Book Review: Moonwalking with Einstein

Our long overdue comeback post is on a wonderful book titled "Moonwalking with Einstein" written by Joshua Foer. The book is described as writings on the "art and science of remembering everything".

While the book does a wonderful job of describing the artistry involved in enhancing one's memory,  and it does provide a (necessarily) coarse introduction to the neuroscience of forming long term memory (and losing it) in a layman terms; it is much more than just about the art and science of memory. It is partly autobiographical and presents the author's arduous journey from a common man who is fascinated with the art of mnemonics to taking part in the finals of the American memory competition. It is partly about the history of how we as a civilization have learned to forget thanks to the invention of external memory devices that are all around us in modern technology. 

The History of Forgetting:

Our memories have evolved to remember things that we can see or visualize - a tiger in a jungle was dangerous to a man living in the medieval ages and he better remember how a tiger looks. Memory was very important in ancient times when man hadn't yet learned to read or write. It was the only way to pass knowledge and our cultural know-how from generation to generation. 

And then something we take for granted everyday in our life happened - writing was discovered. While writing (and reading) was initially more of an art and could only be performed by a small minority, the advent of the printing press made books accessible and reduced the importance of memory due to the presence of external aids. As modern technology continues to pervade our culture and increasingly substitute our memories, the art of remembering has been forgotten.

Memory or rote learning has also got a bad name in society with our education system focusing on the processing of information rather than on remembering it.  While Joshua and a few others do say that knowing how to remember is important in society, I as a scientist, tend to lean on the side of understanding rather than remembering (especially as Google remembers everything). Joshua Foer foresees a not too distant future in which technology directly communicates with our brains and information is directly accessible to our brains - making memory all but a forgotten skill.

The Art of Memory:

The book presents the similarity between the art of remembering and synesthesia, which is a neurological condition that enables a person to visualize (and in some cases smell) every word in a sentence or every event that takes place in his/her life, enabling people with synesthesia to have excellent memories. The memory palace technique that Joshua talks about in his book is based on the mimicking the state of synesthesia by visualizing everything you see or hear.  In order to make a memory stick, these visualizations are performed at a locus that are locations within a house (called the memory palace) that the person has seen in real life.  The weirder or funnier the visualizations, the more it is likely to stick.  In fact, Joshua makes a case for seeing sexy women in weird poses in order to make memories stick (an example that did stick in my mind even a week after reading the book is visualizing Claudia Schiffer in a bath tub filled with cottage cheese to remember that he has to buy cottage cheese in his to do list). 

The technique also involves visualizing numbers (or playing cards) one wants to remember. A person develops a system whether a two digit number (or cards) is typically replaced by an image in the memory palace. So, a person has a particular image for each number from 00 to 99 (or all 52 cards).  While this is tedious at the beginning,  Joshua takes us on an amazing journey in this book during which practice converts him (average in terms of memory at the beginning of the book) to one with extraordinary recollection.  Joshua is himself able to remember a whole deck of cards in about a 100 seconds by the end of the book during the memory competition.

The Competitive Spirit:
Joshua also provides a glimpse to the mindset of a person who is competing in a national event. He talks about practicing and hitting the okay plateau beyond which he found it difficult to improve. The okay plateau is what we invariable hit after repeating a particular sport or action many times and we cannot improve beyond this point just by normal practice. Experts use deliberate practice in which they practice on the bits they keep failing at (or are not an expert on) to improve beyond the okay plateau. He also talks about his efforts to improve his concentration, diligence to practice every single day, and his attention to compare his performance with his competitors add to the aura of the book.

Conclusion:
What I found fascinating about the book was Joshua Foer's attempt at what he calls participative journalism.  In the sciences, this would be equivalent to a journalist performing the experiments to have a deep understanding of its caveats in the findings before writing an article about it (and wouldn't that kind of science journalism be refreshing in this day and age? - this could solve a number of problems in science). No doubt he has shown that the art of mnemonics is accessible to the average human being provided (s)he is ready to practice and put effort into becoming an expert at it.

From birds and fish to cells [*]

A physicist’s view of collective transport in biological systems

Let me motivate what I want to say today with a couple of videos. First up, an amateur video of a flock of starlings in Scotland.

Or see this one, where the flock cohesively responds to a predator. A Starling is a small bird, shown in the picture alongside, about the size and shape of a “myna” if you are familiar with it. They fly in flocks that do amazing things as a collective entity as you just saw in the above example. Understanding how they do this is field of active research as indicated by this cover of the October issue of Physics Today. I will tell you a little bit about how they do that subsequently. But you might say to me, “They are birds, and they have brains, albeit “bird brains”, so they see, process that information somehow and do stuff. Why would a physicist concern herself with that?” So, to make my point even more clear, let me show you one more video.

This one is a microscopic movie of a bacterial swarm (obtained from here). Do you see the complex flow patterns they exhibit? These guys clearly do not have brains. It might be that this rich collective behavior originates in more chemistry than physics, but clearly not biology. And to make my point that it is indeed just physics, I ask you to look at this other video (obtained from here)

Do you see the similarity in the flow pattern to that seen in the bacteria? Can you guess what you are looking at? It is just a vibrated monolayer of some centimeter long metal rods! Whatever is going on here is clearly just physics. Moreover, one can mathematically represent the motion of the bacteria/birds and those of the rods by the same set of equations! What I want to do in the rest of this post is to give you a flavor of some of the physics behind these and other collective phenomena in biological systems.

When looking at fish schools or bird flocks, the first postulate that comes to mind is that the phenomenon is “follow the leader”, with one bird/fish doing its own thing, and the others following. But as stated above, the things that birds/fish do is “mathematically similar” to what the bacteria do. So “follow the leader” seems an unlikely scenario. The logical next question to ask would be, “What are the minimal rules that can give rise to this kind of behavior?” We have known the answer to this question for a while now [1]. The rules are the following – Each member of the flock does each of three things a) Alignment : Adjust my direction of motion so that I am going in the same direction as my neighbors, b) Velocity matching : I adjust my speed so that I am going with the same speed as my neighbors c) Cohesion : I try to keep the distance from my neighbors the same at all times. With these basic rules and simple boundary conditions for entities at the edge of the flock, like “If there is food, turn towards it” and “If there is danger, turn away from it”, most of the complex patterns exhibited by these groups of organisms can be reproduced!

But these are just rules. So, the next question to ask would be, “Can these rules come about from just physical interactions?” Let us ignore the boundary conditions associated with food/predator for the moment, they clearly are chemistry and other higher processes and focus on the bulk flocking rules. What is a unifying thing between the birds, the fish the bacteria and so on? What they are, are objects that have a non-spherical shape that actively move through a medium (air/water etc.). Now what does that mean? They exert a force on the medium [2]. The medium responds, i.e., the fact that my bird/fish/bacterium is pushing on the fluid induces a flow in the fluid itself. This response now propagates through the fluid. So, a bird/fish/bacterium that is elsewhere will feel this change in the fluid, in terms of the local flow field and pressure gradients. And it will adjust its own force on the fluid accordingly, and this whole things feeds back to the other entities in the flock. This phenomenon is called hydrodynamic interaction. And this is the dominant interaction that produces the three aspects of flocks that is listed in the previous paragraph!

Further, I want to make the case that this quest for minimal mechanisms for collective behavior is not just restricted to animal group behavior on the different scales encompassed from birds to bacteria. For this, see the famous video below.

This is a video of a neutrophil chasing a bacterium and then gobbling it, the immune system of your body at work. I know what you are thinking, “This is one cell chasing one bacterium and the primary thing at play here is chemotaxis, so what is collective about this?” The collective aspect lies in how the cell crawls, i.e., at the sub-cellular scale. The interplay between membrane fluctuations, the stresses in the actin-microtubule network that makes up the cytoskeleton of the cell, the interaction of this network stress with the medium that the cell is in and many other things go into understanding how the cell crawls. The mathematical paradigm and the physics aspects of this question are not so different from those one uses to address animal group behavior we considered earlier! But for now, this is just a teaser. A separate post on this to follow later.

[*] Cross posted from my personal blog, where this really belongs. Will write a really SC post in a couple of weeks

[1] Actually the first instance in literature was in the context of an algorithm for computer graphics, available here.

[2] If we want to be careful, then clearly third law tells us that the swimmer must at least be a force dipole. Since there is no mathematics displayed here, I fudge this point.

On Rubber Bands, Entropic Springs and Elastomers

Let us begin by doing a Gedanken experiment. Or you could literally do it if you have the ingredients. Imagine a ring made of some thin wire or may be a regular piece of string. And now imagine stretching it between you fingers. What would happen? It would just become stiff and you cannot do much more. Now imagine doing the same experiment with a rubber band. You will be able to extend the rubber band to at least a few times its original size. And if you let it go, it goes right back to its original shape and size. So, the experiment shows me that rubbery materials can be “deformed reversibly” to many times their original size. This post tries to tell you how and why. Also, in the concluding paragraph, the context of such considerations as applicable to biological systems is mentioned (feel free to skip the section named theory, the rest is sufficient story by itself).

Microscopics of a regular solid

As our starting point, let us consider the elasticity of a regular material, the string or the metal wire in our earlier experiment. What is elasticity anyway? It is the theory that tells you how much force you need to exert in order to deform/extend a material by a given amount. Let us simplify even further. Let us consider a spring attached to a wall and pulling on it. How much force do I need to exert to do this? This is given by the “Hooke’s law” that we all know and love. Let F be the force and x be the extension of the spring from its rest length. Then F=-kx is the statement of Hooke’s law, i.e., the amount of force I need to exert grows linearly with the extension. And the force law is characterized by a single constant, the spring constant of the particular spring I am using.

Now, why did we start there? The reason is that you can get a reasonable theory for the elasticity of a solid if you considered it to be made up of atoms connected by springs (see picture along side), whose spring constants are determined by the electromagnetic interaction among the atoms [1]. What is the typical energy scale of this spring like interaction, i.e., how stiff are these springs? It is about a few electron volts typically, which makes them really stiff springs (but for us to realize that the spring is really stiff, I need to compare this energy scale to something else right? We will come back to this later).

Microscopics of rubber

Note that the picture above is how a string or a wire looks like in the microscopic scale. Next let us ask what does a rubber band look like? [2] It looks like the picture along side. A rubbery material is made of coiled up flexible polymers that are “crosslinked” by some chemical agents (the big black dots in my picture) [3]. For clarity I have caricatured the three polymers shown in the picture with different colors. The message here is that the polymers are in some coiled state. When I stretch such a material, what I am doing is pulling the black dots apart. What will this do to a polymer? It will uncoil it some. You already know that uncoiling a string costs much less energy than trying to pull at a fully extended string in an attempt to lengthen it. This is primarily the difference between rubbery materials and regular crystalline solid. These kinds of solids, to which class the rubber band belongs, have a special name, they are called “elastomers”. Note that I can say that I understand the elasticity of rubber-like materials, if I can get the equivalent of “Hooke’s law” for this system. And for this purpose I need to understand what happens when I pull on a polymer. In the rest of this post, we will try to explain how to describe this theoretically.

The theory

Before we proceed with the theory, let us pause for a moment. How can what I said above be right? If I had a string coiled up on my table and I pull it open to its full length, it does not cost me any energy at all. It comes apart nice and easy. If I were to take the analogy above seriously, the rubber band should not offer me any resistance at all. Pulling at a rubber band must be like pulling on water, it should just come apart. But this is clearly not true. So what did I miss? What I missed is called “entropy”. Suppose my rubber band is at zero temperature (no no, not 0C or 0F but 0K). Then the analogy with the macroscopic string holds and the rubber band should indeed flow like water till the polymers are completely extended. But at all finite temperature, the polymers in rubber are jiggling around with some kinetic energy. And that makes all the difference as we will try to show below.

In order to quantify this notion we need to ask what makes physical systems happy (some of these notions are developed in a slightly different context in this post). Let us consider a regular spring again. If we just let the spring be, it has a characteristic length, let us call this the rest length of the spring. This is the length in which the spring is happiest. Now I pull on the ends of the string. I have to do some work to pull it because I am moving the spring away from the state it is happiest in. This work gets stored in the spring as potential energy. Alright, now let us ask at finite temperatures what is the state in which the polymer is happiest? It is happiest when it has the largest entropy.

What is this entropy? For a polymer, we can understand it as follows. Now suppose you have a coiled up string. It looks like a disc right? The size of this disk is called the “radius of gyration” of the polymer (a measure of the lateral extension of the polymer). Suppose the length of the polymer is L. I ask you, “how many ways can you make an object of length L with it?”, you will tell me, “Exactly one way, stretch the polymer out to its full length”. Similarly, if I wanted to make an object of some length A which is much much smaller than L, again we can do this in exactly one way, namely make a tight coil out of the polymer with each turn of the coil having a radius A. But, if I wanted some intermediate sized object, then I can make it in many many ways, in each of these ways, the polymer will be coiled in a slightly different way. The entropy of a polymer of radius of gyration R is the number of different configurations the polymer can have given this radius as its lateral dimensions. As stated earlier, the polymer wants to have maximum entropy. Hence, from the arguments above it does not want to be fully extended or tightly coiled, but rather be coiled up in some intermediate state. This intermediate state has a size equal to the square root of its length L [4].

So in my unstretched rubber band, I have polymer coils that are happy, i.e., in the state of maximum entropy. Now I pull on the rubber band. What happens? I stretch the polymers. Their radius of gyration increases above the optimal value, their entropy goes down and they are unhappy. So, just like you pay an energy cost to stretch a normal spring, you pay an entropy cost to stretch a polymer. If you calculate this cost, you can derive the equivalent of Hooke’s law for these polymers [5]. Then you find that if I stretch a polymer by an amount x, then the force I need to apply is F=(CT/L)x, where C is just a constant, and T is the temperature of the system. So, a polymer behaves like a spring with a spring constant determined by the temperature of the system! And the reason it behaves like a spring is because of a loss in entropy rather than a gain in energy. This is what people call an “entropic spring”. Now, suppose I compare this spring constant with the spring constant associated with atomic solids we considered earlier, I find that it is 0.00001 times smaller! Thus polymers form very loose springs. And rubber is exactly like a regular solid but with a really itty-bitty spring constant that scales with the temperature of the solid [6].

Conclusion

In summary, rubbers are solids made by crosslinking polymers. Polymers form entropic springs whose stiffness increase as the temperature increases. And this explains why rubber bands become brittle and break when you try to stretch them on hot summer days! But at the start of the post, I said that such considerations are biologically relevant. How is that? The cell wall is a rubber! It is a crosslinked polymer mesh made of polymers that are called actins and microtubules. You will now say to me, why would I want to know about the elasticity of a cell wall? The reason is that it is the elasticity of the cell wall that allows a cell (those that are not swimmers that is) to crawl. And all questions associated with the motility of such cells boils down to understanding the elasticity of the cell wall. And you need to start by understanding the elasticity of the plain old rubber band first!

Jargon, Caveats and Disclaimers

[1] Coulumb, screened coulomb, Lennard-Jones, you take your pick.

[2] I am fudging scales here, mapping Angstroms to a fraction of a micrometer, but for simplicity we ignore this difference here.

[3] This process is called vulcanization, that turns a complex fluid into a solid, gives it a finite zero frequency shear modulus.

[4] You can see this easily if you think of the polymer as a 3D random walk of length L. Then the RMS distance the walker would travel is Square root of L right?

[5] For the experts, note that you can derive this readily. Take a Boltzmann definition for the entropy as S=k_BLogW, where W is the number of configurations of a polymer of radius of gyration R. Using the random walk analogy earlier, this is W=exp(-R^2/L). So the loss in entropy due to stretching must be S(R)-S(R+x). And force F=-T(dS/dx)x (just a standard response relation).

[6] You should ask me now how come I ignored entropy when considering elasticity of an atomic solid. The fact is that entropy strain independent for harmonic solids and plays no role in the elasticity (a brief note on this is here).

A layman’s tutorial to the dark side II

In the previous post, we tried to answer the question “What is dark matter?” In this post, in the same reductionist spirit, we try to answer the question “What is dark energy?” [1]. For a number of years, I kept thinking that dark energy was just “E =mc^2” type energy associated with dark matter. It was only in my second year in grad school that I realized how hopelessly wrong I was.

As in the context of dark matter, dark energy is postulated to exist to solve some problems associated with explaining observed phenomena. So, first let us talk about what the problem is. The problem is that the universe is expanding and the rate of this expansion is increasing, i.e., the expansion of the universe is accelerating! The first question you might ask is “Are you sure?” or “How do we know this?” I do not want to discuss red shifts and the Hubble constant here. So I refer you to wikipedia for more info on this. What I would like to do here is to take as a given that the universe is expanding and accelerating and ask how can we understand this?

The theory of classical gravity is General Relativity. A layman’s minimal picture of what GR is with respect to the familiar Newtonian picture can be summarized quickly enough [2]. But, for the purpose at hand, it suffices to say that one of the consequences of GR is that matter and energy exert a pressure on space time much like a gas in a chamber exerts a pressure on the piston [3]. Now, suppose we use this piston analogy for minute. If I have gas under pressure, kept that way by putting a weight on the piston. I suddenly remove this weight and I ask you what you expect the motion of the piston to be like. You would tell me that the piston would first instantaneously accelerate to a large speed, then decelerate slowly as the gas in the chamber expands. Yes? This same picture is what you would expect to apply to the universe as well. You can think of the total mass and energy of the universe as N in some units. Suppose the volume of space time is V(t) at a given time t, then the pressure exerted by this mass and energy will be proportional to N/V. At the time of the big bang, i.e., t=0, this was enclosed in a very very small volume. Hence it must have exerted tremendous pressure and the universe must have expanded rapidly. As time increases, universe expands, V increases, N/V(t) decreases, and so the universe should expand more slowly than before. If this was the case, then there would be no problems and we would not have so many cosmologists so worried so much of the time.

But, observations of far away galaxies tell us that the universe is expanding faster than it was at earlier times! The question is, how can this be? Clearly, it cannot be from the regular mass and energy that we talked about earlier. So we have to think of something else. One of the possible “something else” is that there is an (as yet mysterious) energy associated with space time itself. If this was the case, then as the universe expands, the number of space time points increases in some sense. Then, this intrinsic energy associated with the space time points increases as well and so the pressure builds up and the universe expands faster. So, the existence of such an energy, the dark energy, could be one possible explanation for the accelerating universe. But where the heck does this energy come from? We have no clue at the present time. Hence the name dark energy.

[1] A more complete and erudite discussion is here.

[2] This is fishing. If you ask me, I will tell you kind of thing.

[3] I know that relativist cringe at such statements, but I do not see how else to say this simply.

A layman’s tutorial to the dark side I

I am a condensed matter theorist. So, I know next to nothing about gravity or cosmology. But, this week, I attended a few Cosmology seminars and hence was motivated to write this post, which is intended to be a layman’s answer to the following question: WTF is dark matter and dark energy? I have been meaning to do this for a while, just because of all the press these things get, for example this old article on dark energy in NYT that was featured along with the cosmologist involved on David Letterman. There is even a movie by this name. This Friday evening is the time to get it off my chest! [1]

Alright. First let us begin with dark matter in this post, which in many ways is the simpler issue. There is all kinds of evidence that all the matter we see in the universe is not all there is. What is some of this “evidence”? Let me try and give you a couple of examples. One of them is associated with “galactic gravitational potentials”. What does that mean? Now, suppose I was observing a galaxy in my telescope. I saw a star that was far from the central bright core of the galaxy. Then, I expect that the star will have a velocity (GM/R)^1/2, where R the distance of the star from the center and M is the mass of the bright stuff in the middle [2]. Now I measure the velocity of this star, it is moving much faster than this estimate. You might say “Aha! You are just underestimating the mass of the bright object in the middle!” But, if you believe that the universe is homogeneous (same everywhere) and isotropic (same in every direction you look), you have no choice but to conclude that there is just some universal parameter that you have to fit observed data, called the “Mass to light” ratio (and you have no choice but to believe this hypothesis unless you want to also believe that the earth is the center of the universe somehow). And I urge you to go and play with this applet to see that you CANNOT fit the observed curve with just one such parameter. So, there must be something else. What that something else could be is some mass that I cannot see, that I don’t know anything about so far, such that the star that I think is far away from most of the mass in the galaxy is not so far at all, for this stuff I cannot see is filling the intermediate space that appears to me to be empty. So, the postulation of the existence of this “dark matter” is one possible explanation for these weird velocities of apparently far-flung stars in galaxies.

One more piece of evidence is associated with the mass of clusters of galaxies. This is rather involved, but if you want, you can go read about it in this post on Cosmic Variance [3]. So, let me move on to another piece of evidence. This is associated with large scale structure in the universe (this is just jargon for stars, galaxies, you and me). The way this argument works is as follows. We know how the universe is today. We use our telescopes, optical or otherwise and know the mass density in the universe everywhere. We also know what the mass density in the universe was when it was only 400000 years old (that is very young on cosmological time scales). This info comes from the cosmic microwave background [4]. Then knowing the mass in the universe, and knowing the laws of gravity, I should be able to go from the scenario 400000 years ago to now. But, I cannot. It turns out that if I try to so this, I get a mass inhomogeneity much smaller than what we have today, to the extent that you and I cannot be here. But we are here. So, one possible explanation could be that there is this “dark matter” we invented earlier is there in the early universe and the information about its distribution is not in the cosmic microwave background and hence we are not able to get to the present structure of the universe and the fact that you and me are here.

Do you see? The postulation of this “dark matter” solves many problems that are around in astrophysics and cosmology. But, the problem is that we don’t know yet what this “dark matter” made of and how it talks to the regular matter that you and I are made of. We have ideas as theorists and we have experimentalist out there testing to see if any of these ideas hold water. But until then, we just have to live with “dark matter”! [5]

[1] There are a whole bunch of erudite articles on the web, for example, this one by Sean Carroll. I will try to be very minimal here, no way near as erudite.

[2] You can do an itty-bitty circular motion calculation to see this, given that gravity leads to acceleration GM/(R^2) on a particle at a distance R.

[3] There is also an interesting comment thread here that is a back and forth on dark matter, that we (readers and writers) at scientific curiosity will do well to emulate! :)

[4] This cosmic microwave background comes from an event in the past of our universe that is called decoupling. But I have to do a lot more work to get this point across. I will provide an update with some appropriate reference subsequently.

[5] But this, namely the postulation of the existence of dark matter is not the only way out of the many problems that cosmologists face. But the other ways out are deferred to a subsequent post coming up shortly, so stay tuned.

Classification of Protein Structure

Structures of proteins: In all modern day organisms, proteins play a wide variety of roles in the cell. At the molecular level, they are responsible for performing all the mechanical work done by your muscles (as known until now) in addition to the chemical catalysis that they perform on nearly all biochemical reactions inside the cell [1]. The immediate question that begs to be answered "How are these complex molecules able to perform this work? How are they so specific in what they do and specific to the reactions that they catalyze?" While the answer to these questions are nontrivial and the subject of research of more than half the biophysics labs world wide, the common theory going around in the scientific world is that the function of a protein is determined by its structure [2]. This statement is only partially accurate and the reasoning behind this statement is that a protein functions because it is able to have a certain 3-dimensional configuration of certain atoms or functional groups in the amino acids that make up the active site of the protein and these functional groups are then able to catalyze the reaction. The specificity of the reaction they catalyze comes from the specific 3-dimensional configuration of these functional group that it is able to catalyze only when the substrate is able to interact with it in a certain manner. While it is true that the structure does determine how a protein will go about performing its function, these structures are static pictures of the molecule which is otherwise in motion [3]. In addition to the global motion of the molecule, there are relative motions of the atoms that make up the protein which leads to slightly different configurations of the important functional groups and hence they are neither completely specific nor is the function completely dependent on its structure alone.

Classification of Protein Structure: The average protein consists of about 20000-30000 atoms and in order to make sense of the structure of the protein, it is necessary to simplify the protein structure. There are more than 30000 structures in the protein database [4] and to go about looking at each structure would be horrendous. Hence, one needs to come up with a classification scheme for protein structure.

One way of simplifying it is to break it into parts called the secondary structure of the protein. There are various courses/books [1,2] to explain the secondary structure, but for our discussion, it is sufficient to know that certain configurations called alpha helices and beta sheets are common structural motifs found in nearly all proteins. While these secondary structures do help in understanding the local structure of the proteins, they give very little insight about the chemistry that the protein is able to perform and about it's active site itself.

A second and more meaningful attempt at classification of protein structure would be to find certain common structural motifs that can exist independently and classify the proteins based on these structural motifs. For example, a protein can be multifunctional but each function can be carried out independently by different parts of the protein even if you split them up. It could make sense that one can split these multifunctional proteins up based on what function they perform and if you can find the same structure/function motif in different proteins, club them together as a single group. In a protein, the part of the protein that can maintain its structure and function independently is called a domain [5]. Quite often domains of one shape combine with domains of very different shapes to form quite different proteins (very much like building blocks can come together in various different configurations giving walls of various different shapes) [6].

To give meaning to the classification scheme, it would also help to know which proteins perform related function (for example, perform the same reaction on different substrates) or have active sites in the same region of the protein structure. In order to give meaning to this classification, it is better to form groups of more closely related structures that perform similarly. Great minds have always argued that evolution should be the guiding principle while studying biology and it does make sense to classify proteins which have a common evolutionary origin from those that have achieved the same structure independently (also called convergent evolution).

There are various different databases that divide proteins into individual domains and divide these domains up into evolutionarily related groups heirarchically. These databases include the SCOP (Structural Classification Of Proteins) [7](manually divided), CATH (Class,Architecture, Topology, and Homologous superfamily) [8], and FSSP (Families of Structuraly Similar Proteins) databases [9](automatically performed). However, these databases are often flawed and corrections to these databases are often suggested in literature. Part of the problem is it is very difficult to say when similarity in structures occured due to homology (evolutionarily related), or convergence (evolutionarily independent origins). The trivial relationships are those that are apparent in the sequences of the two proteins. When two proteins have very similar sequences (measured by the number of times they have the same amino acid or a slightly related amino acid in the same position of the structure), they are related and statistics based on extreme value distributions can be used to find the probability of both proteins having a common origin [11]. However, the structure remains conserved (does not vary much) much more than sequences and below a certain sequence identity, it is very difficult to prove that there is a relationship between the two proteins without a structure [10].

Other problems that come up are related to the process by which structures are obtained (X-ray crystallography or NMR spectroscopy). These methods are inherently noisy because of various problems such as Heisenberg's uncertainty principle onto the crystallization conditions and the substrates that interact with the protein. So there is never a completely correct structural alignment (that is finding one to one which residues in the structure overlap each other) that also causes minor problems in the classification procedure.

But the most important problem is the level at which to classify structures. While domains are the most commonly used level of classification (because a domain is basically independent), during the evolution process, domains might not have been the basic level at which proteins were constructed. Rather subdomain level small structural units called structural words [12] or foldons [13] (because they could be independent folding units) could also be the smallest level of proteins that had evolved from the RNA world. The theory is that these foldons could come together and form various different domains and then evolved further to form proteins with different functions.

References:
[1] - Biochemistry by Stryer.
[2] - Introduction to Protein Structure by Branden and Tooze.
[3] - A perspective on enzyme catalysis by Stephen Bankovic and Sharon Hammes-Schiffer
[4] - RCSB protein database.
[5] - Domains.
[6] - Multi-domain protein families and domain pairs: comparison with known structures and a random model of domain recombination by Gordana Apic, Wolfgang Huber & Sarah A. Teichmann.
[7] - SCOP.
[8] - CATH.
[9] - FSSP.
[10] - How far divergent evolution goes in proteins - Murzin.
[11] - Maximum Likelihood Fitting of Extreme Value Distributions - Eddy..
[12] - On the evolution of protein folds - Lupas, Ponting, and Russell.
[13] - Foldons, Protein Structural Modules, and Exons by Anna Panchenko, Z. Luthey-Schulten, and P.G. Wolynes.

About Rainbows

In this post, what I would like to do is illustrate scientific methodology and scientific curiosity in the context of the simple natural phenomenon called rainbows that we are all familiar with. The choice of this system is only because we all think we understand it and the physics involved is simple ray optics that we all learnt in school at some point (and of course it is pretty as in the picture along side). Now, the first step in scientific methodology is the collection and categorization of facts that we want an explanation for. In the context of rainbows, I want to be able to explain the following facts that I have established by watching rainbows in the sky. Apart from the obvious one about the colors, they are

1. Rainbows are seen when there is sun and rain (incipient or actual). That is why it is called a rainbow.

2. When I stand facing the rainbow, the sun is always behind me. I never see a rainbow on the same side of the sky as the sun.

3. The rainbow is a bow.

The next step is to look into my knowledge bank from the past and see what I already know that would be useful for me to explain the above facts. And I have to do this piece by piece. Now, I remember something about seeing dispersion, the breaking up of white light into its constituent colors when the light moves from one medium to another. Water glass held appropriately in bright sunlight, prisms I played with when I was young and so on. Yes? So, I begin my quest to understand a rainbow by quantifying this vague notion in my head [1].

Willebrord Snellius and the one and only Rene Descartes figured this out for us 400 years ago. They found that if a monochromatic (just jargon for one-color) light ray is incident on the interface between two media (say air and water), then light is refracted (jargon for “bent”) so that if the angle that the incoming ray makes with the interface is u_i , then the outgoing ray comes out at an angle [4] u_r = sin^-1((n1/n2)sin(u_i)), where n1 and n2 are properties of the two media in question called the refractive index of the material (again it is just a name, I could have called the property Karthik or Pradeep, but for the sake of conformity I call it by the name already given to it). Don’t worry about the formula if it looks complicated to you. Think of it as follows. If someone told you that they shined light at the interface of two media of given refractive indices, you can just tap some keys on your calculator and know where to put your eye or your camera so that you can see the refracted ray. So much for that. But how does this explain dispersion? The key is that the properties n1 and n2 depend not only on what the medium in question is (i.e., water, air glass etc) but also on the color of the light in question. Different colors will have different values of n1/n2. So even if they all come in at the same angle u_i as in the case of sunlight, they will come out at different angles and hence I will be able to see all the different colors. So that is why I am able to see different colors in a rainbow, because there is air and water involved. As an aside also note that the above paragraph tells us that the fact that a straw in a water glass looks bent and the colors of the rainbow come from the same underlying physical equation! Cool isn’t it? This is another aspect of scientific methodology, i.e., link together as many apparently disparate facts as possible as arising from one underlying phenomenon.

Wait a minute, this cannot be right. What I said above cannot be the whole truth. Why is that? It is because of fact 2 above. The sun is on the opposite side of the rainbow. So I cannot possibly be seeing bent light, I must be seeing reflected light that bounced off something. So, what did I miss? What I missed is hidden in that messy formula in the previous paragraph. Recall that sine function takes values from -1 to 1. So if n1/n2 is bigger than 1, that equation can never be satisfied for all values of the angle of incidence. What is wrong here? Clearly I can shine light at whatever angle I wish, so placing a restriction on u_i makes no sense. So, ask again, what did we do wrong? What we did wrong was to assume that there is always a refracted ray, i.e., a ray that goes into medium 2. What the “impossible to satisfy” equation above tells us is that beyond a particular angle all the light will be reflected back into medium 1 if n1/n2 is bigger than1. This phenomenon is called “total internal reflection”. If medium 1 is water and medium 2 is air in the earlier picture, then n1is bigger than n2 and light incident at large angles will be reflected back into the water. So, in the context of the rainbow what is happening is along the lines of the figure shown below. The light from the sun enters the raindrop, gets refracted at the front edge of the drop, travels through the drop, gets internally reflected at the back edge of the drop (i.e., the back edge of the drop is acting like a mirror) and then comes back out of the front edge again. And this is the light that you and I on earth see as the rainbow. So we have established that we need refraction and total internal reflection to account for the colors and the fact that the rainbow is on the opposite side to the sun with respect to the observer (jargon for you and me).

Still with me? Just hang on for a little bit more. We only have one fact remaining that we have yet to explain, the fact that the rainbow is indeed a bow. Again the answer lies in the discussion earlier. We just have to tease it out. Let us do this by first noting that the picture in the previous paragraph is clearly an oversimplification. What is really happening is more like this picture below. Light rays from the sun hit the drop and they are reflected and refracted at each interface. And you are standing in such a position that you get only one of a total of four outgoing rays from the drop. So the amount of light that is reaching you is a pretty small fraction of the light that fell on the drop. That is why we made such a big deal about the total internal reflection thing earlier, for it cuts out one of the outgoing rays and increases the intensity (brightness) of the one we get to see. Secondly the reflected light is diffuse. What does this mean? The sun is far enough away that all the light coming from the sun can be thought of as parallel rays. If the interface at hand was flat, then all the reflected/refracted rays will be in the same direction, yes? (Just generalize the picture in the first part of the discussion to many rays to see this). But our interface is a spherical water drop. So, even though the incoming light is all in the same direction, the outgoing light is going to be all over the place. And my eye is a pretty small hole in the scheme of things and I am only going to get a ray or so of the reflected light, not enough to see anything [2]. But I do see the rainbow. How?

This part is slightly more messy to state so bear with me. Let us revisit the picture in the paragraph on total internal reflection for a moment. Since the sun’s light is all parallel, the angle of incidence is going to change depending on where in the sphere the light hits. The angle at which the light comes out to the observer u_f depends on the angle of incidence as u_f =u_i -u_d where u_d is called the angle of deviation (just another name). Now, clearly, by repeated application of Snell’s law, I can express this angle of deviation as a function of the angle of incidence right? The details are unimportant for us. So let us just say u_d = f(u_i) for some known f. In order that I see as much light as possible, I need that u_f change as little as possible when u_i changes, yes? Which is of course the same as saying u_d or f must change as little as possible. Now, I remember from some calculus class I took ages ago that a function is “stationary”, i.e., changes as little as possible near the points at which it takes its minimum or maximum value. Do you remember this as well? So, I am most likely to see enough light to make out my rainbow when u_d is a minimum (you can easily convince yourself that you have to be on the moon or something to see the region when u_d is a maximum). For a drop of rain water and for red light this turns out to be a position such that your eye is located at an angle of 42 degrees to the direction of sunlight (look at the picture to see what I mean). Now, I clearly cannot change where the sun is or where the water is. So I just see all those water drops that make this angle with my eye. And viola! It is a bow!

Phew! We are done. We succeeded in explaining all the things we set out to explain. But just to throw a wrench in the works let me point out why you should not be happy yet. I can think of a 100 reasons but let me state the first couple that come to my mind. In all of the above, I thought of light as a straight line (ray optics). But I remember somebody telling me light is made of photons, little blobs of energy. I even remember learning that light is a wave just like the wave I can make in a string by oscillating it. WTF? How is it a straight line, a blob and a wave? On a totally different front (a front on which I don’t know the answer), I “know” that I see VIBGYOR when I see a rainbow. Hey! But white light is a “continuous” mixture of wavelengths (colors). So what this VIBGYOR business must be telling me is the degree of resolution in the cones of my retina? On the same note, what is it in the processing of images in my brain that leads me to see rainbows around light bulbs when I am drunk or sleepy but not otherwise? That is scientific curiosity for you and there is more than enough stimulus for it from the world around us to keep me occupied for the rest of my days![3]

[1] You can do the simplest of things, go to wikipedia and read this and this.

[2] It is over and beyond my patience levels to make a picture illustrating this. So I recommend that you go and play with this Java applet to see for yourself that this is true.

[3] Apologies on the length of this post. I “cross my heart and hope to die” when I say my future posts will be way shorter!

[4] After I uploaded everything in blogger I see that it has made all my theta's into u's. So the u in the text corresponds to theta in the images.