[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 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. So this is a two compartment model, and the assumption in these two compartment models are that the. Components within a compartment are the comply with the well-stirred assumption. While the compa-, movement while the component that moves between compartments has equal access when it is in one compartment or the other. So, these kinds of, multi-compartment ODE models, are a relatively reasonable solution to deal with, spacial issues in biological systems. But it doesn't always work. 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 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 the reaction but also the movement of RAS within within the. System of interest. In terms of the characteristics of the system suggests the radius and so on. So, this representation of both the, the action term and what is called the diffusion term or the movement term allows can be done partial differential equations. And this sort of explicitly allows for the computation. Oh a spatial characteristics of the cyst. So, here is an example of a, a, partial differential equation model that can be used in biological systems. So, in this particular example, this is an experiment and model done by Suzanne Neves. 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 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. 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. In contrast, stochastic systems require that the progress of the system is determined by predictable Both by the predictable actions of some reactions and by a sequence of random variables that regulate the ability of the system to move from one state to the other. For biochemical systems, stochastics. 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.