complexity of physical entanglement

For knots tied in real stuff, such as molecules or rope, the filament has a non-zero cross-section, and so some length is required to complete a knot pattern. That this length would differ for different patterns has been understood for as long a people have been tying knots. We formalized this notion by requiring that rope to have a circular cross-section of constant radius 1, and we call the length required to complete a given knot type under this constraint the rope-length. The rope length is a natural measure of the complexity of the knot type (it is similarly defined for links, or any conformation of filaments) -- more complex topologies require longer rope length. So the immediate question is: what is the relationship between the rope length and other measures of complexity?

In particular, what about crossing number, which is the minimum number of crossings in a planar diagram of the knot-type? The answer to this question would presumably be important for both theory and applications, since rope length is perhaps the most basic physical notion of knotting, while crossing number is the basic measure for most topologists. We have made considerable progress on this problem. We showed that, depending on the knotting or tangling process, the crossing number varies at least with rope length to the P, where P varies between 1 and 4/3, and the 4/3 is a sharp upper bound. One way to think about this result is that it gives the possible growth rates in topological complexity (as measured by crossing number) in terms of the length of the filament. The primary open question here is whether the exponent 1 is sharp. This is inviting:

Are there are tangle patterns that require a lot of rope for relatively little entanglement? …The trick is in the word “require” – no matter how you tie it…

To briefly describe the 4/3 proof, first note that the average crossing number (hereafter ACN) is the average number of crossings in a 3D conformation when viewed from an arbitrary viewpoint. By modifying Gauss's formula for linking number, the ACN can be given as an inverse square distance double integral over the knot. For the experts, we note that the writhe, as well as some knot energies, such as the occlusion (see below) can also be written as inverse square double integrals. The proof combines the volume exclusion given by the rope length and the nice properties of inverse square laws to arrive at these bounds for any inverse square law. In particular, since the ACN bounds the crossing number, we have the bound on crossing number. Constructions exhibiting the various growth rates, including the 4/3, give the existence and therefore the sharpness.

We believe we have also come to understand the processes that lead to the different growth rates.

In any knot or tangle, a given strand can pass over one strand then under the next then over the next – we call this weaving, or it can pass under several strands, then over several strands – we call this cabling. Two tangles, one primarily woven, the other primarily cabled, have fundamentally different structures, yet might have the same topological complexity. We introduce measures of weaving and cabling for any given filamentary conformation, and for the knot or link type in general. We find connections between the weaving and cabling and the ropelength (the length of a unit radius tube) required to tie knots of a given complexity. In particular we find an estimate for the ropelength of a knot-type given its weaving and cabling.


In this terminology alternating knots are entire woven, and they appear to exhibit linear growth of crossing number with length. On the other hand, our examples of 4/3 growth appear to be maximally non-alternating, in the sense that a strand has as many over passes in a row as possible, so are primarily cabled.

Secondly, and this is vague, but we think there is a nugget in here somewhere, there seems to be a sense in which the maximal growth rate, L to the (4/3), can arise only through a kind of global knotting, while linear growth is the hallmark of local knotting dominating the topology.


A major emphasis of our efforts is applications of these ideas. In any given application we normalize so that the filament has radius 1, and then we can compare the growth in topological complexity in the application with the rates we have identified. We anticipate that finding the proper normalization will in some cases will be a challenge, but this approach has many potential applications. We mention two. First, an argument along these lines tells us that we expect the topological complexity of a random walk on the cubic lattice (and likely a Gaussian walk as well) to grow linearly in the length of the walk. This is an important result, because these walks are the most common models of polymers. A second class of examples arises in the context of magnetohydrodynamics, and other field and fluid theory contexts where topological measures such as helicity are of importance. The helicity can be a topological measure of the energy of the field or fluid, and so we hope to use our results to provide new analysis of the energy in these situations. Other investigators have developed nice results using a different approach. In particular, we could perhaps offer simple methods for building conformations with maximum or minimum energy. When we consider that, for example, the energy in the solar corona is thought to be chiefly in the topology of the magnetic flux tubes, we realize that this could be an exciting application.


A fundamental question is: exactly how is it that knots and tangles are formed? Are there knot-types that are relatively easy to construct by physical means, and others that are not?


We present a first attempt at a dynamic theory of entanglement, an attempt to model the methods by which strands become entangled. The model is reminiscent of Newton’s law of cooling, in that a given conformation will begin at some entanglement and will approach an entanglement equilibrium. We assume the conformation is under forces that induce random motions, giving the possible changes in entanglement. The rate of change of entanglement depends on both the distance from entanglement equilibrium and a geometric factor that measures the degree of freedom of motion of the conformation. The entanglement equilibrium is different for different categories of knotting processes, but is constant for a given process. The geometric factor depends on the particular conformation so may change as the conformation changes, but in some cases can be reasonably approximated with a constant.


What is the relationship between the entire conformation space and the knots that actually appear in nature? The natural assumption is that the probability of occurrence of a knot-type ought to be equivalent to its volume in an appropriate conformation space, but two different methods of creating knots might lead to different sorts of topologies.

A nice question along these lines, raised by Andrzej Staziak, is: why is the knot 5 sub 2 much more likely to appear (in numerical random walk models) than the knot 5 sub 1, when they have virtually the same complexity (by most measures)? Whittington and van Rensburg and their collaborators have done several experiments where they manipulate the writhe of the random conformations, and Staziak seems to concur that writhe is an important element in this story.

As a first step in addressing these issues, we would like to develop at least one, perhaps several, sets of 3D building blocks. The model we have in mind is Conway's tangle constructions -- though we do not necessarily expect such a nice algebraic structure. What we would like to do is describe 3D knotting and tangling as a sum of elemental blocks. We note that minimal rope-length conformations in many cases seem to be constructed of sections of circles, helices and linear segments.

There are some recent developments that portend some exciting applications of this theory, if it can be developed. Taylor has identified deep knots in proteins. Exactly how these knots formed is an essential question, and a solution would give a considerable information about the folding of the protein. It is expected that complex proteins could carry all sorts of topological structure. What we could hope for here is that the knotting and tangling, the topology, would provide a kind of macro guideline for the folding structure, which may simplify the folding problem.



A basic challenge in the applications of topology is that many of the objects we wish to model do not in fact have any topology. In knots and tangles, the problem is that the filaments or filaments are often not closed. This is an old problem -- we clearly consider our shoelaces knotted, but they are not in a topological sense. The usual approach is to consider some method of closing the loop, but this can be problematic for many reasons, not the least of which is that the topology can depend on the method chosen to close the loop. The collection of methods employed is largely ad hoc. What is needed is a theory of the closing, a theory that tells us the strengths and weaknesses of the closing method.

This area of research naturally intersects with questions about the robustness of topological complexity raised above. For example, there may be a way of showing that some closing methods are nearly equivalent, in the sense that, on average, they give knots of similar structure and complexity.

The general point of this effort is that knotting and tangling often has important consequences on open strands -- in fact most tangles in day-to-day life, such as extension cords, are of this type. What we could aim for is a theory that includes a kind of friction, a cost to untangling, which lets the topology or pseudo-topology manifest itself.

For numerical experiments to address some of these questions, we would like to try different sorts of dynamics, to see how much difficulty a given open strand has in untangling itself. We would like to see, for example, which of the energies we now employ are most effective in turning an open tangled filament into a segment of straight line. In fact this method may give us an approach to the knot-building problem described above. We cut a completed knot at various places along the knot, and let the knot simplify to the unknot, by following the gradient of the energy. (We will run this experiment with several different energies). This gives us many different paths for constructing the knot. The total change in energy will of course be the same along each path, but we can measure other aspects of the paths to differentiate them, and also consider other than strictly gradient paths. The point is that if we reverse the direction of time, this method may lead to natural methods for tying the knots.

A final thought in this area. It is an important result in polymer theory that a long filament tends to be knotted. The breaking strength of a polymer is an important question in many contexts. In particular, there is the somewhat open question of the bond strength: how difficult is it to break the polymer? It is an old tailor's trick to knot the thread before breaking it. This weakens the thread, and the thread always breaks at the knot. Recently Wasserman and collaborators have reported similar results for polymers. Our view is that the theory explaining this phenomena is as yet incomplete. We have run several preliminary experiments in breaking large-scale filaments (string and rope) knotted in different patterns (as have others).

dynamics of filaments

In our earlier work, we learned quite a bit about the local analysis of inverse distance laws on filaments. We noticed that the energy functions we were interested in were closely rested to physics problems that had arisen in various fields. If one postulates a uniform distribution of mass or charge along the one-dimensional curve, then we have what seems to be a natural physical question. Unfortunately, for the mass and charge case this construction is well known to be divergent.

For the repelling charge problem, the potential would have inverse power 1. We could attempt to employ some kind of cut-off approximation, since in any application the filament would have a non-zero radius, and so at some scale the interaction would be bounded. One reason we have not pursued this as yet is that the global characteristics of inverse power 1 are not what we were hoping for -- it does not prevent self-crossing.

We may have observed this effect in the laboratory: in simple experiments with J. Schnick, a physicist at Saint Anselm College, we were unable to get much of a repelling dynamic in charged thread, since they would easily degenerate into self-contact. We were able to make two closed loops repel each other a little.

We intend to explore this cutoff approximation further -- it may be that we can discover a good, physically realistic, theory here. We can hope for high (but not infinite) potential walls separating the knot-types. There are many knotting problems that exhibit self-contact of the filament. A prime example is DNA, which carries some charge in the phosphate backbone (see below).


For the gravitational problem we have inverse square attracting forces. In this case we were able to good use of the local divergence. In studying the problem it became apparent that the divergence of the acceleration for the gravitational filament was closely related to the divergence of the Biot-Savart law along a vortex filament (from fluid dynamics). In the gravitational problem the divergence is along the principle normal to the filament, while in the Biot-Savart problem it is along the binormal. The vortex problem is first order, so another difference is that in the vortex problem one studies the divergent velocity (as opposed to the acceleration). Proceeding analogously to the local induction approximation of vortex dynamics, write the acceleration (suppressing constants) as log(a)K, where K is the local curvature vector, and log(a) represents the logarithmic divergence.


Now consider many equal masses equidistributed along a closed space, moving at constant speed along the curve. Since the mass density is uniform, the acceleration due to gravity is approximately proportional to the local curvature. On the other hand, a particle moving at constant speed along a given curve has acceleration proportional to the curvature in the direction opposite the principle normal, so at the proper speed these components of acceleration will (nearly) balance. In this sense we called these axial motions along the curve (sort of like a roller coaster except that the entire track is covered with cars, and there is no connection between cars and for that matter no track either) approximate solutions of the N-body problem. Since this principle holds for any non-intersecting smooth closed space curve, this approach gives a large new family of approximate solutions of the N-body problem -- a problem of notorious difficulty. Joe Keller pointed out that in this balance approach there remains an error of order one – it is a local theory.

We intend to extend this work in several directions.

The approach should not depend on the power of the inverse distance potential, as long as it is greater than 1, so we believe we can extend the result to all other inverse potentials (power greater than 1). This would include, for example, time averaged dipole interactions and the 6-12 molecular potential.

The original derivation required approximate equidistribution of mass along the filament. We hoped to generalize the approach to include variable mass density. If we could, we would gain a general equation of motion for filaments and strings of particles under attractive force laws. The idea is that variation of the mass density will locally give forces in the tangent direction, so the equation will have components in both the normal and tangent directions. In preliminary attempts we are able to recover recent results of Peter Lindstrom’s on the mass distribution of a uniformly collapsing straight-line filament.

This is now done, and we have a general equation of motion for filaments under attractive inverse distance forces. It is: x’’(t)=A(p(x)K(x)+p’(x)T(x)/R(x)). Here K(x) is the curvature vector of the filament at x, T(x) is the unit tangent vector at x, R(x) is the radius of curvature of the filament at x, and p’(x) is derivative of the density function p(x) with respect to arclength – the rate of change of the density along the curve, and A is a constant which depends only on the force law and the cutoff distance.

There are many possibilities for additional applications along these lines. The original result gives us that the filament has a natural speed for this axial motion -- a sort of natural frequency. The construction appears to be related to solutions known in elastica (axial motions include the cowboy's lariat) and fluid dynamics. So it may be that the construction, and the generalized equation, represent a sort of general principle for filamentary objects.

In astrophysics, cosmic strings are filamentary distributions of matter. In preliminary work, it appears possible that sections of spirals may be solutions for the generalized equations, which would be an interesting result for the theory of spiral galaxies. We also hope to consider this approach on surfaces.



There is another dynamical question we plan to address. The technique of gel electrophoresis is one of most useful tools in molecular analysis -- the idea is that different molecules will have different velocities when propelled through a gel by charge. It has been observed that for knotted DNA of the same length, velocity seems to be an increasing (even linear) function with complexity. As an ongoing research project with undergraduates at Saint Anselm College, we have built several different macro models of gel electrophoresis. An example is bead chains dropped through nail boards. In these experiments we can vary several parameters. We find regimes where the relationship between complexity and velocity is the same as is observed in the gels, but we also find regimes where the situation is reversed (a more complex knot is slower than a less complex one). From our preliminary study, we are not able to explain this using the three most common models: reptation, diffusion, or sedimentation. So this merits further study, and invites applications of our physical knots techniques.

accessibility, occlusion, radiation, and temperature of filamentary conformations

History: From the beginning of our study in physical knot theory, we have been faced with bit of a conundrum. The techniques of applying inverse distance laws to curves are at least pseudo-physical, so we would naturally like a true physical interpretation of the energy of a knot. But, as we have noted above, the inverse power 1, a natural physical choice, does not have nice properties, even with regularization, as a knot energy. An inverse square power is better, being scale invariant and preventing self-crossing, though we still have to worry about the interpretation of the regularization. But in the repelling case, the inverse square would be the force, and double integrating the norm of the force has no physical interpretation. We expended considerable effort in search of a good inverse square interpretation. J. Schnick reminded us that the inverse square is associated with radiation through surfaces. A little while later, we recalled this notion when considering an energy functional which seemed appealing to us because the regularization had a particularly pleasing form, and we found that this was the first inverse distance type energy with a straightforward physical interpretation -- in fact it has several.


A tightly tangled long filament has high occlusion -- from an arbitrary point of view much of it cannot be seen. A straight linear filament has high exposure -- its entire length can be seen from almost every point of view. We have found an expression for the occlusion as a double integral over the center curve of a tubular filament.

The occlusion is given by an inverse square law, and we found direct relationships between the occlusion and the topological and geometrical characteristics of the filamentary conformation, such as crossing number, the ACN, and ropelength. This measure of occlusion leads naturally to a gradient flow that can used to find minimally occluding conformations of each topological type -- the energy surface has infinite barriers to topological change.

The occlusion is an elementary notion, and is likely to have applications in many different contexts. We mention two. It can be interpreted as the self-irradiation of a radiating tube. Consider an incandescent light bulb filament. Depending on the geometry of the filamentary conformation, more or less of the thermal radiation leaving the filament will impinge again upon other parts of the filament. If we imagine that the filament reaches thermal equilibrium, then the amount of energy leaving a sphere containing the filament must be equal to the energy entering in the form of electricity. Therefore the filament with a lot of self-irradiation must burn hotter. So we can interpret this energy as a natural temperature of the conformation. The computer graphics technique of ray-tracing gives us a virtual laboratory for this interpretation. Secondly, imagine an agent needs to operate on specific sites on the filament, such as an enzyme on DNA. Then the occlusion is a first approximation of the difficulty in access presented by the topology and geometry of the conformation. It could be related to, for example, the speed of cell division. Also we note that radiation is one of the chief causes of mutations. So it is possible that certain conformations would be beneficial in that parts of the filament in the center of the knot or tangle would be protected from exposure to radiation.

We are doing several computations here. First, our preliminary claim is that the actual radiation, which is best modeled as a surface to surface integral, can be well approximated by a line integral along the center curve of the tube. We would like to check this with numerical experiments. Very preliminary computations, performed by J. Schnick, R. Scharein, and ourselves indicate that we are on the right track. Note that the surface to surface integral numerical complexity grows with the fourth power of the elements, which means that accurate approximations will be very large calculations -- in fact this is one reason why we like our approach -- the curve integral is an n-squared computation.

Next, working with Rob Scharein, the author of KnotPlot , we have implemented a preliminary version of the gradient of the energy, which seems to do a very nice job of simplifying complex conformations. We would like to do extensive numerical explorations of the energy surface for this energy, identifying minimum energy conformations for all of the knots and links up to say eight crossings. We would also like to identify families of minimum conformations, hoping to understand how minimum energy conformations are built from elemental pieces, if in fact they are.

As an application of this approach, we would like to define a measure of radiation accessibility, and determine what role it plays in molecular evolution and DNA packing.

We have found a natural measure of self-occlusion for a thick filament, and it is sensible that this could be interpreted as radiation accessibility, under the assumptions that the radiation approaches the filament from an exterior source in an arbitrary direction, that the radiation particles move in straight lines, and that they are less likely to impact interior or occluded regions of the conformation, presumably because impact with the outer regions diffuses the energy. We have developed excellent numerical techniques for evaluating this measure, as well as found a connection with the computer graphics technique of ray-tracing, which allows for both a powerful visualization of the process and provides a simple simulation strategy. If it is the case that biological molecules do have regions that are thusly protected from radiation damage, then we can identify such regions, and identify maximal packing strategies.

There is another possible geometric defense against radiation damage. It seems that close double strand DNA breaks are most likely to cause cancer. Single strand breaks preserve order and are therefore relatively easy to repair. Distal double strand breaks are usually lethal. Isolated DSBs can usually be repaired. But close DSBs create a free small segment, providing a good opportunity for both misrepair and survival. We note that on some scales DNA packing can be quasi-crystalline, utilizing parallel straight segments. It is a reasonable guess that one could show that such packing minimizes close DSBs

random entanglement

Given the importance of polymers in nature and technology, it is not surprising that there has been considerable interest in describing polymer states of tangledness. The degree and type of tangledness of polymers can determine everything from a plastic's flexibility to the difficulty in replication for DNA. The most common approach to modeling polymer knotting and tangling is with random knots -- the topology in random walks.

One of the really nice results in physical knot theory is the theorem by Whittington and Sumners that says that a random walk will, with probability 1, become knotted as its length tends toward infinity. A nice piece of numerical work is that of Deguchi and Tsurusaki, where they provide good numerical results of knotting in random walks, together with a model that fits the data very well. The basic idea is that the probability of any given knot-type as a function of the length of the walk will peak at some length, then decline exponentially as compound knots and more complex knots come to dominate. This description is widely accepted by researchers in the field.

In our view the next step in this analysis is to predict the frequency of given knot types under various knotting processes. For example, from the random walk work we have that a very long walk is likely to be a compound knot with many factors -- and any given factor is likely to appear with some non-zero probability. But what is the relative frequency of the factors?

Our guess: the relative frequency declines exponentially with the minimum rope-length of the knot-type. This relationship has a pretty implication. It is natural to suppose that rope-length would be additive on factors, that is, that the rope-length required to tie the sum of two prime knots is approximately the sum of the rope-lengths of the factors.

It is also natural to ask whether the appearance of knot factors in a long random walk is independent events. If the relative frequency is an exponential of the rope-length, then these are in a sense the same question (by the product law of exponents). It would be nice if the appearance of factors were independent events, since then we could define the relative frequency of a knot product as the product of the frequencies of the factors. Preliminary investigations seem to give evidence that both observations hold. Simple construction gives that the rope length is at most additive, and recent results on stick number give that it is at least additive on products of trefoils (meaning that it grows linearly with the number of factors).

We plan to consider some special cases in order to illustrate this principle. For example, if a walk is skewed to favor motion in one direction, we have a line of attack that may give us a proof of the exponential relationship between relative frequency and rope-length in this case.

We have also worked with Rob Scharein to run preliminary experiments on bounded random knotting -- cyclically connecting N random points on the surface of a sphere (hereafter denoted FVM for finite volume method).

One way of looking at this problem is to consider the spectrum of a parameter, the rate of growth of the volume of space the filament inhabits. Our guess is that at one end of the spectrum are the factorian processes, where a long filament tends to become a product of simpler components or factors. At the other end is the gordian regime, where a long filament tends to become a single knot of arbitrary complexity. Random linking behaves similarly. It may be that the differences can be explained in terms of the independence (or lack thereof) of the appearance of factors. In the extreme factorian regime, it is likely that the frequency of a factor declines exponentially with its length, (so as we pointed out above the independence of factors is equivalent to the additivity of length for compound knots).

As we mentioned above, we hope to estimate the growth rate of the topological complexity of random filament processes. Perhaps we can use this approach to describe the shift between the factorian and gordian regimes. We have found a method for analyzing data from random knotting experiments to see if it is consistent with the assumption of independence of the appearance of factors. In preliminary study we have found what one would expect: in the gaussian random walk experiment of Deguchi et al the data is consistent with independence, while in the bounded method studied with Scharein (described above), we appear to have clear failure of independence. We hope to make these arguments rigorous. Very interesting numerical studies along these lines have been performed by Millett, Katrich et al, and van Rensburg.

There is at least one more way of looking at these issues. Volume exclusion, meaning the fact that the filament actually has some width (such as in walks on the cubic lattice) or some average separation (such as Gaussian walks) is an important consideration. A basic fact is that volume exclusion implies that each region of space can contain only so much arc-length, which limits the topological complexity. As an illustration consider the nice result of Diao [20], who showed that if a large number of circles are randomly placed in a finite volume, then you can expect them all to be linked in the sense that there is a chain from any loop to any other.

We believe we be able to show that this does not hold if the circles have any thickness (of course in this case there is a limit to the number of circles which can fit in any finite volume.

Similarly, in the walk the FVM method, we believe we will be able to show that in the long run there is likely to be at least one component with complexity (crossing number) that grows at least linearly with the length of the walk. This can be compared with a random walk on the cubic lattice, which will also (eventually) create components of arbitrary complexity, but is likely to require exponential length in N to achieve an N-crossing factor.

Our guess is that the FVM method will in fact in the long run create a single prime knot of complexity that grows at least linearly (likely quadratically) with the length. We are not sure yet how to prove this. Our plan of attack on the weaker statement begins with a similar problem.

But any sort of volume exclusion will give some factorian knotting – more detail needed here.

Let there be a single stick running from the north pole to the south pole on the sphere. (Think of the sphere as a rotisserie chicken). Run an FVM experiment, and ask how the resulting loop is linked with the pole stick. A simple winding number argument gives that the linking complexity grows with the length. We will try to apply this sort of argument to the FVM case.

These observations may have many applications. In applications requiring non-zero width, such as molecular models, gordian knotting is likely not possible in the limit, but still there are many situations where we have a lot of length in a small volume. Gordian knotting is essentially global, so a gordian knot or link has a sort of rigidity -- in general no strand can be manipulated without effecting the rest of the conformation. Factorian knotting processes have flexibility -- the topological obstructions are localized, so a strand has a significant degree of free motion.

In biological applications, tangling is often a problem to be avoided, and a general strategy seems to be to employ controlled linear entanglement in order to avoid higher order entanglement. This, it can be argued, is one of the chief benefits of supercoiling in DNA. Just as it is also one of the chief benefits of braiding long hair.

entanglement of DNA

Long strands in small spaces tend to become entangled, but cell division requires the spatial separation of the daughter chromosomes. Type-2 topoisomerases cut a DNA strand, pass another through it, and then rejoin the first. Because it appears that arbitrary topoisomerase-mediated DNA strand passage events are as likely to increase entanglement as to decrease it, the question is: how does the enzyme choose where to act?

Working with Lynn Zechiedrich, we found a solution:

We propose that type-2 topoisomerases act on juxtapositions of DNA strands where the curvature is such that, locally, the strands are ‘hooked’ together. We show that this model accounts for the known experimental evidence, including the remarkable efficacy of the enzyme in unlinking and unknotting DNA. It is also consistent with a commonsense interpretation of the cellular role of type-2 topoisomerases: if there are forces pulling the daughter strands apart, then topological obstructions will be manifested in hooked juxtapositions (one can pull on any tangle to see them).

In this work we showed that there is local information available to the enzyme, which tells the enzyme which juxtapositions to act on. Previously a thermodynamics argument had been proposed, that somehow the enzyme used ATP, an energy source that it requires, to give the detangling a direction. We think this is misguided, but then the question remains: what is the role of ATP in topoisomerase II action? Why does it need energy?

In this work we came across a curious fact. What is usually not understood about charged strings is that there is a sense in which they are not self-repulsive: global savings in potential will generically encourage isolated self-intersections. If DNA was a charged string dissipating energy, then our numerical experiments show that these energetics would provide the topological directionality -- when we allow the string to pass through itself we always seem to end in unknotted unlinked loops. Now of course DNA does carry charge, but the self interactions are screened by the medium (water). But we wonder: what of the interactions survive the screening? Is the savings in potential of separated loops enough to contribute at all to the directionality? We do show that the charge could contribute to the persistence of hooked juxtapositions – contrary to the naïve belief that the topoisomerase must overcome the charge repulsion to effect strand passage. These issues invite further investigation.


Another challenge in this area: determine to what extent the packing patterns of DNA serve to inhibit entanglement complexity.

We have been developing a theory of random and natural knotting. One of our results is that there is a spectrum of growth rates of entanglement complexity, when measured as a function of the length of the filament. In fact, even with a thick filament, we have shown that it is possible, and in a compact space such as the cell, even expected that the entanglement complexity would grow faster than linearly with the length (it would be to the four-thirds power of the length). The perhaps surprising lesson here is that entanglement is inhibited by increasing local curvature, because this only increases local entanglement linearly, but decreases the length available for global entanglement, which is the non-linear contribution. Our plan is to discover whether this effect contributes to the packing patterns of DNA.


There is another natural question that arises in topoisomerase action. How does the enzyme search for likely places to act? This brings up the more general question of how enzymes search. It has been proposed that they might sometimes use one dimensional search – move along the DNA strand, and sometimes use three dimensional search – float free from the strand until they run into another strand. Halford has called this sliding and hopping. It is clear that the question of which method is more effective can turn on the specifics of the geometry of the strand conformation. It may be that the accessibility, defined above can help with this geometric factor. Moreover, we may have found a relatively simple method for modeling the combination of the two processes, proving, mathematically at least, that a combination of the two is the most efficient approach in most cases.

geometry of equilibria in mass and charge systems

In general, it is impossible to solve the equations of many body systems, such as the n-body problem -- they are generically chaotic. One approach is then to look for special solutions, such as equilibrium or relative equilibrium solutions. In the n-body problem, relative equilibria, which are solutions which rotate rigidly -- "equilibria" in a rotating coordinate frame, are given by configurations of the masses called central configurations. A well known central configuration is the equilateral triangle formed by the Sun, Jupiter, and the Trojan Asteroids. The motion of the system is rigid rotation about the center of mass of the system. A natural thing to do is to try to identify and classify the possible central configurations. We have been able to contribute along these lines. We showed that in any central configuration, the masses cannote be too clustered, and found explicit bounds on the clustering. We also made some progress in determining the mass distributions in central configurations of many masses. The approaches we employed here can be applied in other problems involving equilibria -- for example the distribution of point charges in confined geometries. This problem arises in the development of several nano-sensing devices.

protein folding

We can pose several challenges bridging our work on filaments and the problem of protein folding:

Challenge 1: Determine the role of entanglement in protein folding.

Deep knots have been found in proteins. But is this the average case? That is, do we have as much entanglement as we would expect from filaments of this length in this compact a space? If not, does this give us information about the folding process? Entanglement in most cases is a global process, and so its existence carries a great deal of information about the folding pathway. So it appears one could make progress in both directions -- if there is less entanglement than might be expected, then this tells us that entangled pathways are hard to find, which should tell us something about the folding landscape. On the other hand, in those cases where entanglement is found, we learn something about the folding pathway of that particular protein.


Challenge 2: Define a notion of folding order, and determine its relation to the folding landscape, as well as to other structural measures.

In the topological realm, we have found that there is a fundamental distinction between entanglements that are due to local, ‘independent’ effects, and those that give global entanglements. One distinction is that of alternating versus non-alternating structures. In alternating knots and links, when one follows a particular strand, one observes it going over one strand, under the next, over the next and so on. We have shown that alternating or nearly alternating structures are expected wit ´h a long filament having only local knotting. Highly non-alternating structures are those in which a particular strand will pass over many strands, then under many strands. This is the hallmark of global knotting, and leads to a faster growth rate in topological complexity. There appears to be a similar distinction in folding theory. If we take a long thin strip of paper, one can fold it over and back, alternating, accordion style. This is analogous to alternating topological structures, and can be created locally. On the other hand, one could fold the strip of paper in half, and then the doubled strip in half, and so on. This is global folding, and is clearly a dependent process. It would be interesting to know if we could make a useful three dimensional version of this distinction -- the question of dependence versus independence would seem to have natural application for the folding landscape and the kinematics thereupon.



Challenge 3: Develop an analytic approach that provides comparison of structural measures such as contact order, solvent accessibility, nucleation regions, and folding order.

We have had some success in finding an analytic theory that gave a way to place all the measures of filamentary conformations within the same framework. That is, we were able to classify the measures according to their geometric form, which then let us calculate the expected growth rates of measures, depending on the family of conformations. We would like to develop a similar frame work for the various folding measures.



Challenge 4: Using analytic theory, determine the relationship between structural measures and the kinematics of protein folding.

If we are successful in finding an analytic approach to the structural measures, then the next natural step is to relate this theory the kinematics of folding. In particular, we should be able to predict which structural measures should correlate with which kinematic factors. An example is recent work showing that contact order correlates with folding speed. If this approach is successful, we should be able to predict such correlations from theoretical understanding.



Challenge 5: Relate filamentary conformational measures such as rope-length and occlusion to structural measures such as solvent accessibility and contact order. Apply the successful theory of complexity growth rates for filamentary measures to the structural measures.

This problem is closely related to the problem concerning the theory of structural measures. We might state this problem another way: to what extent is the tertiary (and quaternary) structure of proteins due to the (continuous) filamentary nature of the molecule? This is to be compared to those aspects of the structure that can only be seen with proper discretization. We have developed sensitive analytic and numerical tools for conformations of smooth filaments and piece-wise linear filaments. So we wonder: can these tools capture some of the salient aspects of the large scale structure of proteins?



Challenge 6: Develop a symmetry theory for a simplified model of folded proteins, and use this theory in concert with numerical experimentation to search for possible structures.

It is well known that several sorts of symmetries have been found in proteins. We (and others) have found in our work with the filamentary conformational measures that we can find symmetrical conformations by minimizing one or several of the conformational measures. In fact, in signature work Thistlethwaite has used measures and algorithms we have defined to find symmetrical conformations of knots and links which were known to exist theoretically, but had never been seen (that is, the theory implied that such conformations had to exist, but no-one knew how to find them). It seems that a similar approach to proteins would be interesting. That is, searching for minima of the various conformational measures (at times perhaps forcing certain symmetries to be preserved), may give us a catalog of possible structures.

computation -- entanglement as a model problem

This section is speculative -- but we proceed with the notion that a good walk might begin in a fog and end in a vista. It is fairly easy to see that there ought to be local minimums for rope-length, in fact we have tied an unknot in an actual piece of rope which cannot be manipulated into a circle -- it would first need to be lengthened. On the other hand, we have observed over and over again that the gradient flows of the inverse distance laws on smooth knots seem to be almost unreasonably effective at simplifying conformations which appear complex but have simple topology. Some effort has been expended in searching for unknots which cannot be simplified to a circle by the gradient flow, but none have been found which are convincing (at least to us). Any apparent local mins seem to be artifacts of the numerics or the discretization.

There is another, quite important problem, with similar characteristics: protein folding. Here it is the folding process, computed by nature (and we suppose more like tangling than untangling) that appears to be unreasonably effective. Of course, much work has been done on this problem, and many different approaches have been taken, from postulating that the energy surface has a certain sort of shape (usually funnel-like) to studying simplified models of the folding process. Our question is: can we learn anything by asking what the characteristics of this sort of problem are in general? That is, what is the nature of gradient system on filamentary objects under various constraints and forces? For example for inverse distance based energies, we noted that the local mins we have found seem to be artifacts of the discretization. On the other hand, for experiments involving a large number of edges, our guess is that saddles become the more dominant phenomena. This is the generic case. As the number of variables becomes large, saddles, even those with many attractive directions, are generically much more common than minima. They can present a considerable problem to computation. There is no way to tell, in simply watching the conformations under a gradient flow, whether one is approaching a minimum or simply passing close by a saddle which has only a couple unstable directions, and so has a 'narrow' escape route. We think there might be something interesting here, in that there might be a sort of maximally efficient dimension for the problem. If you assume that the idea is to find global mins, you want enough flexibility to obviate local mins but not so much that you necessitate ubiquitous saddles. This principle may have many applications, one of the most obvious would be protein folding. This sort of reasoning provides a good illustration of working in the knot world, where we have developed some excellent computational tools (and plan to develop more), yet have many interesting unexplored lines of investigations. The world is complex enough to contain all sorts of interesting behavior yet sufficiently delineated that we can make progress. It therefore can serve as fertile ground for growing theorems and principles which can be transplanted to less refined territory.

modeling evolution

The Mathematics of Evolution

We have studied many different models of macro-evolution, including traditional theory as well as sandpile, neural net, and other complexity theory related work. We feel there is work yet to be done, for two reasons. First, a really good theory of evolution ought to come from first principles -- using the same elementary terms biologists use -- and spring from biology itself instead of being offered as a complete model from outside. And second, any good theory ought to encompass evolution at the molecular level, since this is where the best data is.


Challenge: Determine the role of close passes to saddle equilibria in macro and micro evolutionary processes.

In traditional macro-evolutionary theory, species inhabit adaptive peaks on the fitness landscape. However, this interpretation has several difficulties associated with it. For example some investigators have looked for mechanisms allowing species to move from one adaptive peak to another, but by definition this requires some kind of motion against natural selection. Secondly, if movement between peaks is forbidden, then changes -- evolution -- is entirely represented by changes in the position of the peaks, that is by change in the fitness landscape itself, and so the nice interpretation of natural selection as a gradient-like flow on the fitness landscape is lost. Also, it is clear that the natural world is a system of many, many variables, and so in most cases the fitness landscape is a hyper surface of many variables. Critical points on such a surface are far more (exponentially more) likely, generically, to be saddles than to be extrema. If natural selection is a gradient-like flow, and there are enough variables, then one can argue that apparent stasis is more likely the result of close passes to saddle equilibria than inhabitation of peaks. Several aspects of modern macro-evolutionary theory, including punctuated equilibrium, systemic volatility, and adaptive radiation, are simply and naturally explained in this viewpoint. This viewpoint also has the advantage of being nearly traditional in the sense that the elements of the model are the traditional ones employed by biologists -- unlike some other new models of e evolution that have been offered.

This leaves us with a similar question for molecular evolution. Are there similar effects, and a similar interpretation available? What is the nature of protein stability? Are observed conserved designs truly local peaks in some fitness landscape? Or might they be designs that can suddenly evolve to something quite different once the ‘narrow’ evolutionary pathway is found?

software development

We developed and co-developed many different methods for computing different aspects of physical knots. Every part of this work has had a computational aspect, which we have pursued vigorously. Our first effort was a knot energy minimizer, co-written with J. Orloff. Some of the methods employed in this software were taken from cc finder, a program we developed for finding certain configurations in the N-body problem, a problem much like the knot problem. We next wrote RandomKnot, a program using various different methods for creating random knots. As computational efforts became more sophisticated, our computational strategy became to build alliances with other software developers, in order to not have everyone solving the same problems. Ken Brakke was kind enough to add modules for computing energies and minimization paths to his wonderful software Evolver, and we would like to think that some of the experience we gained writing the earlier programs informed the implementation. We also wrote several versions of Mathematica and Maple packages for computing various energy functions. Visualization is an important part of this work. We wrote a Mathematica routine for visualizing knots, which in particular produced the graphics which appeared in some of our papers, and we have used it for discovering new families of parametrizations of knots. We took the graphics to the next level by writing a routine in Mathematica which would take output data from Brakke's Evolver and create a suitable data file for the remarkable ray-tracing program Pov-Ray. These graphics appeared in print articles, in particular we used this method to create the images which became a cover of Nature. At the same time, we realized that Pov-Ray could be used to simulate radiation, and began conducting visualization experiments with it. Next we began working with Rob Scharein, the author of Knot-Plot, a really wonderful program with a strong user interface. We worked with Scharein to develop modules for computing energies and gradients in Knot-Plot. Scharein has just released Knot-Plot for several common platforms, including PC, Mac, and Unix operating systems, and the standard distribution has this capability for computing the energies we study. Scharein has also taken up this notion of using ray-tracing for simulating radiation, and has created a nice link between Knot-Plot and the imaging program Alias to accomplish this. Finally, we must mention Thistlethwaite's work on symmetric conformations, one of the most beautiful applications of the energy of knots work, which employs Brakke's Evolver to do energy minimization on energy functions we came up with.