Physics of biomolecules and cells

442

Physics of Bio-Molecules and Cells

Fig. 1. Some of the topological operations performed by topoisomerases. a) Class I topos can nick a strand, rotate around the unnicked one, and religate, changing the linking number of the two strands. Class II topos permit double strand passage and so can b) remove knots and c) link and unlink (catenate and decatenate, in topospeak) circular strands. From [15].

piece of DNA to be replicated and the two daughter strands to be separated from one another, but there are other problems and situations to solve.

Type II topoisomerases perform more sophisticated jobs. All known type II topos allow segments of two-stranded DNA to cross through another segment of two-stranded DNA. To do so, they bind to the DNA, cut through the two strands (while keeping a hold on the ends), and let the other segment through. One specific kind, gyrases, perform this task in a particular way: they bind to DNA, twist around the DNA segment, and then they let the top segment though the bottom segment; in this way, they change the linkage of the DNA strands always in the same direction, rather than in the energetically downhill direction. This direction is that of unwinding, making DNA less twisted; this is important, for all of the machinery that needs to access the letters of DNA needs to open the strands apart, thus pushing the helical turns closer together outside of the open bubble; unwinding the DNA makes it easier to open the strands. Other type-II topos simply permit the passage of far-away segments, allowing, for instance, the disentanglement of knots.

M.O. Magnasco: Three Lectures on Biological Networks

443

Fig. 2. This reaction is “futile” from the viewpoint of a single-stranded polymer; however, it changes the linking number between the two strands of a doublestranded polymer by ±2. This is the reaction catalyzed by gyrases.

Now, it should be clear to the reader that topoisomerases are a matter of life and death to the organism. This may sound like an abstract idea, but it is not abstract at all. At the turn of the eighties, patient care for the treatment of gram-negative infections was completely revolutionized. Prior to that point, gram-negative bacteria, being hard to target with conventional antibiotics, were being treated with large dose intravenous antibiotics; this required a month’s stay in the hospital, where the patient was bound to get more intrahospital infections. The worst, toughest, most resistant strains of any given infection are most easily found at a hospital. A gram negative infection was thus akin to some ancient curse, for patient and caregivers alike. But then, gyrase inhibitors were introduced into clinical treatments– fluoroquinolones, for instance, can treat the same infection in seven days, by taking one pill a day in the privacy of home, with no risk of intrahospital strains and no IV line, at a hundredth of the cost for patient care. A gyrase inhibitor operates by targeting bacterial gyrases, which are evolutionarily distinct enough from vertebrate gyrases that they can be targeted specifically. The fluoroquinolone inserts itself into the gyrase during the time the enzyme is bound to DNA and has cut one of the two strands; it immobilizes the enzyme in that state, not allowing it to move forwards or backwards along its chemical cycle. Thus, the enzyme does not perform its job, and the signal that caused the enzyme to act in the first place stays on, and more and more enzymes are sent to the job and immobilized while attempting. Pretty soon there’s a large number of double stranded breaks in the DNA, held on to by gyrases with little monkey wrenches in their works. As they fail and fall apart, the bacterial genome is blasted into little pieces. This has caused literally a revolution in patient care. Needless to say, quinolones and fluoroquinolones have meant huge incomes to the pharmaceutical companies that marketed them. Raxar, the star antibiotic of Glaxo-Welcome, was a huge best seller for years for its ability to target rare respiratory infections. Cipro has been one of the best selling items from Bayer–originally for its ability to treat otherwise-resistant urinary tract infections. Recently it

444	Physics of Bio-Molecules and Cells

appeared in the cover of the New York Times every day for weeks: Cipro is the only antibiotic cleared by the american F.D.A. to treat anthrax. It became a best-selling item in the aftermath of bioterrorist threats in Oct. 2001 in the U.S., when the anthrax-laced-letter scares caused the population of New York City to deplete the city’s stock of Cipro in a matter of days and caused the stock price of Bayer to soar. Thus we see a clinical application of topology: how knot theory is really a matter of life-and-death, for bacteria and patient alike.

1.4 Knots and supercoils

Knowing now of their importance, let’s review briefly what knots and supercoils are. Knottedness is a topological property of the embedding of circles in 3-space. There are no knots in 2 or 4 dimensions: there is no way to tangle a loop in 2 dimensions without self intersections, while in 4-D any 1-D structure, loops in particular, may be smoothly untangled without ever going through a self-intersection. We know from basic di erential topology the “dimension” formula from intersection theory, which tells us that the dimension of the intersection between two submanifolds is the sum of their dimensions minus the dimension of the ambient manifold; so surfaces and lines in an ambient 3D space typically intersect at a zero-dimensional set: a discrete number of points. So in the case of knots, if we visualize a line trying to move “though” another line that is left static, the first line traces out an object of one more dimension, a surface; and hence they typically will intersect at a discrete number of points in 3D (thus knotting) or not at all in 4D (thus no knotting). Notice that the topological dual of this situation is a point moving through a surface, which is the case of ions moving through an ion channel; thus, topologically, topoisomerases are in a sense “topological dual” ion channels, since a DNA segment bars the passage of another DNA segment in, topologically, the exactly dual way to a membrane barring passage to an ion, and a molecule must be there to permit or deny passage.

The typical way to depict knots is by their projections onto the 2D page, where the projection causes strands at di erent depths to appear to selfintersect; in this case the further-away strand is drawn with a break through it, to give the illusion of an “occlusion” by the nearer strand.

The simplest knot is the unknot, a knot that is not knotted. A circle is the simplest projection of the unknot, but there are infinitely many, arbitrarily complicated projections of the unknot.

I do not think there’s any need to stress here that knot theory is hard–I may just refer the curious reader to the many excellent textbooks [1, 2]. I will just point out a couple of points. There is currently no known “running”algorithm that can recognize the unknot. One may compute the Jones

M.O. Magnasco: Three Lectures on Biological Networks			445

Fig. 3. Knot projections. Top line: two projections of the unknot. The left one is the “canonical” projection, while the right one can be transformed into the left one by lifting the “flap” up and untwisting. Bottom: two projections of the trefoil. Please notice that the right projections of the unknot and the trefoil can be changed into one another by switching the sign of the bottom right crossing.

polynomial, an NP-complete task [3], and check whether it’s trivial; but it has not yet been proven whether the Jones polynomial (or its extensions like Homfly) do classify knots; thus there is still the possibility that a nontrivial knot may exist whose Jones polynomial is trivial, though none is known. On the other hand, there is an algorithm, due to Haken & Hermion [4], that can classify all knots; but there is no running implementation of this algorithm that I know of, and it is unclear whether it stands in the complexity hierarchy–it seems to be a lot worse than NP-complete! Thus, the seemingly innocuous task of deciding whether a projection of a knot is or is not actually knotted is still an unsettled business, and in the best possible current scenario (that the Jones polynomial or a relative are shown to classify) is NP-complete: exponential in the number of crossings of the projection.

An ideal, infinitely-thin object may be endowed with elastic properties. An object with a finite thickness has, additionally, torsional elasticity: the resistance of a rod to have opposite ends twisted in opposing directions.

Supercoiling results from the competition between torsional elasticity and bending elasticity–since both of them are quadratic the total energy is minimized by appointing a fraction to both rather than most to any one of them. Thus if a torsion is embedded in the polymer, it will spontaneously attempt to relieve some by writhing, i.e., twisting its core into space; see Figure 4.

Supercoils are usually seen in desktop telephones, in the cables joining the handset to the body. Taking a message is a sure way to imbed torsion into such a cable: a right handed person will typically take the handset with

446	Physics of Bio-Molecules and Cells

Fig. 4. Supercoiling is produced when an object with torsional elastic degrees of freedom su ers overall torsion; in this case part of the elastic energy from twisting around the axis is relieved by the axis itself writhing around in space. Try this with any cable with appreciable resistance to being twisted, like an ethernet cable.

the right hand, and lift it straight up to his ear. If a conversation ensues in which no use is otherwise made of his hands, he shall hang up by the reverse of the original path and nothing will happen topologically. But if he has to take a message, then he will pass the handset over to his left ear, so that the right hand is free for action, and in doing so turn the handset by half a turn clockwise (seen from the cable). After the conversation finishes, he will hang up using the left hand, since the phone is on the left ear, embedding another half turn clockwise. Thus one can generally learn the handedness of a person by looking at the handedness of the supercoils o her phone.

1.5 Topological equilibrium

A ghost polymer is a theoretical model of a polymer that has all the normal local properties of a polymer, but can freely pass through itself–i.e., it has no interactions which are long-range along the polymer strand and hence can not “feel” self-intersections. This is the easy, “lazy” thing to do if one tries to implement a computer model of a polymer–the nice description of a polymer is along its arclength, and thus a self intersection, which is local in space but nonlocal in arclength, would require, in the most naive implementations, everyone-against-everyone checking. This can be avoided by more sophisticated techniques, e.g. 3D Delaunay tesselations, but they are a pain to implement.

One can then implement the simplest polymer–a ghost polymer with bending elasticity–in thermal agitation. If we made it a closed loop, one

M.O. Magnasco: Three Lectures on Biological Networks

447

could then check to see how often the loop was knotted and how often it wasn’t. In fact, one may do a histogram of how often the loop is found on any given knot. This distribution over the knots is called the topological equilibrium distribution [10, 11]: it is nothing other than the Boltzmann distribution integrated over topological classes, and so it is the most basic object of discussion in the statistical mechanics of DNA loops.

For su ciently short loops the knotting probability is very small, and

vanishes as e− LL0 , where L is the length and L0 is the length at which it first becomes probable to twist so much knots can be made, of the order of 100 persistence lengths. While a circle is obliged to accumulate a total of 2π worth of curvature, and thus the integral of absolute value of curvature needs to be at least 2π, it has been shown that any knot has to accumulate at least 6π in total curvature. (A result derived by John Milnor at the age of 17!). This represents an energy barrier to knotting which becomes steeper and steeper the shorter the polymer loop is, since the elastic energy is the integral of the square of curvature di erential of length, which is homogeneous order −1 in the length.

Experiments can be made to check this theory. They have been done by using equilibrium religation [12, 13] – the nice thing about an equilibrium distribution is that it does not matter how one reaches it, and so the experiments were done by letting DNA loops with “sticky ends” flicker freely between open (i.e., linear) and closed (circular) configurations. At some point an enzyme (DNA ligase) is added which solders the sticky ends, thus freezing the mixture at an equilibrium snapshot. If the DNA pieces are run through a gel, they migrate through it at di erent speeds depending upon their topology, and so it can be quantified with exactness how much of a given topology there is in the mixture. For instance, for 10 kb loops, about 3% are trefoils, 0.1% are figure eight knots, and negligible quantities of higher-order knots; about 97% of the mixture is unknotted loops. For 7kb loops about 1.8% are knots. These experimental results were in complete agreement with the theoretical calculations, and everyone was happy.

1.6 Can topoisomerases recognize topology?

So experiment and theory were in agreement. But then, someone had the idea of checking whether this topological equilibrium was respected by topoisomerases. And thus the trouble started.

The idea people had about class-II topoisomerases was that, by allowing double stranded DNA to pass through itself, they e ectively rendered real DNA into a ghost polymer. There is no topology problem in a ghost polymer. It was hard to think that they could do anything else, since topology is a global property and the topoisomerases are thousands of times smaller than

448	Physics of Bio-Molecules and Cells

the DNA they untangle. But becoming a ghost polymer cannot alter the topological equilibrium distribution, since it is just given by the Boltzmann distribution of the elastic polymers. So, adding type-II topoisomerases to a topological equilibrium mixture should not change the distribution, according to these thoughts.

But of course it did. Type-II topos strongly suppressed knots, as well as supercoil density fluctuations. Let’s look at knotting first.

	Equilibrium	Experimental [5]
10 kb P4	0.031	0.00062
7 kb PAB4	0.017	0.00019
nm P4 Links	0.064	0.04

So from this experiment the suppression of knotting and linking can be as large as a hundredfold. Pretty impressive for an enzyme many thousands of time smaller than the DNA it is unknotting. Furthermore, titration of enzyme concentration showed that full activity was being reached with an average concentration of only one enzyme per DNA plasmid; thus this e ect is not the outcome of collective interactions.

Now, the fact that a deviation from the Boltzmann distribution is effected is no problem, since class-II topos consume energy in the form of ATP to perform their job. The Second Law is not the one at risk here. The problem is that topology is a global object, while the enzyme acts locally. It is even more insulting when we think that we do not have a good solution to the unknotting problem.

1.7 Proposal: Kinetic proofreading

Now, the problem rapidly acquires twists. The suggestion in [5] is that somehow topos recognize a few specific configurations and only e ect strand passage on them. A suggestion then developed further by Vologodskii is that topos may bend DNA locally into a hairpin, and then strand passage into a hairpin would be more likely from a knotted than from an unknotted state. This proposal has its own problems which we will comment upon later. For the time being let us just look at one feature. If the topo binds to DNA in one place, and allows a di erent segment to cross through it, the overall rate at which this happens will be computable as an integral over all possible locations of the first and second segment–the path integral decomposes into a double integral. Now, it is known that no double integral can discriminate a knot from an unknot; the linking number can be computed as a double integral, but it diverges when evaluated in a single loop rather than on two disjoint loops. No equivalent for knottedness, no matter how crude, has been developed. Higher order binding rapidly becomes complex, and disagreeably

M.O. Magnasco: Three Lectures on Biological Networks

449

chiral. Chirality is of vital importance, since there are two trefoils, the right handed and left handed ones, and both seem to be strongly suppressed in the experiment, so we have to find a chirally-insensitive mechanism.

It looks like, simply stated, there is a “ground” state (unknotted) and “excited” states (knots), and a way has been found to focus the system onto the ground state. There is one biochemical mechanism that does just that, and it is called kinetic proofreading. It was discovered/invented by Hopfield [6], and independently by Ninio [7], in an attempt to explain how the accuracy of DNA replication comes about. I will detail the mechanism below, but the key point is that the signature of kinetic proofreading is the squaring of the error rates: if the native mechanism had an error rate of 1%, then by repeating it twice independently in a kinetic proofreading scheme it becomes 0.01%.

Now, examination of the previous table becomes suggestive the moment we add a new column with the square root of the experimental data:

		Equilibrium	Experimental [5]		[expe]
	10 kb P4
		0.031	0.00062
				0.025
	7 kb PAB4	0.017	0.00019	0.014
	nm P4 Links	0.064	0.004	0.063

i.e., the square root of the experimental data is almost unreasonably close to coinciding with the topological equilibrium distribution. Even if our detailed model below is wrong, which it might very well be, it sounds like too much of a numerical coincidence to have close to a square of the Boltzmann number for there not to be some form of two-collision process at work.

1.8 How to do it twice

The kinetic proofreading proposal works as follows. Naively, one always expects that if there is a test to weed out unwanted stu , if the test fails to detect an undesirable with probability p, then repeating the test twice will fail with probability p2. This is, of course, subject to all sorts of caveats, including the tests being fully random and statistically independent of one another; i.e., for any undesirable, it must be the case that it is detected with probability p.

The problem is how to implement this in chemical reactions. In equilibrium statistical mechanics there’s no such thing as doing it exactly twice, since everything is a random walk: any attempt at doing it twice will result in doing it once with some probability, and doing it thirteen times with some other probability. This is evident in the specific setting that Hopfield discussed: assembling a biopolymer such as DNA so that the right letters

450	Physics of Bio-Molecules and Cells

are copied faithfully. Let’s say the global strokes are such: there is a letter soup containing the right and wrong letters. Both must be there because next letter will be di erent and so we need a soupwith all letters. We have a substrate, and then we want to incorporate the right letter to the substrate at some rate:

S + R	↔	SR →ϑ right
S + W	↔	SW →ϑ wrong

the discrimination here is carried out at the level of the double arrow, which establishes di erent equilibria for the SR. Now, in the case of DNA replication, it can be argued that the left arrow in the reaction has the same rate at the top and bottom reaction: the letters in the soup don’t have a clue as to what letter is being copied, and thus arrive at the same rates, without regards for whether it’s the right or wrong letter. What does change between the right and wrong reaction is the back arrow: the right letter sticks longer than the wrong letter. It sticks exactly e∆E/kT longer than the wrong letter, to be precise, where ∆E is the discrimination energy. Then if ϑ is su ciently slow, then enough time is given to the back reaction to equilibrate and the overall rate at which wrong product is being incorporated is

p = e−∆E/kT .

In order to proofread these reactions, there must be a way in which the discrimination can be done twice. But the discriminating step is the unbinding part, the back arrow; how can this be done twice? Only by adding a new state, SR , and allowing this new state to dissociate–otherwise there’s no way to add another dissociation

S + R ↔ SR ↔ SR →ϑ right

S + R

and a similar set of reactions for the wrong substrate. But the problem is that the moment we put a down arrow to allow SR to decay to S +R again, we necessarily must put an up arrow. Therefore, allowing the exit path that permits the reaction to be carried out twice allows the entry into the second stage of the reaction directly, without ever doing the first check. When all is said and done, when all reactions satisfy all necessary energetic constraints, no major improvement is achieved from this mechanism: instead of getting a geometric improvement like p2 we get an algebraic improvement like p/2. This is all good and sound: there should be no way to cheat Boltzmann in an equilibrium situation.

Hopfield and Ninio’s observation is that energy expenditure breaks us away from these limitations. Notice that the is not the only arrow we

M.O. Magnasco: Three Lectures on Biological Networks

451

added to the reaction diagram: we necessarily added a ↔between the SR and SR states. In an equilibrium situation this is the no-win place: all forward arrows starting at the left S + R and ending at the bottom S + R must multiply to one to satisfy detailed balance, and simile for the back arrows. Otherwise we would be gaining energy going through a loop, which is forbidden by the first law. Making SR higher in energy than SR diminishes the chances of entering through the vertical pathway, but in so doing slows down the horizontal pathway by exactly the same amount because of this balancing requirement. But if we couple the transition from SR to SR to degradation of ATP, for example, we shall be exempt of this restriction: we can make the SR → SR reaction e ectively unidirectional.

The e ect of this energy expenditure is deep: we can then make the SR state a higher-energy state than SR, and so make entry through the second pathway arbitrarily di cult.

S + R ↔	SR →ϑ right
S + R ↔	↑
	SR

while the quotient of left and right rates may be the same for the top and bottom arrows, their individual values now will be scaled by a factor of the exponential of the energy di erence between the SR and SR states.

1.9 The care and proofreading of knots

This is all fine and dandy, but the question remains of how we could possibly proofread a knot. In order to know how to do it twice, we need to know how to do it once. Let’s first imagine a topological transition in the ghost polymer setting. The transition between the unknotted U and knotted state

Kgoes through a self-intersecting or singular state S:

κλ

K S U.

λν

But we have theorems telling us that there is no way to know, when we are in the state S, which way lies the knot and which way the unknot. Thus λ must necessarily be the same in both decays; this feature is going to give our diagrams a weird symmetry. In physical terms, this symmetry comes about because our problem is purely entropic. If elasticity was important, then the strands would be pressed against one another trying to untangle the knot and it would be possible to know which way the unknot lied. This is the most important di erence between this model and the standard proofreading models of Hopfield and Ninio: in these models, there was a di erence