 So, moving on from spatial scales this is quickly take a look at temporal scales. So, let us see again I cannot read most of these things over here is evolutionary time scales at 10 to the power of 15, 10 to the power of 18. So, this is diversification of humans and chimpanzees over there you come down this is 72 hours which is the cell line doubling time. This is an E. coli doubling time of 20 minutes. So, when that is basically the cell division time or the cell lifetime protein half-life sort of the order of minutes. This over here is the scale of around protein folding. Let us see these are enzyme turnover rates which go from milliseconds to seconds and over here are very sort of chemical time scales side chain rotation hydrogen bond rearrangements and so on covalent bond vibrations which occur in the scale of nanoseconds to picoseconds femtoseconds. So, all of the in principle all of these time scales taken together would constitute well let us forget about evolution, but at least as far if I cut it off as the level of a cell this whole range of time scales constitute processes that are integral to the proper functioning of the cell. So, if the cell takes is in non-equilibrium at some time scale over here and you are looking at processes which happen over nanoseconds and so on you could use equilibrium approximations to deal with those processes. So, here are some examples of different time scales again taken from a random assortment of systems. So, here is Drosophila development Drosophila is the common fruit fly it is again it is a model system in biology just like E. coli Drosophila is another model system. So, you start from an egg you go to a larva to a pupa to an adult fly this whole process takes around 9 days 9 to 10 days. If you look at the early development when this egg is just sort of starting to differentiate you are starting to get all these wings and this vertebrae divisions forming that takes place over a time scale of hours. So, this thing forms roughly over 10 11 hours and this whole development from the egg to that adult fly takes place over a period of days. So, these are completely different time scales and again the physics that you would use to describe these sort of processes would again be tend to be very different. Here is a nice experimental movie which shows the development of Drosophila this particular one shows the early development. So, this is in this stage over here where you. So, Drosophila is slightly special in that in this initial stage you do not have cell divisions you just have nuclei which are not separated by cell membranes. So, here is my Drosophila embryo you have nuclei over here and these nuclei sort of divide. So, this one nuclei maybe becomes two nuclei, but there is no cell membrane surrounding these nuclei initially after some point the cell membrane forms and you get cellularization. So, this first video is in the stage where you just have these different nuclei which are dividing. So, this is at that stage. So, this is I think 1 hour 35 minutes past the fertilization of the embryo. These blue things are individual nuclei you will that was a cell division cycle and you will see that the number of nuclei has increased the system gets denser and denser right. Also very interestingly you see that this cell division sort of propagates as a wave. So, the color. So, the color the cells depending on when cell division was happening. So, at the time of cell division they color the cells by this magenta color and you will see that the cell division sort of progresses. So, this is one end this is called the anterior end of the embryo that is the posterior end of the embryo. So, the cell division wave sort of progresses from this anterior end to the posterior end and this every time the cell division happens the number of nuclei is becomes twice what it was earlier. So, this is one very nice example of what I meant when I said that this whole field of mathematical or quantitative biology has become possible due to this extremely extremely difficult sophisticated and yet very illuminating experiments. So, here you can track each individual nuclei of this embryo right this whole this will become the whole organism. You have the positions you have the velocities of each individual nuclei. So, you could do a very microscopic level modeling in trying to understand what sort of processes are causing this cell division to happen or sorry this development of the Drosophila embryo to happen. So, that is the clock over there. So, this is very early times if you go to slightly later times. So, this is again I cannot see this is around the 3 R mark I think and these are two different views the dorsal view and the ventral view of the same organism as it differentiates. So, here it is here at this stage the cells have already formed these now these individual dots of the cells they are no longer them nuclei you have formed a cell membrane and as time progresses you will see different features sort of starting to become clear within this embryo. So, you can see the segmentation pattern starting to form in this ventral view right. So, over here this let it play this is the other the ventral this is the dorsal view sorry very see from the bottom and if you let it play it will go on to develop. So, again in this case you have exact information about each individual cell of this whole organism right the positions the velocities is a function of time. You can see how these different cells flow from one point to another. So, this is basically at roughly this larval's this pupil stage rather no not the pupil stage the larval stage. These are very fascinating experiments relatively recent to the last five six years where it has been possible to sort of image each individual cell or each individual nuclei at that resolution and trying to understand what sort of processes are going on all right. Coming back to this array of time scales. So, if you look at cell division what time were you asking about so much? Selflessly ok we will come to that. So, if you think about cell division like E. coli cell division that takes place over a time scale of minutes. So, around 30 minutes is when the mother cell is going to divide into two daughter cells. If you look at cell movements this E. coli moving with the help of its flagella and this is what a typical trajectory of an E coli would look like that happens over a time scale of seconds. So, it moves when it moves like this in a directed fashion all the flagella are bunched up together they move beating in synchronously the cell moves it reaches one point the flagella sort of let go of one another it sort of tumbles around for a bit until it chooses a new direction and again it moves in that direction if you look at the tracks it just looks like random walk trajectories and this happen this process happens over a time scale of seconds. If you look at protein synthesis that takes place over roughly a second. So, this is 0.1 seconds 0.5 seconds 1. So, this is the ribosome which comes in attaches to the mRNA the transfer RNAs come and feed in the correct amino acids as the amino acids get fed in the protein gets better spit spit out and that happens over a time scale of around the second roughly. If you look at transcription so, an RNA polymer is coming on to the DNA and producing the same RNA this growing mRNA transcript that happens over 0.10 of a second. So, 0.4 seconds is what is given here, but roughly of that order. If you look at ion channels remember ion channels of these objects which are embedded within this lipid bilayer they open and close and when they are open they allow ions to pass through that happens over a time scale of milliseconds 0.001 seconds. So, this is very fast process compared to these slower scale processes of synthesis and so on. You could also talk about faster processes where we are leaving more getting more into the chemistry aspect of it for example, enzyme catalysis that takes place over a time scale of around microseconds. So, both of these are proteins substrate and enzyme they come together they do whatever they are supposed to do and this process takes place roughly around the microsecond. It turns out I do not have capsid assembly over here. I think capsid assembly typical time scales is roughly of the order of minutes if I am not mistaken. I will check once more, but roughly I think of the order of minutes I thought I had it over here, but apparently I do not. And in fact, you can show that if you if you just took this proteins of units which make up this viral capsid and you let them be. So, you take these proteins of units and you let you just put them together and you let they will also self assemble actually into they will form whatever like a nice capsid like this. Turns out that the time scales of that are much slower and in order to get it to the biologically relevant time scales you have to introduce this electrostatic forces. So, if you introduce electrostatic forces between these charged proteins in this viral capsid proteins and the DNA that this capsid is going to encapsulate which is negatively charged you can show that you will get to the right time scales roughly. This process of cell division. So, here is a movie again forgive the poor. So, here is a process colony of E coli cells dividing started off from two cells they divide and divide and divide until it sort of fills the petri dish that you had. You can take the frames of this movie and you can analyze that and you can plot the area that is covered as a function of time. So, this is the frames of that movie you sort of calculate what area is covered by these cells and you plot that area is a function of time this x axis is time in minutes and here is the sort of plot that you get. You can calculate roughly a doubling time which is that how fast does this area double double and from this plot it comes to roughly around 45 minutes. So, this is sort of geometric growth right you get you started off with two two bacteria that becomes four that becomes eight and so on. So, I could write if I would write an evolution equation for the number of bacteria how that changes as a function of time. So, I have n number of bacteria at some time again I will try to be very naive and try to write the simplest thing that I can and I will say that well let me say that dn dt will grow depending on how many bacteria you had to start off with right because it is doubling. So, let me say it grows as something like this r times n right. If I wrote an equation like this dn dt is equal to rn and I gave you this curve of this experimental curve which says that my doubling time is roughly around 45 minutes. Could you estimate what sort of an r this implies this data implies how would you go about doing that what is the solution of this equation what is n of t exponential. So, what is this r then the doubling time. So, r is basically log 2 divided by the doubling time. So, you can try even if you do a very simple equation like this you can try to say that well you can explain this data that I have that I observe in this experiment. If I provided I take this rate to be log 2 over this doubling time that I have observed this called Neinhard's equation it is in this bacterial growth paper it is of course, a very naive model it is of course, at some point this model will fail when will this model fail when it become when the numbers become very large and whatever agar or whatever food that you have put in the petri dish it cannot sustain an infinitely growing colony. So, at some point this rate of growth must slow down right. So, it cannot keep growing like this. So, at initial times it might grow very fast, but as you sort of reach the limits of the population that you can sustain that growth is necessarily has to slow down. Can I write down a simple equation that will correct for that they have come across an equation like that yes minus something that is about it what is this sort of an equation logistic equation. So, I can write down a logistic sort of growth model for this typically it is written like R n 1 minus n over k where k is called the carrying capacity or the maximum population that you can sustain right. So, where basically when n reaches k you will see the dn dt goes to 0 right it will not grow anymore when n is much much smaller than k then this dn dt is like R n which goes back to this Neinhard's equation. So, this is called the logistic equation and then you can see how this equation will look like. So, if you look plot the number as a function of time this is what will look like initially it will grow very fast, but then after some time after some time it will as it reaches the carrying capacity the growth will first slow down and then entirely stop this n of t will as t goes to infinity this n will just tend to k. Talking from a non-linear perspective this equation has two fixed points one fixed point is at 0 if you did not start off with any bacteria of course, you would remain at no bacteria, but provided you started off with something. So, this is an unstable fixed point the moment you perturb away from this fixed point you will go to this stable fixed point which is this n star equal to k. So, that is the logistic equation you can solve the logistic equation you can get n as a function of time and then try to see whether that fits your experiment. This is rough. So, the basic idea is this is what we will start off with we will start off at looking at these biological processes growth movement and so on. So, for next class I think we will start with diffusion and movement and we will try to see what is the sort of simplest model that we can write down. We will try to analyze that model and see what are the shortcomings of that in what regimes are those models correct in what regimes are they not and then can we do anything better than this. So, for example, this R n was the simplest possible thing that we could write it works in a certain regime when the number of cells is small it will fail at a certain regime when the number reaches close to the carrying capacity and then you would need to readjust your model ok. So, that is the sort of spirit this is a very trivial example of course, we will try to do slightly more complicated things as the course goes on. So, this sort of glossary of terms what are the molecules what are the numbers and so on those are roughly taken from these first two references Rob Phillips chapters 2 and 3 and Nelson's chapter 2. If you are interested even more these two books Albus and Loddish these are sort of the Bible for molecular biology these have all this information in a lot lot more lot greater detail and depth. So, for those of you are interested you can take a look at these books over a long period of time these are really thick books. So, ok. So, I think I will stop here today I will start off with movement and diffusion starting with next class and how to do modeling of diffusive processes in biology good. So, I will see you again on Tuesday then and we will sort of get into it proper.