Simulation of Biological Processes phần 8 potx

when it is not attached to the lattice. There are e⁄cient analytical expressions for

computing this e¡ect (Lagerholm & Thompson 1998), and it would be very

interesting to combine such equation-based methods with our individual-based

stochastic approach.

Noble: When Raimond Winslow was presenting his work on combining

stochastic modelling with di¡erential equation modelling, as I understand it this

leads to greatly increased computational times. When I recently heard Dennis Bray

present some of this work, he gave the impression that the stochastic com￾putational methods that you are using actually go extremely fast. What is the

explanation for this?

Shimizu: If it is the case that there are certain complexes that have a large number

of states, so that a large number of equations would need to be integrated at every

time point, then stochastic modelling can be faster.

Noble: So it’s a matter of whether each of those states were otherwise to be

represented by kinetic expressions, rather than by an on^o¡ switch.

Winslow: The reason this is di⁄cult for us is that we are describing stochastic

gating of a rather large ensemble of channels in each functional unit. Another

confounding variable is the local Ca2þ concentration, because this is increasing

the total number of states that every one of these channels can be in.

I have a comment. We have now heard about models in three di¡erent areas. We

have heard about a model of bacterial chemotaxis, neural models that Les Loew

described and the cardiac models that Andrew McCulloch and I have talked

about. I grant you that in each one of these systems there are di¡erent experi￾mental capabilities that may apply, and thereby make the data available for

modelling di¡erent in each case. But there are a lot of similarities between the

mathematics and the computational procedures used in these systems. In each

case, we have dealt with issues of stochastic models where the stochastic nature

comes in through the nature of channel gating or molecular interactions. We

have dealt with ordinary di¡erential equations which arise from systems that

are described in laws of mass action, and we have dealt with partial di¡erential

equations for systems where there are both reaction and di¡usion processes

occurring on complicated geometries. Perhaps this is one reason why Virtual

Cell is a useful tool for such a community of biologists: it covers so much of

what is important in biological modelling. We should see how much overlap

there is in these three areas, and whether this is a rather comprehensive class of

models de¢ned in these three areas.

Noble: A good way of putting the question would be, ‘What is it that is actually

missing?’ Part of what I suspect is missing at the moment would be the whole ¢eld

of systems analysis, which presumably can emerge out of the incorporation of

pathway modelling into cellular modelling. One of the reasons I regret not

having people like Bernhard Palsson here is that we would have seen much more

178 DISCUSSION

of that side of things. Are there tricks there that we are missing, that we should

have brought out?

Winslow: I would say that this is not a di¡erent class of model; it is a technique

for analysing models.

Noble: Yes, this could be applicable to a cell or to an immune system.

Subramaniam: I think the missing elements are the actual parameters that can ¢t

in your model at this point, based on the molecular level of detail. We don’t have

enough of these to do the modelling. Tom Shimizu’s paper raised another

important point, which is the state dependence. Our lack of knowledge of all

the states clearly inhibits us from doing any model that is speci¢c to a system.

We are coarse graining all the information into one whole thing.

Winslow: Again, I didn’t hear anything in what you just said about a requirement

for a new class of models. Rather than new methods of data analysis, you are saying

that there may be systems or functionality that we don’t yet have powerful experi￾mental tools to fully probe in the same way we can for ion channel function in

cardiac myocytes. I agree with that.

Loew: One kind of model that I don’t think we have considered here is that of

mechanical or structural dynamics, in terms of the physics that controls that. Part of

the problem there is also that we don’t completely understand that at a molecular

level. Virtual Cell deals with reaction^di¡usion equations in a static geometry. It

isn’t so much the static geometry that is the limitation; rather it is that we don’t

know why that geometry might change. We don’t know how to model it because

we don’t know the physics. We know the physics of reaction^di¡usion equations,

but the structural dynamics issue is another class of modelling that we haven’t

done.

Subramaniam: The time-scale is a major issue here. If you want to model at the

structural dynamics level, you need to marry di¡erent time-scales.

Loew: Getting back to Raimond Winslow’s point about the di¡erent kinds of

modelling, this time-scale by itself does not de¢ne a di¡erent kind of modelling.

The issue is whether the physics is understood.

McCulloch: I agree with both of those points. It seems that what is missing is an

accepted set of physical principles by which you can bridge these classes of models,

from the stochastic model to the common pool model, and from the common pool

model to the reaction^di¡usion system. Such physical principles can be found, but

I don’t think they have been articulated.

Winslow: Yes, we need these rather than our own intuition as to what can be

omitted and what must be retained. We need algorithmic procedures for quanti￾fying and performing that.

Paterson: The opportunity to use data at a level above the cell can provide very

powerful clues for asking questions of what to explore at the individual cell level.

If we are trying to understand behaviour at the tissue, organ or organism level,

MODELLING CHEMOTAXIS 179