 One more a different a different sort of motor that you could think of. So, these were polymerization sort of motors. A different sort of motor that you could think of are these translocation motors for example, you talked briefly about this for example, if you are packaging DNA inside a vital capsid screen is very bad. But for example, let us say if you are packaging DNA inside a vital capsid this is again an ATP dependent process where ATP has to come and bind and then you push this DNA inside this bacteria phase. Or similarly in reverse when a when a virus is trying to infect something let us say a bacteria it has to push out the DNA into the bacterial cell. So, for example, this is actually an experiment on this pushing out DNA from viruses vital capsids. So, these points are different viruses and you will see that these vital capsids are being ejected. This is this is stuck for sometime it will get ejected ultimately and once it gets ejected. So, there is a flow this is a solution this is this is a solution where you take various many many viruses and the virus sort of ejects its DNA and once ejected the DNA will zip out. So, you can actually. So, this is experiment, but you can actually measure speeds in these sort of experiments. So, these are what we call the translocation ratchets. So, if you measure speeds under different conditions. So, these are for example, experiments on two different viruses lambda C 160 and lambda B 221. This is DNA of different lengths this is 48.5 kilo bases this is 38 kilo bases and what you are seeing is the ejected DNA as a function of time. So, these are time traces of the fluorescent DNA as a function of time under two different salt conditions. So, one is a monovalence salt NaCl another is a divalence salt magnesium sulphate. Again because these DNA are charged depending on the amount of salt that you put you can control the sort of speed with these with which this DNA is ejected. So, with NaCl it is sort of with monovalence salt it sort of gets ejected very fast very quickly all the DNA is ejected it reaches 48.5 whereas, with divalence salt it is sort of very slow. The x axis is time seconds. So, you can measure the sort of velocities. So, this is what is plotted over here the DNA instantaneous DNA velocity as a function of the amount of DNA that remains back in the capsule. And you can see that the salt has a major effect on the DNA velocity the salt concentration. The length of DNA not so much. So, these two plots are for these two DNAs the two viruses. So, the length does not play it plays some role, but it is minor on the other hand the salt plays a major role. So, if you were to sort of model something like this that I have let us say a cell which is being infected by this virus. So, it needs to push its sort of DNA inside over here can I come up with estimates of the speed in this case in this sort of scenario. And again if I do a 0th order modeling. So, often what happens is that if I have this sort of a translocating polymer. So, this is how it is going in let us say and this DNA often will have spaces where proteins can bind. For example, ribosomes can come and bind to start the transcription process. So, the idea is this that I have the idea is this that I want to sort of get this thing in into this cell let us say. I could just try to diffuse it in and it would take some time. So, if this polymer is of length l it would take some time l squared by d to diffuse this whole thing in. On the other hand if it has sites for binding proteins every some distance every d distance then once the protein comes and binds this thing cannot slide back out anymore. Let us say a large protein comes and binds like a ribosome then that can the DNA cannot slide back anymore. So, that will then speed. So, once we have these proteins binding over here that will then speed up this translocation process and you can estimate by how much. So, for example, the diffusion. So, if this length of the polymer I call it as n times d n being the number of binding sites and d being the space between these binding sites. So, l if I just write as n times d then the diffusive timescale for this whole polymer is n square d square by capital D of the order of. On the other hand if these things bind every d base pairs or every d sites whatever then the time taken to diffuse one unit of d base pairs is basically d square by d. And once it goes in the assumption is that a protein comes and binds and it cannot come out anymore which means and there are n such sort of segments which means the time in this case is n d square by d and in this case it was n square d square by d ok. So, you get a speed up of around factor of n n being the number of binding sites. So, diffusive timescale grows as n square this translocation timescale with the help of these ratchets this binding proteins grows as n. So, you can get significant speed up simply by this sort of a simple mechanism yes because the virus for example, for the case of a virus the virus wants to infect the bacteria right. So, the way it infects is that it injects its genetic material into the bacteria and it injects it through these very small pores on the membrane like ion channels. So, there are not ion channel, but there are pores on the cell membrane. So, it has to inject its DNA through these pores ok. So, if it were to just inject by diffusion that would take a long time on the other hand through some mechanism like this it can be injected at much smaller timescales ok. It is some salt the buffer salt that is being flowed from left to right. So, which is why once the virus gets ejected it just lifts through because there is a background flow of this salt in the solution ok. So, we can do this a little more formally again this sort of a calculation. So, to formalize the model again what we say is that I consider a rod that is diffusing through a pore. So, I neglect the polymer degrees of freedom I just say it is a straight rod that is diffusing through the pore. I say that the binding sites are de-distance apart proteins are present only on the inside not on the outside ok. The binding is irreversible. So, again infinite k on 0 k off in that sense and this binding implies that the protein has this polymer rather has translocated by a distance t an irreversible translocation because this does not come off. So, it cannot come back out over there. And then I could ask therefore, what is the translocation velocity? What is the velocity with which this polymer gets pushed in? So, again I can frame it in terms of the kinetics of a binding site that is last emerged from the pore ok. So, a binding site that is last emerged from the pore. So, I call p x t the probability that this last site is between x and the x at time t and again I can write down a drift diffusion sort of equation. So, the flux of protein binding sites is like this there is a diffusive term and there is an advection term if there is some sort of a force. So, for example, the force again could be some ellipso static force it could be some other force that these proteins exert, but let us say there is some force that is exerted on this polymer. In that case I can write down a current like this. So, again I am going to solve the diffusion equation basically I want to solve del p del t is equal to minus del j del x with j being given by minus d f times k b t times the probability minus d del p by del x. This is again like f by gamma gamma is k b t by and I put the boundary condition again that that p d comma t is equal to 0. So, again the moment it reaches d sites a new protein will bind and it will come back to 0 exactly like in the other case. So, you can solve again you can again solve this in the steady state. So, let us say this is some steady state current you can solve this equation for p the probability in the steady state. So, p s s of x and again I will just write that down. So, this is some steady state current k b t by d f e to the power of f d minus x by k b t minus 1. You can normalize this to find what is J s the steady state current and from there again you can find out what is the translocation velocity. This is very similar formalism to what we did for this polymerization ratchets just. So, this steady state current if you normalize it comes out to be 1 by d square by d k b t by f d whole square e to the power of f d by k b t minus f d by k b t minus 1 which means that your translocation velocity the translocation velocity is thus the steady state current times the distance that you have covered which is the distance between binding sites. So, that is nothing, but d by d f d by k b t whole square and then this object e to the power of f d by k b t minus f d by k b t. So, again you can find out the translocation velocity in this sort of a model. In the presence of some sort of a force f positive is an assisting force if you have a repelling force and f would be negative and accordingly the translocation velocity would be. So, you can have all of these sort of different things molecular motors, polymerization motors or even these sort of translocation motors all of which depend on ATP, but they do very different things. And for these different objects you can calculate these different quantities like velocity of translocation or the velocity of polymerization for example, using this sort of advection diffusion formalism that we learned way back. I think I will stop here for today. There are many other things that I did not cover for example, I did not do rotary motors which I said I will, but I do not think I will have time. So, if you are interested in these sort of flagellar motors the rotary rotary sort of motors you can look up Phillips as a good discussion on this sort of rotary motors. What I will do for the next two or whatever classes remain is that I want to move on to these reaction diffusion systems and see how patterns form in these reaction diffusion. So, we have done chemical kinetics. So, reactions we have done diffusion. We will put these together to look at reaction diffusion systems and how patterns form via these reaction diffusion systems and how these patterns affect various biological processes in particular development of organisms, early embryonic development of organisms ok. So, that is what I will do for the next topic of this session.