Curvature, contractility, tension and the shaping of surfaces

Definition of curvature: A curved line gradually changes direction from one point to the next. The rate of this change in direction, per unit length along the curve (deltaAngle / distance) is called the curvature. If you use units of radians to measure the angles (one radian = 180 degrees/ Pi), then it turns out that 1 / curvature (that is to say: distance / deltaAngle ) is the radius of curvature, in other words it is the radius of the circle an arc of which would most closely approximate that part of the curve. To state this another way, if the direction of the curve changed at a rate of one radian per inch (distances being measured along the curve), then the radius of curvature would be one inch.

Note that a small circle has a large curvature, a large circle has a small curvature, and a straight line has zero curvature (and an infinite radius of curvature).

Curvature = 1 / Radius of curvature ; Radius of curvature = 1 / Curvature

When a curve bends back on itself, one says that its curvature has "become negative", or "reversed its sign"; but it is arbitrary which part of the curve one says has negative curvature and which part one says has positive curvature.

The shapes of lines (and also surfaces) can be entirely defined in terms of the variations of their curvature from point to point, independently of any kind of external coordinate system or axes.

The drawing above shows a curved line on the left and a graph of its variations in curvature on the right. The line starts out with a short straight (zero curvature) region (labeled a), then the curvature rapidly increases in the clockwise direction (in the sections labeled b), then decreases back towards zero (c), passes through zero (at d), then becomes "negative" (in the sense of curving counter-clockwise, in the region labed e), then passes back through zero (at f) to become positive again (g), before becoming very small (in the region labeled h). Below are shown six different curves all having this exact same variation of curvature as a function of distance along the curve, but starting out at different locations and in different directions.

There is an entire field of mathematics, called "differential geometry", that is concerned with defining curves, surfaces and other shapes in terms of their local properties, independent of systems of coordinate axes. Since most biologists are so much more familiar with the alternative field of analytical geometry, based on defining shapes in terms of one or another system of coordinate, we are prone to accept the (very mistaken!) idea that regularity of geometric shape implies the existence of some equivalents of x, y and z coordinates playing a part in the causation of biological shapes.

When a curved rope is subjected to lateral pressures and also to longitudinal tension, these forces will come into balance where the local pressure, curvature and tension obey the equation: P = C * T

And since the radius of curvature is the reciprocal of the curvature: P = T / R

Many specific types of curves can be defined much more simply in terms of the rules by which their curvatures vary with position along the curve, as compared with defining them in terms of x and y coordinates and algebraic equations. Examples include many kinds of spirals, such as the clothoid (whose curvature increases linearly with distance along the curve), and also the elastica curves (the curvatures of which vary with the sin of distance along the curve).

One example of a clothoid are the spirals formed by old-style bimetallic strip thermostats; since one metal strip exceeds the length of the other by a constant fraction, and therefore by some constant amount per unit length of strip; because this difference in length produces and amount of curvature proportional to the difference in length between the base and each different point along the strips, this curvature increases linearly. This type of spiral is also identical to the Fresnel integral, which is important in optics. It is sometimes called Euler's Spiral or Cornu's Spiral. In railways and highways, curves should ideally follow sections of this curve to maximize smoothness. For example, when driving, if you turn the steering wheel at a constant rate to the left and then back to the right, your path will have approximated two sections of a clothoid.

The elastica curves likewise arise in several physical situations. If you compress a long thin metal rod, when it eventually kinks its shape will approximate one of the elastica. This is said to optimize the spatial distribution of bending stress, in that the rate of change of the rate of change of this stress in proportional to minus the local amount of this stress; with this stress being proportional to the local amount of curvature. It is also supposed to be true that river meanders approximate the elastica; this is supported both by theoretical predictions and by observations of actual rivers. These are among many cases in which the curves generated by mechanical causes are ones have simple definitions in terms of differential geometry, but much more complicated ones in terms of analytical geometry.

The curves discussed above are plane curves; so are parabolas, hyperbolas, catenarys etc., all the points of which are in the same plane. There are also such things as space curves, of which a helix is an example. They can likewise be defined in terms of local properties, but in addition to curvature they also have a property called the torsion; one way to think about torsion is as the rate at which the curve twists away from the plane tangential to it. For a helix, both the curvature and the torsion remain constant at each point. In much the same way as you can specify the shape of a plane curve, intrinsically, purely in terms of the variation of curvatures as a function of distance along the curve, it is equally possible to specify space curves in terms of equations stating the curvatures and the torsions at each point along the curve. Although it is a matter of arbitrary convention which directions of curvature and torsion you decide to call "positive" and which "negative", it may be worth realizing that two space curves will be mirror images of each other if their curvatures and torsions have the same absolute values at each equivalent point along the curve, but with either the curvatures or the torsions having the opposite sign (e.g. negative instead of positive).

The shapes of fibrous proteins, including bacterial flagella, are space curves. So are the paths of swimming animals. But the subject does not seem to have much relevance to the shaping of embryos, and will not be further considered here. Conceivably, the torsion concept might eventually be worth pursuing in relation to the shapes of hairs, or the paths of blood vessels, or something like that.

Surfaces also have curvatures. In fact, every point on a surface has not just one, but two different curvatures. The curvatures in exactly perpendicular directions vary independently of one another, and both of them need to be specified in order to specify the overall shape of the surface itself. If you lay a straight-edge along some irregularly convex surface, like some part of the outside of an automobile, then at each point where the straightedge touches, the surface will recede away from it at some angle. Except in areas that happen to be evenly dome-shaped (shaped like a sections of a sphere), the amount of this angular bending-away will vary with direction; it might be twice as much in the anterior-posterior axis as in the left-right direction. Just as in the case of plane curves, the curvature of surfaces is defined as the angular change in direction per unit displacementalong (in this case) the surface.

In the case of a cylinder, the curvature is zero in the longitudinal direction, but equal to one-over-the-radius in the direction perepndicular to this, i.e. in the circumferential direction.

If there are sharp corners or creases, the curvature is not really defined at these places, to the extent that the angular direction of a tangent line would change instantanously at those special points. Alternatively, you could choose to regard the curvatures at these points as merely being very large, whether at a single point or along a line. Also note that curvatures in exactly opposite directions (e.g. north as compared with south, east as compared with west) are always necessarily exactly the same as each other. Since curvature is angular change per unit distance along the, to calculate a curvature in two opposite directions is simply to change from the plus to the negative direction for both angle and displacement, so that these two changes in sign cancel out.

Vectors are familar to most biologists, and so is the concept of a vector field. The curvature of a surface is not a vector, however, but belongs to a category of variables one step up from vectors in complexity. Curvature is a symmetrical tensor of the second rank, in contrast to vectors which are tensors of the first rank (except for certain "vectors" like the one called "curl", which are actually antisymmetric tensors of the second rank). As we will see, both stress and strain are also symmetrical tensors of the second rank. And there are varaibles that are tensors of still higher ranks: elasticity, for example, is a symmetric tensor of the fourth rank.

We can think of the curvature of a surface in terms analogous to a vector field, except that it is a tensor field. A vector-valued variable, for example the strength of an electric field, has an amount and also a direction at each point in the field. Comparing this to a tensor-valued variable (of rank 2 or greater, of course since vectors themselves are first rank tensors), we find that a tensor-valued variable has an amount in each direction.

At each point there is some direction in which this amount is a maximum, and another direction in which the amount is a minimum, and these maximum and minimum directions are always exactly perpendicular to one another. These are called the "principal directions".

The variation of such a second rank tensor property with direction obeys the equation below, which for the particular case of the curvature of a surface is called Euler's equation

The same equation also applies to both stress and strain. It also applies to the directional components of second derivatives. In the diagram above, the property (curvature, etc.) in its maximum direction is 100 units and in its minimum direction is 20 units. When that is true, then the relative amounts of the property mesured in all the other directions are proportional to the lengths of the lines drawn from the center out to the "figure-8 shaped" line. On the right, these directional variations are drawn in the manner of a collection of vectors.

Below are drawn the equivalent directional variations for curvature (etc.) for a situation in which the value of the property in its maximum direction is 100 units, while in the minimum direction is minus 20 units. Notice that, in this case, there are 4 directions (2 pairs of opposites) in which the property in that direction is zero.

The diagram below shows the sorts of directional variatons that occur (according to the equation): Tensor properties of higher orders can vary with direction in more complicated patterns; specifically, the equations have powers of 3 and 4 for tensors of the third and fourth rank:

e.g. Cos4 (angle). These matters have never become part of the "common sense" of most of us.

Very much in the same way as the amount of lateral pressure exerted by a stretched rope depends on the amount of curvature of the rope at each point, the pressure exerted by a stretched sheet or film of material also depends on both the surface's curvatures. The equation is usually considered to be:

delta P = C1T1 + C2 T2

where C1 and C2 refer the curvatures in the principal directions, and T1 and T2 are the tensions in these two directions. Note however that it need not be true that the principal directions will be the same for curvature as for stress, much less that the direction in which one property is a maximum will be the same direction in which the other is a maximum. In a cylinder subject to outward pressure, the principal directions do happen to correspond for these two properties: curvature being a maximum in the circumferential direction and minimum (zero) in the longitudinal direction, while tension is likewise a maximum in the circumferential direction, and exactly half this maximum in the longitudinal direction.

A more familar equivalent to this equation is the following (Laplace's Equation), which applies to the situation in which T happens to be isotropic (the same strength in all directions)

delta P = T( C1 + C2 ) or delta P = T( 1/R1 + 1/R2 )

This applies to soap films, as well as to any situation in which a flexible sheet is stretched equally tightly in all directions.

In the situation where the pressure difference across the surface is zero, then the equation becomes:

C1 + C2 = 0 or 1/R1 + 1/R2 = 0

Which is the same thing as: C1 = - C2 or 1/R1 = - 1/R2

In other words, the maximum and minimum curvatures are exactly equal and opposite everywhere.

Surfaces that have this property are said to be surfaces of zero mean curvature and have been of much interest to mathematicians.

Soap films stretched across a wire ring will automatically shape themselves so as to have this property of zero mean curvature. This is because the contractility of a soap film is isotropic and homogeneous (the same strength in every direction at every location). It is commonly said (correctly) that zero mean curvature shapes have the minimum possible area , and that the soap film is gravitating to the spatial arrangement in which its energy is minimized, we should realize that any isotropically contractile sheet would adopt the same shape. The shape results from the forces being balanced; and the nature of these forces is irrelevant.

There is a classic problem in higher methematics called "Plateau's Problem", one of those sorts of "holy grails" that mathematicians like to set up for one another. In this case, the problem is to find equations that will accurately predict the shape of a soap film, given the geometry of the wire loop on accross which the soap film is stretched. To state Plateau's problem another way, what shaped surface has the smallest possible total area?

 A pair of such shapes are shown above, drawn by a simple computer simulation.
The two drawings on the left show different perspective views of the same shape, while the two drawings on the right show the equivalent perspective views of a different shape. In the left-hand case, the wire frame has a straight line across the front; but in the right-hand case, the wire frame differs in that this front wire loops downward. Notice that the surfaces are everywhere curved: zero mean curvature doesn't mean flat (although flat surfaces do have zero mean curvature), the curvatures in perpendicular directions are equal and opposite. The drawing below was generated by a more complicated variation of this same computer program, in which the directional variations in curvature are also drawn for the 8 neighboring points of each place where the lines cross.

Note that the problem is not the more mundane physical or experimental one of finding out what equations soap films obey. It is assumed that the soap film simply contracts with a constant, isotropic force, a force directed in the plane of the film at each point, having the same strength at each point, and likewise equally strong in all directions parallel to its surface. All that is assumed. The problem is to know what will be the net result of these isotropic and homogeneous forces. Into what shape will these forces mold this layer of soapy water; or to pose this same question in reverse, "What combination of surface curvatures will the soap film need to have adopted in order for all these contractile forces to become exactly counterbalanced against one another?". It is a harder problem than that of the individual forces themselves, just as it is harder to calculate the paths of the planets than to know that their mutual forces of attraction vary inversely with distance, etc.

Usually, however, the form in which this question is posed is some variation of the following: "What shaped surface will have the smallest possible minimum total area, for a margin of any certain prescribed shape?". A variant of this is"What shape will minimize the energy of the soap film?". It might seem that this last variant form should be bringing us closer to understanding the real causation of shape, considering how fundamental a principle the minimization of energy seems to be in other contexts. However, I would argue that the direction in which we would be led by such an approach would take us away from biological and embryological applications of the principles involved. Imagine that we have a sheet of cells, in which each of the cells is actively contractile; suppose that the strength of this contracility is the same for all the cells, and that it also happens to be isotropic, in the sense of all the cells contracting with equal strength in all directions parallel to the sheet (just as strongly in the north-south direction as in the east west direction, and as in the n-n-w direction etc. If you stretched such a sheet of cells across the same kind of a metal ring you had previously used to stretch soap films, into what combination of curvatures would it be molded by the forces of its own contractility? The point is that it would be molded into the very same shape as the soap film. The same equations have the same solutions, to quote Richard Feynman.

Such a sheet of cells would not be minimizing its energy in adopting this shape. Nor would the fact that this shape happens to be the one with the minimum possible total surface area have much of anything to do with the reason why the cell sheet would gravitate to that shape. The real cause has to do with the contractile forces, and the fact that the strengths of these forces are the same for all the cells, and in all the directions (homogeneity and isotropy). To the extent that these active forces are approximately homogeneous and isotropic, their net effect will therefore be the same as it is in the case of soap bubbles.

Since Plateau first proposed his problem more than a century ago, several mathematicians have had considerable success in establishing the principles by which soap films could be calculated. My impression is that the problem is still not regarded as completely solved, however. It may thus be premature to propose an extension of Plateau's problem, in fact what amounts to a converse version of this problem, extended to a much wider variety of shapes. This extended or converse problem that I would like to propose is basically the following: "Given the geometric shape of a surface, a cylinder for example, what combinations of rules of contraction would a sheet of material need to obey in order that it would gravitate to that particular (e.g. cylindrical) shape?"

Plateau's problem: given the rules of behavior of the parts (homogeneous, isotropic contractility) into what shapes will the material contort itself.

The converse problem: given the shape into which a sheet of material is observed to have contorted itself, what rules of behavior do its component parts need to be obeying so as to have created and maintained this shape?

It should be obvious that the motivation for this converse problem lies in embryology, and developmental biology. Cells and groups of cells contort themselves into all sorts of geometric shapes, and frequently restore themselves to these shapes and arrangements if disturbed. Cylindrical shapes are particularly prevalent, either solid cylinders (such as hairs and, the notochord and the shafts of long bones) or hollow cylinders (such as blood vessels, intestines, urine ducts, etc.). Developmental biologists are faced with the task of figuring out what causes cells to arrange themselves into one shape rather than another. Attention has focused mostly on the physical origin of the forces responsible for shaping tissues, with evidence pointing in most cases to forces of active contraction generated by cytoplasmic actin and myosin (i.e. proteins very similar to those that cause the contraction of muscles). Much less attention has been given to questions about what rules need to be obeyed by these contractile forces, such that they will create one shape rather than another.

Continuum mechanics:

Starting with Galileo, engineers have been busily figuring out mathematical equations to predict the spatial distributions of forces (stresses) and the resulting distortions (strains) within walls, floors, roofs, bridges etc. Textbooks for engineers are full of very sophisticated and impressive equations for predicting stress distributions within all sorts of shapes of beams; likewise, other engineers measure the breaking strengths, Young's moduli, Poisson's ratios and other properties of steel, concrete, wood and other building materials. Very few mistakes have been made in either the mathematical predictions or the empirical measurements, which is one of the reasons so few buildings fall down spontaneously; the other reason is the practise of calculating the maximum force that their planned structure can withstand without breaking, and then putting in 5 times as much material as the amount that would just barely break. A whole field of Biomechanics has been developed devoted to explaining anatomical structures in terms of engineering concepts; this is often a matter of calculating why something the observed distribution of fibers in a notochord sheath is pretty close to an optimum solution to its mechanical load problems. Increasing attention is also being given to what sorts of physical forces need to be (and are) developed within embryonic tissues for the purpose of arranging cells and fibers into their normal anatomical patterns.

Sometimes, mathematicians have been able to perdict spatial distributions of stresses and strians by actually solving the equations. The result is called an analytical solution, and much of the higher mathematics of the later 1700s and early 1800s had this as its motivation. On the other hand, the equations are often insoluble, so that people have to resort to what amount to successive approximations; such approaches are called numerical methods, and have been extensively developed by mathematicians going back to Newton. The development of computers has given this approach a big boost; indeed, the main original motivation for mathematicians to invent computers was for the purpose of doing numerical method calculations.

Finite element methods are one means of calculation of forces and distortions within materials. These are almost always done on computers, and there are many "packages" or sets of programs by which one can apply such methods to physical problems of your own choosing (alas, their prices tend to range upward from $103).

The basic idea is to idealize the continuous material as if it were made up of a finite number of little points, each interconnected to its neighbors by some sorts of elastic springs, or stiff rods, or viscous dash pots (shock absorbers) etc. The computer creates sets of numbers that obey (=are repeatedly recalculated according to) rules meant to be like the rules obeyed by these springs, rods, etc. You can observe some simple home-made examples in the "positional information" folder on your computer disk.

Different classes of quantities.

The simplest type of quantity is the scalar variable. Such variables as chemical concentration, temperature and hydrostatic pressure are scalars. These simply have an amount at each spatial location.

The next more complicated type of quantity is the vector. Speed, acceleration, and forces, such as gravitational, magnetic and electrical forces, are vectors. You an either think about these as having an amount in each direction at each spatial location, or you can think of them as having an amount and a direction at each location.

Beyond vectors, there are an unlimited range of more complicated types of variables called tensors (with vectors being considered as first order tensors, and scalars as zeroeth order tensors). These have an amount in each direction. stress, strain and the curvature of a surface are all in the category of second order symmetrical tensors. There are also higher order tensors, and anti-symmetric tensors; for example, the elastic modulus of a material is a fourth order symmetrical tensor, while the curl of a vector field is a second order anti-symmetric tensor (which turn out to behave enough like vector fields that you can get away with pretending that's what they are!). Much of practical engineering (and "common sense") depends on finding ways to pretend that various higher order tensor variables are really scalar variables. The whole concept of energy and its minimization can be considered to be a way to deal with vectors in terms of scalars. Many requests that biological measurements be made more "quantitative" amount to demands that we pretend like higher order tensor variables are really just scalars.

Near the beginning of all engineering texts, just before they start with 300 pages of increasingly long and impressive equations, can be found the explicit caveat that these equations have been derived on the assumption that the engineering materials are

These idealizations tend not to apply to biological materials; actually, they don't even apply too well to concrete or steel, let alone wood; this is why engineers need to make buildings 5 times as strong as their equations tell them is needed (but note that airplanes are only made 2 to 3 times as strong!).

Do you think that the problems of creating biological structures (during embryonic development for example) will require biologists to learn more mathematics? ...or mathematicians to learn more biology? ...or both?

Perhaps mathematicans will even have to invent some more mathematics, as was the case with the historical development of physics and chemistry.

to the next section, Symmetry