 So, before I will do some simple calculations, but before I do we will just sort of emphasize once more how and when equilibrium considerations might be appropriate. We know overall all biological systems are non-equilibrium, but in certain cases one might get away with an equilibrium approximation. So, we will sort of discuss how to ascertain whether an equilibrium approximation is good or not. Before we do so, let us just start. So, energy inside cells comes in various forms of course, what we have been discussing is in some sense the manifestation of thermal energy right diffusion, viscosity and so on. So, Brownian motion and so, all of these these are manifestations of thermal energy, but there are other forms of energy that are relevant to a living system. For example, chemical energy right ATP is form of chemical energy. So, you store chemical energy in ATP, use the ATP hydrolysis to get that useful energy out and do some work. What other form of energy would be relevant? Electrical energy ok. Let me just generalize it and say electromagnetic. Where would that be applicable? Nerves. Nerves ok that is good. What else? Pressure. So, let me say mechanical energy and then one can look at different manifestations of these. For example, electromagnetic energy one of the most basic manifestations where electromagnetic energy is important is for example, photosynthesis right. You take in sunlight you convert that into useful energy which is basically the basis of all energy for life forms on a plants do that and then different things eat plants and eat different things and so on. Also for example, these fireflies which are glowing in the dark these also use stored chemical they convert stored chemical energy into this light energy and they blink and they give up light. So, this is sort of the reverse of this where you take in light and store it in the form of chemical energy. This fireflies or other bioluminescent animals in the sea and so on you convert chemical energy into a form of light. Chemical energy we talked of all this ATP, GTP and so on. You go from an ATP to an ATP and in the process you get energy. Mechanical energy for example, you can think of these muscles flexing in your arm or even molecular motors which are pulling on different proteins. So, in this case this is a myosin motor which is acting walking on an actin filament and pulling an actin filament and then of course, there is thermal energy of diffusion random books and so on. So, that is ok. So, that is to do with energy. Before I actually get into equilibrium and so on, let me just. So, energy we are at least all familiar with. What is entropy? What is the micro what is the statistical definition statistical mechanical definition of entropy from a micro canonical ensemble? It is KB times log of omega right where what is omega? Number of microstates that correspond to a given macrostate. So, this is the number of microstates. What is temperature? Yes, I am going to forget what is temperature 1 by T is del S by del E. What is the Helmholtz free energy minus KB T log? Good. What is what is the partition function? Z. It is the sum over all states e to the power of minus beta times E of a state. Then this F is minus KB T log Z or you can also write this F is E minus T S. It is just good to keep in mind in the manner. So, here for example, two processes which are definitely non-equilibrium. Here is a worm which is crawling across the surface. Here is an RNA polymerase which is reading a DNA molecule and producing the corresponding mRNA. Both of these are reliant on energy. The RNA polymerase uses energy to slide along the DNA. The worm of course, uses energy to crawl along the surface. But depending on the question that you are asking the non-equilibrium nature of this overall process may not for example, be important. For example, if we ask that well given certain concentration of RNA polymerase molecules in the cell, what is the probability that the RNA polymerase will be bound on to the DNA? So, here is my DNA. Here is my double-stranded DNA. It might have different binding sites for RNA polymerase which are upstream of the transcription gene sites. So, RNA polymerase can bind, come and bind here and then it can read this DNA sequence and produce an mRNA. This sliding is along the DNA is active or a driven process which requires anything. But for example, if I am not interested in this full problem, but if I just want to ask that what is the probability that an RNA polymerase molecule is bound to this DNA backbone? Given a certain number of for example, binding sites, given a certain concentration of RNA RNA polymerase and so on. This question as we will see that is one of the questions we will try to answer. You can sort of deal with within an equilibrium approximation ok. The probability that a certain site is occupied by an RNA polymerase is can be obtained or can be predicted fairly well within equilibrium approximations. Similarly, for example, if you are looking at this sort of deformation of the cell shape, underneath the structure of this form you have whatever you have this cytoskeleton which is composed of actins and microtubules and so on. These exert forces on this cell wall on the cell membrane and they can lead to deformation of this deformation of this of this membrane. The forces that are generated by the cytoskeleton are explicitly an active process. These are driven by motors which walk on these filaments and generate forces. These, but if you are looking at the structure of this the shape of this you might say that well if the time scales of if this feels a force it responds to this force and assumes some shape. If the time scale of that process happening is very different from the time scale of this motion of this organism of the cell as a whole, then we can consider this as an sort of equilibrium problem. I can say that well I apply a force what would be the shape that this membrane assumes ok. So, again if you look at some sub part of the problem where the time scales are different from the time scales where these non-equilibrium factors become important. It is often reasonable to use a use an equilibrium approximation. To show this you can explicitly consider sorry let us start give an example of a chemical reaction. So, for example, if I take a chemical reaction like this right I have a going to b with some rate k plus, b converts back to a with some rate k minus and it also goes to c with some rate r there is no reverse thing over here. So, for example, this could stand for consider this as this RNA polymerase thing only. You could have DNA plus RNA Paul going to this complex formation where this DNA is bound to the RNA polymerase right. This is inter convertible the RNA polymerase binds and unbinds right. So, it is a reversible process it goes forward and backward with some rates right. But this one will also give you with some rate r let us say the corresponding mRNA plus the DNA plus the RNA. So, it is something this is of course, it is not to say that this is just represents this process this is just an example you could consider many other reactions which is something like this. Now, if the these rates the time scales that are associated with these rates k plus k minus and r are very different. For example, if this k plus k minus this A to B interconversion is a very fast process whereas, this B to C which is given by this rate r is a very slow process ok. If you would consider something like that and then if you try to solve that if you try to see how these molecules will change the concentrations of these molecules will change with time that will look something like this again you take this is a given we will solve it along the way as we go along. So, for example, this red is the concentration of A this B is the this blue is the concentration of B and the C is the concentration of this green is the concentration of C right. So, in the limit where these interconversion of A and B are very fast processes right k plus and k minus are much faster than this rate r, A and B will come to an equilibrium amongst themselves ok. This does not mean that A becomes constant with time as you can see the concentration of A changes with time B also changes with time and therefore, C also changes with time that happens over a much longer time, but over this very very quickly over this very short time scale A and B have come to an equilibrium amongst themselves which means that the ratios of A to the ratio of B assumes a constant value that constant then changes with time, but A to the ratio of A to B stays constant ok. That is a reflection of the fact that you have very two very separate time scales in the process. One time scale which is given by 1 by k inverse k inverse k plus or k minus which is this time scale let us say 2, 3 seconds. The other is the time scale which is given by r inverse which is a much longer time scale say 50 it is 100 seconds or so on ok. So, if you have well separated time scales like this then and if you are interested only in a sub part of this whole process if you are only interested in this A converting to B sort of B process and not this whole B going to C thing then you might be ok with considering this A A inter converting to B as an equilibrium process and then using equilibrium arguments to calculate statistical quantities associated with that. This depends of course, on whether these time scales are well separated if this k plus k minus was not the much faster process which is given in this right panel then of course, it does not come to an equilibrium you will see that A you know varies on its own B which is the blue curve varies on its own C varies on its own and so on ok. So, the same so, depending on whether you can separate your time scales for these sub processes you can consider these sub processes as part as an equilibrium under an equilibrium approximation ok. So, that is that is the basic idea it always or mostly has to do with separation of time scales if you identify time scales that are widely separated and one is much smaller than the other then you might be justified in using the equilibrium approximations. What can you do with such things? One of the classic problems equilibrium problems that one attempts is finding the folded structure of proteins. So, you start off with the chain of proteins which is some amino acid sequence and then the protein manages to fold into this final native structure or the functional structure very quickly. So, some 2500 or 3000 nanoseconds. This is still of course, an open problem it is an extremely hard problem as to how given a sequence you can end up at the structure, but we will again look at how to at least start thinking about this sort of problem. Similarly, another example where you might consider free energy minimization is to do with gating of ion channels. So, the ion channels are these channels on the membrane of a cell which lets in ions. So, here is my cell, here is an ion channel, here is an ion channel and so on. And you can have different ions for example, potassium, sodium which pass through these channels. These ion channels can be open or closed if they are closed they do not allow ion exchange if they are open they allow ion exchange. For example, here is one mechanism which you might which is one of so cells control the opening and closing of these ion channels in different ways. So, for example, this on the right hand panel is a way where you control it using mechanical tension in the membrane right. So, here is a here is an ion channel, here is the membrane in which it is embedded this is remember the lipid bilayer membrane. Depending on what forces or tensions you apply on the membrane you will result in a different free energy. So, for example, here is some membrane which has an ion channel which is closed let us say ok nothing can get in. Due to some reason you apply some you apply some tension on this and maybe it switches to a by a free energy minimization under the effect of this tension maybe the shape changes to something like this where the ion channel becomes like so. So, the structure changes from here to here and it goes from a closed to an open state. So, again you can think of this in terms of a free energy minimization in the presence of some forces or you could have voltage gated ion channels where you change the voltage across on the two sides of the membrane and that again causes the ion channel to go from a closed state to an open state. Then of course, we have discussed this earlier the shapes of red blood cells is a classic problem where you can get the canonically observed shapes or close to the canonically observed shapes through a free energy minimization. You write down the Hamiltonian which takes into account the physical forces that are involved bending surface tension and so on. You minimize that free energy you minimize that free energy for different values of these parameters which might be the bending energy strength the surface tension strength and so on. For different strengths of these parameter values you get different shapes as the minimum of the free energy and you can show that the shapes that is these different shapes that you obtain closely correlate with the shapes of red blood cells that are seen in vivo. So, underlying all of this is the competition between energy and entropy. So, this free energy encompasses two terms the energy term E and the entropy term minus Ts. So, for example, this is a simulation of a polylephalite which is so let me just describe yes. Can you see the protein folding happening in real time? I do not think so right. You can get extra crystallographic structures which are the folded structures and then you can reconstruct at least you can get that for small domains and so on you can reconstruct the 3D folded shape. I do not think you can see the as the sequence folding into the shape in real time. You can see that well you cannot see see, but to a certain extent maybe in molecular dynamic simulations if you trust your force fields that they are the accurate force fields up to a few nanoseconds for not too large proteins you can see how it starts folding, but there is no guarantee that that is how the cell is doing it. So, for example, here let us see I take a polymer it could be any charged polymer DNA or even proteins so on. So, which has some backbone. So, these are the monomers of my polymer and these are these have some charged groups sitting on them right let us say negative charge is sitting on them. If these so, these are called the counter ions. So, these are my electrons these are my counter ions. One could ask that what is the what is the effective charge of this polymer. So, for example, if with each monom if each monomer is plus 1 it is associated with the counter ion which is minus 1 overall this polymer is neutral right. But this counter ions can dissociate and go in solution right when you put it in solution this counter ions can dissociate with leaving a polymer which is say very strongly positively charged and the counter ions are all floating around freely in solution. How much of these counter ions will so, how much of these counter ions will dissociate this sort of a competition between energy and entropy. So, for example, if the counter ions sit here then it gains some energy because there is an interaction between this plus charge on the backbone and the minus charge of the counter ions right. So, it lowers its energy by sitting here on the backbone which is good it wants to lower its energy. But on the other hand if it goes into solution it can increase its entropy right because here it is fixed over here it does not it cannot explore around. If it goes in solution it has this whole solution space to explore around. So, it gives up some energetic it gives up some favorable energetic cost which by associating with this plus, but on the other hand it gains some entropy cost. So, which or how much of these counter ions will go into solution then is determined by this free energy which is E minus Ts right. It is a competition between the energy and the entropy. So, for example, in the simulation these are two polymers where carrying opposite charges and initially the counter ions were very close to the polymers. What will happen is that this polymer will come in and it will display it has a stronger binding affinity to these to this first polymer and eventually it will displace all the counter ions which will start going into solution and these two polymers will form a complex. But this is also true for example, in the case of proteins. So, for example, if you have a protein right the amino acids the amino acid groups on the protein can be charged or not depending on what is the pH of the. So, let us say I have some folded protein which have some amino acid residues depending on the pH of the solution you can change the effective charge on these amino acids right. So, depending on the pH you can change the effective charge on these amino acids once you change the charge you might also change the structure right because that will change the forces that are felt by different parts of the protein and therefore, you will get a different free energy minimum right. So, that is why often you will see that there is not a unique need of state for proteins if you change the pH of this of the environment that this protein is living in it can go from one folded conformation to another ok. So, these would be yes.