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[SOUND].

In lecture 5A I talked

to you about the criteria for good models and also started to describe

the strengths and sort of the limitations.

Of ODE models.

One of the limitations of ODE model is

the so-called well-stirred assumption when one has multiple compartments.

So, when one has multiple compartments, one can

actually build what I call multiple compartment ODE models.

And in this type of model eh, a, a typical example is for the MAPK signaling pathway.

Of how map kinase activation in the

nucleus needs to transcription factor, transcription factor a-.

Excuse me.

Map kinase activation in the cytoplasm leads

to transcription factor activation in the nucleus.

In this cartoon shown on the left, one can see this.

Very clearly.

Raf activates MAP kinase to act, Raf activates MAP kinase to

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MEK to activate MAP kinase and these two reactions occur in the cytoplasm.

The activated MAP kinase, ERK, now translocates into the nucleus.

Nucleus and then it can, it can

then phosphorylate the transcription factor or the kinases.

So, there are two compartments here.

The cytoplasmic compartment [SOUND] and two the nuclear compartment.

The cytoplasmic and the nucleic and

map kinase moves between these two compartments.

So, in addition to the biochemical reactions of RAF

activating MAC and MAC activating ERK one needs a trans.

Put reaction of activated ERK moving from the cytoplasm into the nucleus.

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The more explicit way of dealing with spatial

issues or movement across compartments and biological systems.

Is, is to use PDE, or partial differential equation based

models, where one can explicitly attribute to each

protein both Sort of a reaction capability

and sort of a diffusion capability based on its characteristics.

So, in this particular mo, model, again taken from that review, that

John Ingdram Ron and I wrote back in 2004, one looks at

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Movement occur within the cytoplasm to understand how Ras

might be activated at different locations at different times.

So, the question that we were trying to answer with these kinds of model is how

the EGFR receptor could activate Ras over different

time scales at different locations within the cell?

Experimental data, produced by, several laboratories

that shown the this indeed occurs.

That there was an early, and a late phase of activation

of the last, and many cells died in the early phase involved.

Meh, activation of the cell's surface or plasma membrane,

and a later phase in world activation and ecology.

And these reactions here.

Allow us to sort of describe how these two kinds of activation

might occurs and the, in these reactions one calculates not just the

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A former, Suzanna Nebbits, a former graduate student

of mine as part of a thesis research.

As part of her thesis research and what she did in this model was to

sort of look at the production cyclic AMP in both the cell body.

In both the cell body [SOUND] and the long extension of this neuron.

I thought if I should draw this thing better not obscure it.

So, she did imaging experiments looking at cyclic mp production upon

addition of the ligand, I think this was, actually the protein that binds to

the beta-adernergic receptor and the increase in cyclic AMP causes a decrease

in fluorescence and so you can see here that the outline of the.

Prove k in the extension can

be seen and that sort of starts to disappear when it's activated.

The model actually Captures or the experiments can

capture the prediction of the model that there is differentially higher expression

of higher expression of cyclic A-M-P or higher levels of

cyclic A-M-P in the dendrites as compared to the cell body.

And this graph here shows the comparison between the

experiments, all these dotted squares, the red and the blue versus the

simulations and showing that the model actually or the experiment actually

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Shows the prediction, Sort or confirms the prediction of the model.

That, even though the receptors, in this

case, are evenly distributed across the plasma membrane.

Due to, a variety of reasons that I'll deal with later.

There is more cyclic AMP formed in the cytoplasm here than in the cell body.

So, this kind, this model here, [SOUND] this competition was done in

a program called The Virtual Cell

using a, a partial differential equation-based model.

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In addition to ordinary differential equation-based

models and partial differential equation-based models.

The intrinsic uncertainty in some of the reactions

within cell biological systems requires us to consider.

Stochastic representations.

So, let us deal with deterministic versus stochastic systems.

Deterministic systems, are systems in which progress

that is time-evaluation of the system can be

fully computed from specification of the initial

conditions that is the concentration of the reaction.

Concentration of the reactants, and the reaction rates.

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Become important when one reactant is present in a very low concentration.

For example, when a transcription factor is in the nuclears

and needs to bind to the promoter region of a gene.

There are only two genes.

There is only two copies in the nucleus.

There's, there may be a large number of transcription factor molecules, but only

two copies of the promoter region and the reaction becomes stochastic.

Stochastic models typically are computed using what is called the master

equation that describes the progress of the system with respect to time.

The system can be modelled in a defined state in a given.

At a given time and moving to another state in a probabilistic manner.

The differential equation that describes the variation of

the probability is called the master equation, shown here.

For biochemical systems for biochemical systems, the

biochemical reactions the stochastic processes are often solved.

Using the Gillespie algorithm.

The Gillespie algorithm enables us to discretely

simulate each reaction between two reaction, reactants.

The interval time and or the space between reactions can

be, can follow a probability distribution function given by the master

equation and thus for every reaction, one can have.

Sort of a probability of whether it can or cant occur.

And when it can occur in space, so this kind of

probability distribution captures the interim [INAUDIBLE] of the system.

So, the take home message if for, from this lecture is follows.

Mathematical representations of biochemical and biophysical systems help

us understand systems behavior and predict input-output relationships.

This is kind of an important ca,

capability for which mathematical models are used.

For instance, in the example that I showed you.

The input from the imaging experiments, the

input had stimulation of the beta [INAUDIBLE] receptor.

And the output was really, a a spatial or a space dependent

accumulation of cyclic A and P in the, dendrites versus the cell body of neurons.

And this out, input output relationship could be predicted from the.

[INAUDIBLE] PDE based model.

Biological systems can be both deterministic or stochastic

and deterministic models can use either ODEs or PDEs.

So, the nature of the biological process

being studied determines th type of model that.

Be, that is being used, if one just want to study raf

activation of map kinase in the cytoplasm and ODE model will suffice.

If one wants to study cyclic KMP

accumulation in different parts of the cells,

a PDE model might be required if one wants to study transcription factor act.

[UNKNOWN] Of gene expression, one might need a stochastic model.

So, the nature of the process being studied

define the kinds of mathematical representations that is required.

This concludes lecture five.

Thank you.