 Hi all, I am Ranjit Padignati. I am a faculty in the department of Bioscience and Bioengineering at IIT Bombay. I will be teaching this course, Biomathematics. So, essentially this courses will deal with various mathematical methods that is used in life sciences. So, we will take mainly, we will mainly concentrate in biology. So, different mathematical methods that is needed and useful to understand and explain various biological phenomena will be discussed in this course. So, the first lecture of this course is essentially an introduction to Biomathematics and we will discuss why we need mathematics at all. So, that is the first lecture is our introduction. So, the question, when we say mathematics for biologists or mathematics for life sciences, you will have this question that why at all we need mathematics to understand biosciences. As you all know, we can do certain experiments. We can understand results from that experiment and reach some conclusion. Isn't that enough? Do we need to know any mathematics at all for studying biology? Conventionally, we do not learn mathematics in biology. For example, BSE biology or in biological courses. So, what we want to first discuss is why we need mathematics at all. So, the idea is that mathematics is like a language. It is like a language to understand scientific natural phenomena, to explain natural phenomena and using mathematics, we can describe also biological phenomena. So, as we go along this course, we will. So, learn many, many things related to why we need mathematics for understanding biology and today we will discuss some examples where mathematics is used and in the next other parts of the course, we will describe more and more examples or biological phenomena where mathematics is helpful to understand it and we will learn the mathematics behind all this phenomena. So, when we say mathematics, the first thing comes to everybody's mind is some set of equations. So, mathematics is always related to equations. So, what are these equations? So, essentially equations are like some kind of statements. So, they are much more very precise statements. So, when we say statements in English, we make lot of statements. So, to describe an event that we see in a day-to-day life, we use English or some language, English, Marathi, Hindi, all kinds of languages we use in India and this can describe some event that we see in day-to-day life. But mathematics also then why we need mathematics. So, the advantage of mathematics is that you can describe things in a very quantitative way, much very precise way. So, let us take an example. So, let us say you want to understand a biological phenomena of bacterial growth. So, you are doing an experiment of growing bacteria. So, you have a patchy dish and you have some bacterial colony which is growing. Now, you see the you record this experimental observation that this bacterial colony is growing fast or bacterial colony is growing slow. That is all we can say with a language. How fast it is going? More precisely, what is the speed with which it is growing? And so on and so forth can be much easily described using mathematics. So, this that the statements like growing fast or growing slow etcetera are very qualitative statements. On the other hand, using mathematics one can make much more precise, much more quantitative statements. So, let us go and take an example of bacterial growth itself. So, this is one equation which is valid in some particular regime of bacterial growth that is n is equal to 2 power k t. This is an equation which is valid in one of the phases of bacterial growth. Now, what is this equation convey to you? So, this equation essentially tells that how the n which is the number of bacteria at a particular time t. So, what is the how many bacteria are there in this battery dish at a particular time t is what this is describing. So, using this, this is just one equation like what were like 5, 6 letters. One can convey a lot of things like this already conveys what is the speed with which it is growing. After 5 minutes, how many bacteria are there? After 10 minutes, how many bacteria you can find? So, many, many lot of information is embedded in this one little line. So, this is the power of mathematics where you can you can convey a lot of information in just very tiny state equations like this. And we will our aim of this course is to understand this equations. And if you do an experiment, how to express those experimental results in using appropriate equations or in the language of mathematics that is precise and quantitative. So, with this aim we will study, we will go and understand and study this course. So, let us say in this course we will learn how to use mathematical equations to make precise statements. So, now let us go and understand how and learn how to use mathematics in some certain examples. Let us go and take some examples. So, in biology what we typically do is we usually do lot of experiments. So, and as how do we represent this experimental data? For example, as I said we you measure the bacterial colony. So, or when you do some other experiments basically as an experimental data what you get is some table. So, let us say number of at a particular time how many bacteria are in there in the battery dish or something is growing or polymer is growing. So, what is the length of the polymer at a particular time? So, this is an experimental data. So, you have basically you will have basically time in one column and the length of polymer in another column. So, you basically you get a table in a lot of experiments as a result of the experiment you essentially at the end of the experiment you make some table, but a particular concentration you have this value at some other concentration you have some other value. So, you vary concentration in units in interval of 0.1 micro molar and you get some results. So, essentially you get a table at the end of each experiment and you plot this table as a graph. So, basically experimental results are typically presented as a graph and not a set of statements and why is it? Why are we presenting the tables? Why do not we present just this table and not any graph at all? For all experiments why do we need graph? The answer is that the graph can convey much more information than a table can convey. So, that is the idea. So, basically this is the and the graph is much more quantitative. So, instead of more than what we say in a set of statements the graph can give much more quantitative graph can convey much more quantitative information in a quantitative way. So, that is why we use graph and this very idea of plotting graph is kind of the first step or first use of mathematics. Essentially, you are using while you are when you are plotting a graph, essentially you are using some ideas of mathematics already. So, graph and mathematical equations are very closely connected. So, as you know that most of the graphs in principle can be represented by a mathematical equation. So, actually all graphs in principle can be represented by a mathematical equation. The equation may be very complex equation, but however it is possible to write down an equation for almost all graphs that you see and the understanding that equation but through and by understanding that equation we can learn much more information about this data and about this phenomena and about this biological system. So, in this course as we as I said we will learn how to write down this equations and we will learn how to understand from this equation how to understand this biological phenomena in a better way. So, the next question obviously comes to mind is using mathematics how do we extract more information from experimental data. So, let us say you have an experimental data and how do we get more information that what we know from the graph. So, let us take a simple example of fish growth, growth of fish for example. So, this is some experiment done by very famous biologists many years ago. So, let us take an example that you have a lake and in this lake you are growing fish. So, you put lot of fish and then it is like a fish farm. So, your aim is that you want to grow lot of fish and sell this fish. So, you will be happy if you get lot of fish every day per day you want to get let us say this many kg or this many number of fish every day and then you will be happy. So, as a person doing a farming or growing fish your aim is that you should have maximum yield. So, the question comes to your mind is that how can we get maximum yield. So, let us say you are doing an experiment of this fish growth and you get a particular curve as we see in the graph here. So, let us see this graph this is a typical growth curve many of you might have seen already. So, what is plotted here is population density in the y axis versus time. So, as we start growing fish at the early stage you have very little population. So, by population density we mean how many number of fish per square unit area or unit volume. So, that is the population density. So, at the early time you have very small population density and the population density increases and as the time goes it reaches a maximum value and kind of saturates here. So, this is a typical growth curve which is applicable for many different kinds of phenomena including bacterial colony, yeast cell division and so on and so forth. So, let us take in this fish case now let us ask this question as what population density you will get maximum yield. So, to understand this that when do we get maximum field one has to understand the mathematical idea called derivative. So, this is a mathematical idea in the field of mathematics called calculus which is which can be used here to understand this question to answer this question. And by answering this question you can basically learn you can basically get lot of yield and you can also if it is applied to a different experimental context you can understand more about the experiments in the system. So, this is one example. So, as we go along in this course we will answer this question we will answer we will learn about derivatives and then we will have this we will we will get the answer to this question that where do we get maximum yield where do we get maximum speed and so on and so forth. So, this is one aim of this course for you to do the derivative and understand this experimental data in and out. So, the next aim is basically so now what we do next is let us see some examples some biological examples where a graph a curve we used to represent some data or some phenomena. So, we will take about 5, 6 examples all related to biology or our day to day life where we have some experiment or some inform some phenomena going on and this will be described using a curve. So, let us go to the first example. So, the first example is a seasons. So, as you can see in this graph here. So, whenever is something season comes to your mind what comes whenever you listen about seasons what comes to your mind is basically something in periodic nature. So, summer every year may is like the peak likes very hot very warm. So, it is may every year is like repeating year by year. So, it is like periodic in nature keeps repeating similarly rain at some particular time of every year the monsoon arrives in India and it rains and similarly in January every year it is winter. So, the temperature for example, this varies very low. So, if you let us take temperature as some kind of a measure for a season and we are plotting some temperature here with this is a very schematic plot of temperature as a function of time and see this plot at some point let us say spring you have some value of temperature. So, this is just to give you some idea I am not taking numbers here and none of these examples we have numbers as of now. I will give you some overview of this idea you I want you to get the idea of this graph how to represent a particular phenomena using a graph and once you get this idea we will put in numbers and try and extract more and more information from this graph. So, that is the aim. So, let us look back this to the graph and this temperature here in the y axis and time in the x axis. So, at spring at a particular time you have some temperature. So, spring is typically March and as the time goes it goes to May and all that. So, this temperature increases and this summer is. So, in summer the temperature is the peak of the temperature in summer and then in most part of India rain arrives at some point July or so and then the temperature somewhat decreases June July the July August temperature decreases and the temperature decreases further and at winter is the you have the minimum temperature and again the temperature increases. So, winter is like January. So, we have the minimum probably of the temperature and then temperature slowly increases and again you have spring March April. So, March in most part of India. So, this is the basic idea of this graph. So, this graph essentially represents temperature over seasons and we can get the we can you get the idea of periodicity in this. So, this is periodic in nature. So, let us go and see the next example. Next example is a biological example where let us say you have a molecular motor. You might have heard of molecular motors like kinesine, dining, etcetera. So, every cell has certain motors like tracks in a real life. They carry cargo from one part of the cell to the other part. So, now here we are looking at the motion of this motors. So, let us say let us have a look at this graph. So, what is plotted in this graph is average position of a particular motor called kinesine in y axis as the time progresses. So, at a particular time it is in a particular position and along the microtubule track it moves. So, the distance from let us say from the center of the cell the distance increases. So, the distance increases with time. So, this is a simplified version of the reality. So, but the aim here again is to convey you the idea that this phenomena of motor walking and the distance from the center of the nucleus as we go along the microtubule to the periphery of the cell the distance increases. So, the position the distance or position increases. So, this is the idea which this graph through this graph I want to convey you and we will learn the mathematics of this graph and we will get more and more information from this graph and similar graph in the context of real experimental in the real experimental context we will discuss similar graph and then let us see what all we can learn from this graph. So, again the next example is basically a free energy. So, in thermodynamics you might have heard of free energy and you might have heard that proteins can have two states states A and state B and similarly something else some other thing have two states reaction where A going to B and you might have heard that B is preferred because B has less of free energy. So, you might have seen this particular graph let us have a look at this graph. So, where free energy is plotted as of with some reaction coordinate and you have two states A and B and this typical shape of this graph you might have seen in various courses that you have studied and to understand this very well one need to understand some mathematics behind this why this particular shape what is the meaning of this particular graph what is the meaning of state A and B and all that to understand all this basically one need to understand the mathematics behind this. Again in general thermodynamics itself to understand thermodynamics itself one need to get some understanding gain some understanding of mathematics it is good to know some mathematics. So, that we can have a better understanding of the thermodynamics which is an important theoretical framework thermodynamics an important theoretical framework to understand biology in general and biophysics in particular. So, basically one need to to understand thermodynamics one need to know some simple math at least elementary mathematics and through this course we will learn all mathematics that is required to understand thermodynamics for example. So, now let us go to the next case which is basically the case of gene expression. So, let us say you have an experiment you have lot of cells and each of the cell you are measuring how much a particular gene is being expressed. So, let us take the example of each cell and take the example of gal gene or any any gene your particular your favorite gene and the question we are asking is how much this gene is being expressed what is a how much the amount of gene expression. So, some cells you can imagine will have lot of gene this gene expressed some other cell will have only very little of this gene expressed. So, there is some variability in this gene expression some cells will have lot of gene expressed some cell less as gene expressed. So, basically we are counting how many cells have this amount of gene expressed how many cells have lot of gene expressed how many cells have little gene expressed if we do this counting and we do this plot we will see this graph that you can have you have here. So, let us have a look at this graph where the x axis is the amount of gene expression and the y axis is the number of cells having that amount of gene expression. For example, so take this value around this peak. So, you have some some amount of gene expression and this peak means you have lot of cells. So, the number of cells is very large lot of cells with this particular value of amount of gene expression. If you go to the left and right of this peak the gene number of cells decreases what does it mean is that only few cells you have only a few cells with lot of gene expression and a few cells with little gene expression. So, there is an average gene expression amount and lot of cells have this many this much of gene expression. So, basically this again is a well known curve in biology in mathematics and to understand a lot of things one can learn by understanding the equation of this curve and the mathematics behind this curve one can learn much more thing about this experiments. So, now let us go to the next example which is basically force velocity relation. This is also a modern experiment done in a case of growth of actin or microtubule. So, let us say you have actin and microtubule going. So, let us think of a typical case in a biology where you have actin or microtubule growing against the membrane of the cell. So, as they grow they will exert some force take for example, the acrosome reaction where actin needs to grow and it has to protrude it has to go through some. So, it has to apply some force against some membrane. So, the more the rigid the membrane it is difficult to push this. So, imagine such a situation in a in vitro experiment. So, that is shown here in this cartoon here. So, you have a filament here and the filament is growing against a wall here and you are applying some force here. So, if you apply a lot of force F it is difficult for this filament to grow. So, now if somebody can plot there are many such experiments in such experiments one can plot growth velocity versus force. So, if you apply force lot of force the filament cannot grow. So, the in so that is clear. So, the growth velocity will be very small. On the other hand if you do not apply any force at all then the growth velocity will be large. So, this is what this graph shows and again by studying about this graph and the idea the mathematics behind this graph one can understand a lot about this experiment and one can understand in general about the mechanism of actin or microtubule growth itself. So, this is another example where one represents their idea something that we observe in our experiments through a graph and this can be studied through mathematical equations and as we go along we will try and understand how to try and study how to understand this experiments and the mathematics behind it and using mathematics how do we gain more insight to this experiments. So, the next experiment next experiment is measuring the membrane potential. So, you know that ions pass through membranes the pores of the membrane and this ion charged ions will cause some membrane potential and this is important in various different different contexts in biology. So, now and you also might have studied at some point that this membrane potential depends on the concentration of the ions. So, the concentration difference or the ratio of the concentration is more the more the membrane potential is. So, now the if we represent. So, if we do an experiment of measuring the membrane potential as a function of by varying the concentration of ions we will get some data and if we plot this data we get this particular curve as seen here. So, let us look at this graph again. So, here it is membrane potential on the y axis and concentration of ions on the x axis. So, you can see that as the concentration increases the membrane potential increases and kind of. So, this is some sort of increase like a logarithmic increase. So, this is called logarithmic increase in mathematics, but we will understand this what does this mean what does it mean to say it is logarithmic increase. So, for and what exactly what precisely it means we will learn as we go along this course, but at this point what I want to tell you is that this curve the curve that we see in this graph is an experimental results and one can learn a lot about this experiments and the phenomenon behind this by learning the nature of the curve the logarithmic nature and so on and so forth. So, that is the and this is one example in biology where this is another example where one biological experimental result is described using a graph. So, now again we saw growth curve. So, we had seen the fish growth again similar thing applies for yeast cell. So, you can get a growth curve for example, have a look at again here the number of cells versus time you get a typical growth curve again one can learn a lot of information by understanding the equations behind this the mathematics behind this and all that. So, now let us go. So, what we have learned so far is basically something called an idea. So, idea of a function. So, have a look at this slide. So, each curves that we saw here can be represented by a mathematical equation. So, we had about 6, 7 examples each of the and we saw different kinds of curves growth curve we saw we saw a force velocity relation we saw a position as a function of time we saw a concentration velocity. So, membrane potential as a function of concentration. So, all this can be represented by some sort of an equation. So, basically by representing this through an equation one can understand a lot about this experiments on this phenomenon. For example, let us say y is equal to m x plus a constant this is well known equation many of you might know this is an equation of a straight line, but in the next course we will go and learn about this equation. So, and another equation is could be v the potential could be log of concentration over c and another equation be the number of bacteria is a times a minus exponential of minus k t. So, all at this point I am just writing down some equation. I do not expect you to understand this equation, but the aim of this course is to make you understand all this equations and getting you familiarize with the equation and enable you to write this equations yourself and explain the experimental data using this equations. So, basically when I say an equation. So, m x y is equal to m x this is a function. So, this is the here I want to introduce you the idea of a function. So, when I say a function when you say a function in mathematics it is a relation is a relation between two quantities. So, when you plot a graph it is basically a relation between two quantities membrane potential versus concentration that is what you plot in x axis you plot concentration in y axis you plot membrane potential. So, basically a table or a graph gives you some relation between a quantity in the x axis and a quantity in the y axis. So, and you plot this in a graph. So, the function is essentially this relation how this particular quantity is related to the other quantity. So, how this y axis is related to the x axis. So, this is the mathematical idea of a function intuitively you already know this that is why you drew a graph. So, this graph essentially tells you about this function the nature of the function. Now, what we need to understand is the equation behind this graph behind this curve in the graph. So, in this course we will go along. So, let us relook couple of examples we looked. So, let us look the next slide where we have next curve we have we already saw this kinesine position versus time. So, this is like a straight line you can see it is like a straight line. So, you can see the position y is equal to velocity times a time t plus some constant. So, this is I call y the position is a function of time. So, y is equal to v t is a relation and like one can say the position y is a function of time. What does it mean? It means as the time varies the position also varies. In this case the average position y increases with time. So, that is why we represent y is equal to v t the more the t the more the y is the numerical value of y will be increased by increasing the by as we increase the value of time. So, we will see this in the next as we go along this course in the next lecture. We will have a closer look to all these equations, but again what I wanted to say is that what I wanted to tell you here is the idea of a function. The idea that the position changes with time and this is called a function. So, now let us look the next example the membrane potential. So, in this membrane potential we had we vary the concentration and as we vary the concentration the potential difference across the membrane increases. So, again if we look at the graph one can say that as we increase the concentration the potential increases. Therefore, the membrane potential is a function of concentration of ions. So, the question is how the potential across the membrane varies that is what this function describes. When I say an equation this equation describes to you how the membrane potential across the potential across this membrane varies with concentration of ions. And we can say that membrane potential is a function of ion concentration. So, I hope you get this idea of a function here also how the membrane potential varies with ions. And this idea the relation between concentration of ions and the membrane potential is described by a mathematical function. So, this is the idea of a function. So, let us look the next example of a growth curve again. In the growth curve we saw that as we go the time the curve the number of cells increases. So, if you look at this curve this graph the number of cells again increases as a function of time. So, how the number of cells increases with time they can increase in different way linearly they can increase logarithmically they can increase exponentially. So, there is a particular way with which this number of cells increases. And this precise information that how exactly whether it is logarithmically whether it is linearly whether it is exponentially. So, when I say exponentially when I say linearly there is always have some precise meaning. So, aim of this course again is to tell you and make you understand what exactly is the precise meaning of this when somebody says something is increasing exponentially. When somebody else says something is increasing linearly when somebody says something is increasing logarithmically what exactly one means. So, this course will help you to understand this. So, all this logarithm log exponential linear all these are functions. So, linear function exponential function logarithmic function. So, in the next course again we will see more closely all these functions. And we will do an exercise where to plot all this function by you can plot yourself. And there will be a clear exercise where you plot yourself this function. And understand that how exactly this function behaves in different in each and each case case by case we will learn this. So, again we can go to the force velocity relation again. And we can low we can learn how exactly this force velocity relation varies with how the growth velocity varies with force. So, these are the examples which we learnt so far which we had a look at so far. And the function in each of these examples we will have a closer look. So, to summarize this part of what we said we studied the idea of a function so far. And a graph represents basically a mathematical function. And the function is a relation between two quantities. So, we had some quantity which we plotted in the x axis. And some other quantity which we plotted in the y axis. And equation basically relates these two quantities. And this is what we are basically plotting in a graph. And by doing in an experiment we are measuring basically two quantities by varying one quantity we are measuring some other quantity. So, this relation between these two quantities can be described through a mathematical function. So, this is the basically the idea of a function. So, now one can ask so we got this idea of function. Now, the next question is there are many other use of mathematics in biology. So, another example is if you want to ask this question why mathematics another example is structure of biomolecules. So, you have you know that proteins have a three dimensional structure. So, this sequence amino acid sequence specifies the structure of a protein. And it takes a particular structure in 3 D in three dimension. And this structure is related to the function. To have a particular function you have to have a particular structure. So, basically the structure of a protein is crucial. And to understand this structure one has to understand some mathematical coordinates 3 D geometry and so on and so forth. So, this general is a biophysical thing is a biophysics is a part of biophysics to learn about the structure of a protein. And as you can see in this slide for example, you have a typical structure of some protein. So, you can see some helisers and so on and so forth. And you can see this x y z axis here. So, the 3 D geometry and to understand about this 3 D geometry one has to know some mathematics some basics of mathematics. So, coordinates Cartesian coordinates and so on and so forth. And idea of a helix and so on and so forth. So, we have plane planar and all that. So, basically what I am trying to say here is that one need to understand some basic mathematics again to understand the 3 D structure of a protein. So, that is another example clearly where we need mathematics. And another case where we need mathematics statistics. So, as you know that most of the experiments are most of the biological processes can only be described statistically. Almost all measurements that we do involves some kind of statistical variability. We saw gene expression and we will show this we all the statement which I say here this 3 statements. We will explain one by one at later stage. And we will at the end of this course you will precisely you will understand what all the statement precisely means. But at this point just read the statements. So, almost all measurements we do involves statistical variability. So, we will understand what this statistical variability means. And it is important to understand what this variability means to understand this understand various systems and to get some meaningful information from this data. So, one need to understand mathematics and statistics is a subsection as part of mathematics. And you can understand a data meaningful to get meaningful information from an experimental data. One need to get one need to understand this in a better way about distributions probabilities and all that. And so let us look at couple of examples where it is obvious statistics is in play. So, let us say the size. So, size of anything size of let us take the simplest example size of human beings right the height of a people in a classroom it is a statistical quantity. And it can be size of a protein it can be size of an organism it can be size of chromosomes it can be size of anything. And it will have some kind of a distribution. So, this idea of distribution is a statistical nature. And that distribution conveys a lot of information. So, let us have a look at this slide one typical distribution called normal distribution is plotted here. So, this normal distribution what so in the x axis you have size let us say height. And the probability that this many people have this size the probability of having certain size is plotted on the y axis. And it will have such a curve and such a curve is basically what describes this probability. Now, this idea that so the idea of variability of size is included is embedded in this graph. So, by understanding the mathematics behind this graph one can get a lot of information. And one can understand what do mean by variability what do we mean by average. So, average standard deviation etcetera, etcetera are quantities that we use very often used to describe experimental data. And one need to understand statistics basically to understand what do we precisely mean by all this. So, again we will go if you have a section which describes statistics. And how to understand this statistical how do we statistically interpret biologically biological data the experimental data. And another example is probability you could ask this question what is the probability you might have seen bioinformatics some of you might have seen. What is the probability that a particular enzyme binds to the target. So, known the sequence you can ask this question what is the probability that this enzyme will bind to a particular sequence. Again this is the question of probability. So, one needs to know statistics. So, again it is mathematical in nature. And this is the idea of mutation itself has is a mutation is the random event. And the randomness and statistics and randomness are very closely related with each other. So, as we go along we will go into this idea of how the mutation leads to variability. And how should we how do we understand this. And how it is important in the biological context. So, I in to summarize we had this question we started with this question why mathematics. So, mathematics describe natural phenomena. It can describe 3 D structure of biomolecules. And it one needs to have mathematics to describe 3 D structure of biomolecules. And one needs to learn statistics to do one needs to bio mathematics to do statistical analysis. So, there are lot of it is very clear it is very evident that one needs mathematics to understand. If you want to meaningfully understand some set of data. If you want to gain more insights to some experiments. If you want to extract maximum information from some experiments. One needs to understand mathematics and use mathematics. So, that we can get lot of information. So, as we go along in this course we will see how in this course we will learn how to use mathematics to understand different biological systems. And how we and we will learn basic fundamentals of mathematics. So, we will start with functions simple functions in the next class. And we will describe simplest functions like linear function, exponential functions and all that. And e when we describe we will not use a typical we will not do this like a mathematics course like a typical mathematical course mathematics course in a mathematics department. On the other hand we will take a biological example where for example, where as exponential function appears. And we will explain how this function how this particular function is useful. How does this particular function describes this experimental data and we will go ahead. So, today we will stop here and the next class see you in the next class.