 So, where would I come across something like this? So, a sort of example of an equation that looks somewhat like this where I have this separation of time scales is if you consider let us say enzyme kinetics if I consider enzyme kinetics. So, let us say I consider this set of equations that I have an enzyme plus a substrate which reacts to give me a complex which I call this enzyme substrate complex and then from there I get this enzyme plus a product. So, what is essentially happening is that substrate is going to a product, but through the action of this enzyme which I recover back at the end the enzyme does not get depleted. And again if I think of putting some rates similar rates k plus and k minus and a rate R then this has the same sort of structure as this thing that we are talking about a going to b going to c. Where would I see this? So, for example, if I had let us say DNA. So, let us say I have DNA plus this RNA polymerase RNA polymerase that reacts to form this complex where this RNA polymerase binds on to this DNA DNA RNA polymerase complex. So, and then you get back your RNA polymerase plus you get back your RNA plus you get production of RNA. So, this is like the transcription equation transcription reaction written in this sort of a language. And you can write indeed many sort of wherever you have an enzyme sort of a kinetics you can frame it in this sort of a language that you have some substrate whatever that substrate is in this case DNA and something that transiently binds in order to facilitate the reaction. For example, RNA polymerase and this and ultimately at the end of all of this you get some product out of it and then the enzyme sort of comes back. But once I write down an equation like this once I write down a schematic like this I can again write down the rate equations exactly like I did for A and B. So, let me I will not solve the whole thing, but let me at least write down the equations. So, if I write down the equation for the enzyme D enzyme DT right. So, that has decays with the rate k plus when an enzyme and a substrate comes together it is formed back with a rate k minus from this enzyme substrate complex a second rate R might as well just write it as k minus k minus plus R from this enzyme substrate complex that takes into common basic reaction plus this reaction. For the substrate I would have again a k minus k plus enzyme and a substrate and that is it. Then for this enzyme substrate complex D enzyme substrate DT it is produced at a rate k plus from this enzyme on the substrate and it decays with the rate R over there and another k minus over here k minus plus R enzyme substrate. And finally, I have the product DPDT which is simply produce anything else I am missing the substrate has another term right. So, it is produced back with this minus k plus plus k minus. So, that is my full set of equations. So, given a set given a chemical reaction you should be able to write down the corresponding rate equations. The solution of this is somewhat more complicated, but what I will do is that I will make an approximation just to give you an idea. So, let me say that I make this approximation that I say that this again this sort of a quasi equilibrium that this enzyme substrate complex that is found that reaches some sort of an equilibrium. So, this sort of a quasi steady state quasi steady state approximation that this enzyme substrate complex reaches some sort of a steady state. Then what that immediately means is that I have the ratios of this from this equation the ratio of this e into s by this enzyme substrate concentration that is equal to k minus plus R by k plus and this thing I call as k m. This thing is a name it is called as Michaelis Mentenkoch Michaelis constant. And again if I say that this rate R is much much less than k minus then this k m is simply like k minus by k plus provided this R is small r small. So, then what I have over here is if I were interested in the rate at which product is forming is ultimately what I want to calculate how fast does this product form then I can just write this product equation dp dt dp dt the rate of formation of product was R times this enzyme substrate which in this quasi steady state approximation is e into s by k m is R by k m enzyme time. Let me write the maximum of this maximum rate of this equation as some V max. So, let me call V max is the maximum rate of this equation and we say that this is R times. So, I am looking at this dp dt it is goes as this R times the concentration of the this enzyme substrate complex. So, if I say that all of my enzyme whatever I had was bound to this bound to the substrate and available for this reaction then that would give me this maximum rate of this reaction. So, that is R times the concentration of this total enzyme that I have. So, this gives me the maximum rate of this reaction I will just write it in terms of this. So, I will write this dp dt. So, I will write this dp dt I just want to cast it in some familiar form. So, V max is R is V max by d total R is V max by d total by d total into R into e times s. So, e times s by k m this is V max into enzyme concentration substrate concentration by k m by the total enzyme concentration which is e plus e s e plus. So, this total enzyme concentration again I break it up into two parts one is the pre enzyme concentration and the other is the enzyme substrate complex the concentration of the enzyme which is bound in the substrate. So, this e total I break it up into two parts and this e s I again write using this equation e s is e s plus k m. So, this is again e times s by k m. So, effectively if I if I now cancel out this ease what I get is that this product the rate of formation of product that goes as the rate of formation of this product dp dt goes as the maximum possible rate into s by k m k m divided by 1 plus s by k m. This is a very famous equation this is called the Michaelis-Menten equation this is called the Michaelis-Menten equation or the Michaelis-Menten kinetics. And basically you can interpret this k m this Michaelis constant as the concentration of the substrate at which you get half maximal rate. So, if k m was equal to if s is equal to k m then you get a half over here 1 by 1 plus 1 which will give you a V max by 2. So, a way of interpreting this Michaelis constant is that it is the concentration of the substrate at which you get half maximal rate of this product formation. So, if you look at so, if you now solve this equation and if you look at what it gives this is basically the rate of formation of this product as a function of yeah. So, this is the rate of formation of this this p as a function of time and this is the fraction converted. So, this is p by s. So, let me first look at this these are these various normalized concentrations. So, s divided by s naught e s divided by e naught e by e naught and e by s naught. So, these are all scaled by these concentrations of the initial times. So, this is what it looks like. If you look at this enzyme substrate complex which is where we made this quasi-steady state approximation it is sort of varies of course, but it varies at a slow rate compared to these others which is why we make this sort of a quasi-steady state approximation. So, this is valid not in in this sort of a regime this approximation that we have made and this dp dt grows something like this. So, at the max so, this is scaled by this v max and if I if I look at this so, this is this dp dt again as a function of this substrate. So, that is product as a function of time this is actually dp dt as a function of this substrate concentration. So, the maximum rate possible is v max which is what I have and at this half maximal rate. So, wherever it becomes half that gives me the appropriate value of Km. So, wherever it is 0.5 that gives me the appropriate value of Km. So, v max is whatever around 0.16 then around 0.08 over here is where my Km would be that line has not come. So, the corresponding value of Km for whatever this reaction is some I do not know 0.01 millimole of the sum. You could of course, solve this full set of equations explicitly which is how one gets these curves. I just wanted to cast this product formation equation the rate at which you get the product in this form because this is often a form this Michaelis Menten form is often something that you see referred to in various equations that this rate of product formation goes. So, this is again like this hill function like of a behavior it is s by 1 plus s with some pre factor which is the maximum rate of formation of the product. I think I will stop here for today. What I will do is that I will use this language of rate equations. So, this is sort of very generic you can use it in biological systems, but also in any sort of chemical reaction that you are interested in. But what I will do is that I will use explicitly this sort of a framework to look at cytoskeletal polymerization of microtubules and of actin starting from the next class.