime in MAPs (in ODEs such behaviour never occurs). However in the MAP the direction of change in the chaotic regime is always to the other side of the equilibrium. In the Event-based formalism the direction is not necessarily relative to the equilibrium. This can be considered to be more realistic for stochastic simulations where once in a while something happens.

Next: Spatial models overview


References




(CHANGELOG 2014-2015)
- Extended explanation of Gillespie birth/death
- Added ToDo: add ribosome example of Shah et al (2013)





(NOTES FROM COURSE 2006-2007)

(LECTURE 3)
Model Formalisms CONTINUED

We have taken FSM as center-stage and have looked at how other model formalisms can be derived from this using short cuts on the FSM conditions.

cont time discrete time

cont variables ODE (cont time requires MAP (next time function with interval undefined)
cont variable)

discrete variables EVENT-BASED (time cont FSM (nominal states requires discrete time)
but not everything changing (n-FSMs, CAs, B-nets)
together only certain events

(If a variables changes continuously one needs a continous arbitrary amount to change (cont. var.))


EVENT-BASED models

In comparison to ODE: N' = aN - bN*N

History of interpretation of terms:
a) N*N => direct competition
b) rN ( 1- N/K), where K is carrying capacity => which is a concept derived from the concept population
c) Minimum orgininal interpretation:
King of Belgium calls meeting to discuss Malthus's preditction about human overpopulation (population doubling in 30 years!).
Verhulst (physicist) calms fears by explaining how any function can be approximated by a Taylor expansion: N' = a + bN + cN*N .... nN^y. For the human population the first term (a) can be dropped because there is no influx. Moreover there should be at least one additional term other than bN and it should be negative otherwise the population will grow to infinity, which is a physical impossibliity.
So there is no direct biological interpretation, just dictated by mathematical reasoning.

Points:
i) It still depends on the coefficients whether overpopulation is not alarming!
ii) Interpretations of model terms can be very minimal!

In comparison to MAP: shows period doubling and chaotic behaviour that does not arise in ODE.

EVENT-BASED (or Gillespie algorithm / stochastic reaction kinetics)
Need to reinterpret terms:
aN = birth + death
bN^2 = extra death + reduced birth

then: frequency of all events Eo = (a1 + a2)N - b1N^2 + b2*N^2
b d red b extr d

then time to next event

tau 1/ (Eo ln (1/rand1)) T

T + tau

then next event is (b) N

N+1 if a1N - b1N^2 < rand2*Eo else (d) N

N-1

This model leads to stochasiticity which is comparable to chaotic regime in MAP, however in the MAP the direction of change is always to the other side of the equilibrium, in EVENT the direction is not necessarily related to equilibrium (i.e. probably more realistic for stochastic molecular simulations where once in a while something happens).





[[prebio|]]