The structure of agarose gels, and the accommodation within them of irregularly shaped particles

The Structure of Agarose Gels, and the Accommodation Within Them of Irregularly

Shaped Particles*

ROGER WEST, M.R.C. Clinical and Population Cytogenetics Unit, Western General Hospital, Crewe Road,

Edinburgh EH4 2XU, Scotland

Synopsis

This paper shows how the number of cross-linkage points (nodes) in a random reticulum, such as an ideal polysaccharide gel, may be calculated in accordance with mathematical principles. The influence of nodal configurations upon the statistical geometry of the reticulum is discussed, and it is shown from experimental evidence that the nodal configurations in agar- gel are nonrandom. A method is given for calculating the accommodation probability of an irregularly shaped particle in a reticulum, which is relevant to the theory of gel chromatography and to the distribution of cells in tissues permeating a network of capillaries or veins.

INTRODUCTION

The structure of agarose and similar gels is usually assumed to be a randomly distributed and orientated suspension of gel fibers in a fluid medium. These fibers comprise bundles of polysaccharide molecules,l with some side chains and kinks (caused by unusual linkages between monosaccharide units) that assist the interchange of polysaccharide chains a t nodes and thereby help stabilize the reticulate structure (see Ref. 2). The gel fibers have a finite radius r , which is assumed to be characteristic of the species and concentration ( G , polymer mass/volume) of the gel. The total length of gel fiber per unit volume L is related to G by r r 2 L = k,G, where k , is a parameter (probably varying with the gel concentration and the salt concentration of the solution from which the gel was set) that relates the hydrated volume of the gel fibers to the mass of polysaccharide that they contain. I assume that nodes (fiber junc-

*Notation: G, gel concentration (mass of agar- per unit volume); F, half-length of a gel fiber; Y, number of gel fiber centers per unit volume; x, number of nodes per unit volume; L, total length of gel fiber per unit volume; r, radius of a gel fiber; g, radius of a “rod” or another gel fiber; ti’, angular variable used in the geometry of fiber-fiber contacts; j3, angle between the surface of a particle and the perpendicular to its radius at a given point; 4, orientation of fibers (units of solid angle); P, accommodation probability; &(ti’), probability of the occurrence of angle ti’; R, radius of a particle; R,, radius a t point P on the surface; R,, reference radius; a, major radius of a spheroid; b, minor radius of a spheroid; k , ratio b / a; A,, cross-sectional area of a particle (= area of its projection onto a perpendicular plane); As; surface area of a particle; V, volume of a particle; qa, dimensionless parameter relating A, to Rg for particles of a given shape; qv, dimensionless parameter relating V to Rg for particles of a given shape; 62, a small increment of n; d x an infinitesimal increment of x . Symbols in italics, e.g., R, r, are generally scalar variables. Symbols in roman capitals, e.g., P, are points in geometrical constructions. Quantities in angle brackets, e.g., (Ap), refer to mean values.

Biopolymers, Vol. 27, 231-249 (1988) 0 1988 John Wiley & Sons, Inc. CCC oooS-3525/88/020231-19$04.00

232 WEST

tions, crossover points) occur whenever fibers intersect. In gels where the cross-linkages are formed by the incorporation of a small proportion of a cross-linking monomer to the polymerization mixture (for instance, polyacrylamide gels copolymerized with a small proportion of methylene bi- sacrylamide), this assumption is invalid and the argument of this paper therefore does not apply to such gels.

In the classic theory of Og~ton ,~ which was developed in order to calculate the distribution of void sizes in a gel, the gel fibers were assumed not to impinge upon one another. But if there are fibers of finite length and radius scattered randomly within a finite volume the probability that some of them impinge upon one another must be greater than zero. Moreover, investigations into the structures of polysaccharide gels have revealed the existence of notes,* and the occurrence of a large number of nodes is necessary to account for the mechanical rigidity of gels. I t would therefore be interesting to be able to estimate x, the number of nodes per unit volume in a gel. Since x has the dimension (length-3), the total length of gel fiber per unit volume L has the dimension (lengtv * length-3), and the gel fiber radius r has the dimension (length), dimensional analysis suggests that x is likely to be proportional to L2r only.

ESTIMATION OF v AND x IN A RANDOM GEL

The estimation of v and x is closely related to the determination of the second virial coefficients of straight rods. This problem has received consider- able attention in the literature in connection with the physical chemistry of colloidal solutions. Relevant papers are those of On~ager,~ Zimm' and I~ihara. ' .~ Onsager5 determined the covolume of a pair of randomly located and orientated straight rods of equal radius (i.e., the volume occupied mutu- ally by the interpenetrating rods; (Eq. (33) of Ref. 5). This equation, rewritten in the notation of this paper with Fl and F, as the half-lengths of the rods and r as the radius, is

In a random suspension of fibers the distances between successive contacts upon each fiber may be expected to have a Poisson distribution, and if the points of contact become nodes the mean distance between contacts is equivalent to the mean fiber length, 2 ( F ) . The number of fibers whose length lies in the range 2 F to ( 2 F + d F ) is v . P( F ) - d F, where P( F ) is the appropriate Poisson function for length F. The sum of the covolumes of all the fiber-fiber interactions per unit volume is therefore

-P(F2)[4F1F2 + (T + 3)r(F1 + F2) + ~ r ' ] - dF, - dFl (2)

The division by 2 corrects for counting each fiber-fiber interaction twice in the integration. Since J P ( F ) = 1 and f F - P ( F ) = 2 ( F ) , and 2 v ( F ) = L by

STRUCTURE OF AGAROSE GELS 233

definition, this integral evaluates to

nL2r x = 2 + (n2 + 3n)vLr2 + n2v2r3

Note that the terms of this equation have dimension (length-3) and therefore relate to (number)/(volume) not (volume)/(volume). The equation is therefore dimensionally equivalent to the number of nodes per unit volume, not to the covolume of fibers per volume of gel. Since L = 2 v F and v = 2x, (F) = L/4x. The equation implies that, if we know the relationship, which is not necessarily linear, between G and L, and between G and r ( r is not necessarily constant) we can in principle estimate x and (F) from the gel concentration alone.

If 2( F ) s r this equation may be simplified to

I have adopted this abbreviation to simplify the algebra in the rest of this paper, but I emphasize that whenever in practice the assumption 2(F) >> r is doubtful, Eq. (3a) should be substituted for Eq. (3b) with consequential reworking.

I have also calculated x directly by geometrical construction and calculation of contact probabilities. This method, which takes a different approach to the problem from that of Onsager, produced an equation that agrees with Eqs. (3a) and (3b). This confirms that Eqs. (3a) and (3b) are correct both dimensionally and in their numerical constant.

Alternatively, x might be a function of the number of fiber growth points per unit volume, y, i.e., equal to (twice the number of fiber initiations per unit volume) - (the number of extinct growth points per unit volume). In this case we could conceive that nodes form whenever two fiber growth points come within distance d of one another, whereupon a node is formed by the splicing of fiber ends and two new fibers proceed from the node, allowing the formation of subsequent nodes by the same process and thereby generating a greatly cross-linked reticulum. One would expect that, at the end of this process, when all the agarose in solution has been converted into gel fiber, there would be a rather large number of free fiber ends, contrary to experimental eviden~e.~ If, however, this process is responsible for the formation of nodes, x should be proportional to nd3y2/6 multiplied by 2L/d, i.e., nd2y2L/3. These two models might therefore lead to different scaling relationships between G and x.

If the orientations of fibers are random then all values of 6' are equally probable. But if the points of contact become nodes and if the configurations of nodes are biased by energetic constraints, then the possible values of 6' are not all equally probable. If nodes are the meeting point of four fibers there are six fiber-fiber angles at each node. The probability that lies between 8 and 6' + d6' is likely to be conditional upon the values of the other five angles 8, to

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86. The mean probability of 8, is

with summations over 8, . . . 8, weighted according to Pr( 8,). . . Pr( 06). Even if the physical chemistry were known in sufficient detail to estimate these probabilities, the averaging process would still present formidable problems in computing. Pr(8) should be regarded as a weighting, with a mean value of 1, i.e.,

The relationship between the distributions of fiber lengths, fiber-fiber angles, and the location and orientations of fibers may be examined by a different approach. We note that a reticulate gel, in which all nodes are the meeting point of four fibers and in which each fiber terminates at either end a t a node, is topologically equivalent to a tetrahedral lattice. We may therefore start with a regular tetrahedral lattice and produce the structure of a random reticulum by random topological transformations alone.

In a regular tetrahedral lattice the fiber lengths are all the same (2 F ) , the fiber-fiber angles are all the same (the tetrahedral angle), and neither the location of fiber centers nor the orientation of fibers is random. Let A, B, and P be three neighboring nodes in the plane z = 0 of an undisturbed tetrahedral lattice, and let point P define the coordinate origin. The Cartesian coordinates of the nodes are

A = (X,Y,O)

B = (-X,Y,O)

P = (o,o, 0)

The lengths of the fibers are (X2 + Y2)'/' = 2F and the distance between A and B is 2 X . The angle APB is the tetrahedral angle and its cosine is

(AP)' + (BP)' - (AB)' 2 . (AP) . (BP)

~2 - x2

Y2 + x2 - -

Now add random increments x, y, and z with Gaussian distribution to the Cartesian coordinates of A, P, and B, then adjust the coordinate system so that P once again lies at the origin. The coordinates of the three nodes are

p = ( O , O , O )

STRUCTURE OF AGAROSE GELS

The lengths of the fibers are

235

The increments x, y, and z each have mean value zero, and they cluster around the mean, so that the increments d in the lengths of (AB)2[d2 = (AB)2 - 4F2] have a Gaussian distribution:

(AB') = 4X2 + ( d 2 )

(AP2) = (BP') = 4F2 + ( d 2 )

jm jm jm (2 + y2 + 2) ( d ) - ( & L J ) 3 - m - m - m

1 2 -

The mean value of the angle APB is therefore

Jm Jm JW Jm Jm JW cos-'(C) 1

36a3a6 -m -m -m - m - m - m

236 WEST

The variance of O p is

If we know the location of point P, the directions of PA and PB, the mean distance between nodes, and the underlying structure of the gel (the undeformed structure, in this case a tetrahedral lattice) we could predict the location of nodes C and D, linked to P by fibers, by the algorithm appropriate to the undeformed structure. There is then a probability of 1/2 that node C actually lies within a radius 0.6745~ of its predicted position, and likewise node D. If u = 0 the position of any node in the gel can be predicted with complete confidence from the location of any one node and the orientation of the axes of the structure. Therefore u2, the variance of the lengths of the fibers, determines the variance of the cosines of fiber-fiber angles and the intrinsic predictability (nonrandomness) in the location of nodes (and therefore of fiber centers). These three parameters are interdependent, so that if the distribution of fiber-fiber angles is known, the distribution of fiber lengths and the randomness of the location of fiber centers can be ascertained. Unfortunately, the distribution of angles cannot be deduced from the distribution of fiber lengths since the number of degrees of freedom of the former is greater than that of the latter.

These two approaches to the analysis of reticulate structure can now be drawn together. The Gaussian distribution of fiber lengths that I have assumed during the second approach transforms into a Poisson distribution as u increases from zero. It is a feature of Poisson distributions that their variance is equal to the square of the mean. We may define the “coefficient of randomness” of a reticulate structure C, as (variance of fiber lengths)/(mean fiber length)2. For a regular geometrical structure in which all fiber lengths are equal, C, = 0. For a random reticulum, C, = 1. If the distribution fiber lengths has more than one mode i t should be decomposed into unimodal distributions and the calculation of C, made for each mode.

In a random gel, u = 2(F) and the radius of the sphere within which there is a probability of 1/2 of finding a predicted node is 1.349 . n1I2(F). For n = 2, this is nearly 2(F) . After two steps in a random gel, the probability that a node is located within the largest sphere, centered on the expected position of the node, that does not include the expected position of any other node, is only 1/2, and if a node is found within the sphere i t is as likely as not that it is not the “expected” node. Therefore the positions of nodes and fiber


centers are completely unpredictable a t a distance of only two linkages from any node whose position and orientation are known, and the structure of the reticulum may be considered completely random.

In a previous paper” I referred to the discovery by Waki et al.” of a great difference between the distribution of fibers in agarose gels set from water solution and those set from saline solution. I proposed that the most likely explanation for this difference is that the presence of salt in the solution tends to force the configurations of nodes toward unstrained forms, and that this determined the distribution of fibers in the fields examined by Waki et al. Table I1 of their paper reports that, in a 0.5% w/v gel set from distilled water, the radius of the gel fibers was r = 4.2 X cm (the unit given was “pm” but, for consideratioris of internal consistency in the paper this must be a misprint for “nanometers”), and the length of gel fiber per unit volume L was 2.4 x 10“ cmP2. From Eq. (3b) the number of nodes per unit volume in this gel is approximately x = 3.8 X 1014 ~ m - ~ . The average volume associated with each node is 1/x cm3, which in this case is equivalent to a cube whose sides are 1.38 X cm. The authors reported that L in gels set from saline solution (250 m M Tris borate buffer) was about 14% smaller than in gels set from distilled water, so that x is likely to be somewhat smaller in the “saline” than in the “water” gels.

In Fig. 3 of their paper, Waki et al. show histograms of the number of fibers per field seen in electron micrographs of gels. Data is presented for fields of 2 x lo-‘’ cm2 and 4 X lo-’’ cm2, and in each case 500 fields were examined to compile the histogram. These fields are equivalent to squares whose sides are 1.41 x cm, and therefore by coincidence are of the same order as volume associated with one node. The histograms compiled for the “saline” and “water” gels are strikingly different, and the former have minor peaks at n = 4,7, and 9 fibers per field. Since the sample sizes were large, the presence of these peaks demands an explanation. I believe the most likely explanation is that, in the “saline” gels, the orientations of nearby nodes are

and 2.0 X

Fig. 1. Schematic representation of a node whose configuration agrees approximately with that required for the simulation shown in Fig. 2. The details of this figure are not intended to imply a considered model of the molecular contortions within the node. The significance of the fiber-fiber angles is only that one opposite pair is greater than the other.

238 WEST

0.5-

0.4-

0.3-

significantly correlated due to the constraints placed upon fiber-fiber angles by the salt.

Figure 1 is a schematic, two-dimensional representation of a node whose configuration I would expect to be relatively unstrained, since none of the helices or molecules passing through it are required to be bent through a small radius of curvature. The details of the figure are not intended to imply a considered model of the molecular contortions within the node, nor do the fiber-fiber angles have any special significance except that one of the opposite pairs is appreciably greater than the other. If a plane is defined at random in the vicinity of this node the probability that it transects 1 of the fibers

-Data .----i S i mulat i on

Fibres per field

- Data - S i mulat ion

Fibres per f i e l d

Fig. 2. Simulation of data from Waki et al." of the number of fibers per field seen in freeze-fracture electron micrographs of agarose gels. (A) 4 X lo-"' cm2 fields. (B) 2 X lo-'' cm2 fields. For further details see text.


meeting there is greater than that it intersects 0 or 2 of them, and this probability in turn is greater than that it transects 3 or 4. I therefore sought to simulate the histograms upon the basis of a very simplified hypothesis about the distribution of nodes and fibers, viz., that the number of nodes within the vicinity of plane fields has a Poisson distribution with mean p ; that, in respect of each such node, the probabilities that the plane transects 0 . . . 4 of the fibers meeting at the node are in the ratio x : 0.5-x : x : 0.25-0.5~ : 0.25-0.5x, with x > 0.25; and that the orientations of the fibers meeting at different nodes are uncorrelated. The numerical values of p and x used in the simulation were found by iterative computation to minimize the sum of the squares of the differences between the data and simulated points. These values were p = 1.045 in the 2 X 10-l' cm2 field and 1.705 in the 4 X cm2 field, and x = 0.205, giving, as the probabilities of finding 0 . . . 4 of the fibers, the values 0.205, 0.295, 0.205, 0.1475, and 0.1475. Simulated histograms based on these assumptions are shown in Fig. 2. It is interesting that this simple model provides a simulation of the data in which the minor maxima are correctly placed. The agreement of the model with experimental results might be improved by introducing a more complicated hypothesis, perhaps by introducing correlation between the orientations of fibers meeting a t the nodes (local anisotropy). However, this would increase the number of independent variables in the simulation, which I think would not be justified on the basis of the existing experimental evidence.

I must emphasize that I do not regard this simulation as a proof of my hypothesis, but i t is encouraging that an algorithm based upon a minimal set of assumptions can lead to such a close agreement with experiment. However, the distribution of the number of nodes within a small volume may be more random than the distribution of the configurations a t nodes, although I have argued that these distributions should be related. The relationship is a complicated one and this apparent discrepancy with the theory may not be significant, but I must leave it as an unsolved question.

PROBABILITY OF ACCOMODATING IRREGULARLY SHAPED PARTICLES IN A GEL

A problem related to the structure of gel is the probability of the accommodation by the gel of particles placed at random within it. By this I mean the probability that a particle of a particular size and shape placed at random into a gel of a particular structure does not impinge upon any fiber of the gel. This problem is closely related to the question of the volume fraction of the gel accessible to infiltration by particles of that type and therefore to the properties of the gel in gel chromatography. Ogston3 solved the problem for spherical particles placed at random within a gel of randomly distributed, unconnected gel fibers. However, the particles of interest in gel chromatography are rarely spherical and real gels do not conform to Ogston's model. Rodbard and Chrambach" showed that Ogston's equation could be extended to nonspherical particles, but their proof rested upon the postulate that fiber orientations are random, which is shown here to be unnecessarily restrictive. In this section I develop the analysis for the case of irregularly shaped particles placed at random in nonrandom gels.

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C A D

J B I

Fig. 3. Geometrical illustration of contact probabilities by gel fibers on an irregularly shaped solid. (A) A section through the center of mass of the particle in orientation #. The “capsule” outline represents the volume within which fibers impinge upon the particle, whose outline is the “ellipse.” (B) A transverse section of Part A in the plane AB.

Consider an irregularly shaped convex particle, surrounded by fibers each of length 2 F, whose centers are located a t random in space but whose orientations may or may not be random. The total number of fiber centers is v, and the number whose orientations are 4

In Fig. 3(a) the orientation 4 is left to right in the plane of the page. The line AB represents a plane passing through the center of mass of the particle to which the orientation I) is perpendicular. Figure 3(B) is a section through the particle a t AB rotated by 90” relative to Fig. 3(A). The orientation 4 is now perpendicular to the page. Points lying in the plane of the page and to the left of AB are located vertically above AB in Fig. 3(B). Figure 3(B) is therefore also the projection of the particle in the plane perpendicular to 4.

The “halo” around the particle in Fig. 3(B) represents the result of expanding the particle by d R in all directions. In order that the proportions of the particle shall remain unchanged throughout the expansion, d R is not constant over the whole surface but is locally proportional to R, the distance at that point from the center of mass. Suppose we define a reference radius, R,, as the radius of the particle along a defined axis. Since the shape of the particle remains unchanged during the expansion, the area of any cross section through the center of mass is always proportional to Rk. I t can be proved that for any convex solid the mean area of sections through the center of mass (equivalent to the mean perpendicular projection) is one quarter of the surface areal3: (Ap) = As/4. Let (Ap) = 2nqAR& where qA is a constant that depends upon the shape of the particle and the axis chosen for the reference radius, but not upon the particle’s size. The mean area of the halo around sections through the particle is the differential of this expression, which is

64 is v+.

47~. qA. R, * dR, (11)

Similarly, the volume of the particle is V = 4rr . qvRL/3, where q v is also a constant that depends upon the shape of the particle and the axis chosen for the reference radius, but not upon the particle’s size. The volume of the


“shell” round the particle is the differential of this expression, which is

The factors 2m and 4 ~ / 3 could be included in qA and qv, but it is convenient to keep them separate so that for a sphere q A = qv = 1.

In Fig. 3(A), the thin rectangular strips CDEF and GHIJ are a longitudinal section through a cylindrical shell of which the halo is a cross section. The points, C, D, E, F, G, H, I, and J each lie at a distance F from AB. Therefore any fiber whose half-length is F, whose center lies within the cylindrical shell just described, and whose orientation is J , will touch the perimeter of the particle within the halo, that is, it will make tangential contact with the particle. From Eq. (11) the average number of these fibers is

dn, = 4nv+F. qA . R E . dR, (13)

The cylindrical shell is capped by a surface at each end, of thickness dR, which is an exact image of the half-surface halo of the particle lying to the left or right of AB, and translated through distance F to left or right, respectively. Any fiber whose center lies within either of these caps, whose half-length is F, and whose orientation is J , will make an end contact with the surface of the particle. From Eq. (12) the average number of these contacts is

In calculating the number of contacts with fibers of orientation J , i t is important to remember that we mean all fibers whose orientation lies within a range dJ,, with mean orientation J,. Some of the fibers whose centers lie within the cylindrical shell and whose orientations are J , will end outside the shell, but these will be compensated, on average, by fibers whose centers lie just outside the shell but end within it.

The number of fibers impinging upon or within the particle may be calculated as follows: In Fig. 3(A), any fiber whose half-length is F, whose orientation is J,, and whose center lies within the space of which the closed curve FEHG is a perimeter, will impinge upon the particle. The volume of this space is 2 F . A, + V where A, is the area of the projection of the particle on a plane perpendicular to 4. Since ( A,) = A,/4, the mean number of fibers of orientation J , impinging upon the particle is

V+’ (3 2 + v)

So far I have considered only contacts by fibers of defined length 2 F and orientation J,. If contacts by fibers of different lengths and orientations are stochastically independent the contact probabilities are additive. To obtain the total number of contacts within the shell, add Eq. (13) + Eq. (14) and integrate over F and 4 , replacing 2v( F ) by L. The result is

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The total number of contacts within the particle is similarly found by integrating Eq. (15) over F and $. The result is

Note that the integration over F and 1c, does not require that the distribution of F conforms to any a priori model or that the fibers are isotropic, only that contacts by fibers are independent of one another. In case even this assumption is untrue, as it might be, for instance, if the “fibers” are replaced by a network of anastomosing veins, it would be possible to use an integration method that takes into account the appropriate conditional probabilities. This would generally lead to a different result from the random case, as I show below. Chalkley et al.14 and Cornfield and Chalkley” have shown that, for the purpose of calculating the expected number of contacts, the LA,/4 formula applies to concave as well as convex solids.

The probability that no fiber impinges upon the particle is, from Eq. (17),

L * A,

and the probability that there is a t least one fiber contact within the “shell” is 1 - exp[ - Eq. (IS)]. Since dR, is infinitesimal the value of 1 - exp[ - Eq. (IS)] = [Eq. (16)]. Following Ogston’s argumentY3 the probability distribution of spaces just large enough to accommodate particles of the given shape whose reference radius is R, is

Since this depends upon the shape of the particle, I avoid the word size with its connotation of volume and describe these spaces as isometric to the given particle with tolerance dR,. (This differs from the usual meaning of “isometric” in geometry.) The equation is valid only when the thickness of the shell within which surface contacts are estimated is allowed to be locally proportional to the particle’s radius. But to estimate the probabilities of surface contacts their expected number must be counted within a shell of uniform thickness. In Fig. 3, assuming the halo to be of uniform thickness d R, the number of surface contacts at each orientation 1c, is L+C+ . d r . d$, where L, is the total length per unit volume of fibers of orientation $ and C, is the perimeter of the projection of the particle in orientation $. The number of tangential contacts on the whole surface is found by integrating this expression over 4. If the orientations of fibers and particles are uncorrelated, L, is uniform over the surface and this integration evaluates to L . ( C ) where ( C ) is the mean perimeter over the whole surface. If the orientations are correlated the integral has a different value, greater than that of the uncorrelated case if fiber orientations tend to be perpendicular to the major axes of the particles, less if they tend to be parallel. In general, the probability that the


particle is accommodated with at least one surface contact by fibers of radius r is

P,= (1 - exp - (/L+C+ . d# + v + A , ) ) . r . exp -

Equations (19) aqnd (20) appear similar, but in fact they are fundamentally different because of the different concepts of the shell upon which they are based. The application of these formulae to ellipsoids is illustrated graphically in Fig. 4. I t is evidently important always to choose the correct formula for the problem in hand!

Continuing to follow Ogston’s argument, the fraction of the volume of the gel accessible to particles of the given shape whose reference radius is R, is

whose value between the required limits is

L . As

Equations (19) and (21) take no account of the thickness of the gel fibers, and apply strictly only if the radius of the fibers is infinitesimal. The diameter of gel fibers is perhaps their most difficult characteristic to measure. If the gel fiber radius r has been estimated the correction may be made as follows: Add r to the radius OP of the particle over its whole surface and reevaluate Eqs. (19) and (22). (This correction was suggested by Rodbard and Chrambach.” After this adjustment Eq. (19) remains fundamentally different from Eq. (20) and Eq. (22) is no longer strictly accurate, since the condition that the shape of the particle should remain unchanged throughout its expansion is no longer true. If r is fairly small compared to the smallest radius of the particle, the error arising from the application of Eq. (22) as the definite integral of Eq. (19) will not be large. In practice, experimental errors in the estimates of r and the dimensions of the particle are likely to be greater than those introduced by approximations in the algebra.

Equation (22) is equivalent to Eq. (7) of Rodbard and Chrambach.12 These authors stated that for any shaped object there exists an equivalent sphere with the same accommodation probability. But Eqs. (17) and (18) contain both the surface area and volume of the particle. Since the sphere is that solid for which the (surface area)/(volume) ratio is minimal, there cannot exist any nonspherical particle equivalent to a sphere in this sense. For a given nonspherical particle, a sphere can be defined that yields equal values in Eqs. (18) and (19) only if the values of both L and v are specified, or if the uV term is assumed to be negligible. This approximation may be convenient in practice but should not be incorporated into the theory or terminology, and I suggest that use of the phrase “equivalent sphere” be discontinued.

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1 .o

0.9

0.8

0.7 h

c h .c -

0.6 v L ‘f

0.5

3 c 0 0.4 a ’i;i >

0.3

0.2

0.1

0.0

pa>

Ac /// E-

l I I I I I I I

1116 118 114 112 1 2 4 8 16 Axial ratio, a/b (a.b2 constant)

Exemplary plots of some accommodation functions of spheroids of constant volume plotted vs log( a / b). The central vertical line marks the sphere. Oblate spheroids are to the left of this line, prolate spheroids to the right. The numerical constants used in the simulation were those found by Waki et all’ for a 18 agarose gel: L = 3.0 x 10” cm-’ and r = 3.3 x cm for the reticulum, and R = 1.17 X cm for the sphere (giving the volume 6.7 x lo-’* cm3 for the spheroids). For these values of L and r, x is estimated by Eq. 3(b) to be about 6.5 X 1015 cm-3, SO that the number of “apparent end contacts” 2v has been assumed to be considerably less than x. The scale of the ordinate is correct for the accommodation probability (Ac) with these constants, and for the plot of probability of isometric spaces (Pa), for which the value d R = cm was chosen to make the Ac and Pa curvb coincide at the sphere. The Pp and Pm curves represent the probability of accommodation with at least one surface contact, using the maximum perimeter and mean perimeter formulae, respectively. These curves were multiplied by an arbitrary factor to bring them within a convenient range of the ordinate, so their position with respect to the Ac and Pa curves is without significance. Note, however, that the Pp and Pm curves coincide at the sphere.

Fig. 4.

OTHER FACTORS CONCERNING THE CALCULATION OF ACCOMMODATION PROBABILITIES

The following points should be borne in mind when evaluating accommodation probabilities by the method described above. For the purpose of calculating contact probabilities, gel fibers may be regarded as probes testing the particle at all points on its surface. Such probes cannot resolve features smaller than their own diameter. Also, certain features, even if their linear dimensions are quite large, contribute little or nothing to contact probabilities. For instance, fibers have a very small (or zero) probability of making


tangential contact within a groove or depression on the surface of a particle, since they are very likely to encounter the rim of the depression first. Nodes also have a very small probability of penetrating a depression without making tangential contact a t the rim. Mapping the surface of a particle into the depth of a depression would undoubtedly overestimate the true contact probabilities. It would therefore be correct to smoothe over depressions and features smaller than the diameter of gel fibers before mapping the surface, since the errors arising from this procedure are likely to be quite small. Representing the complex shape of a real protein molecule by a geometrical solid or other simple shape that is a reasonably close-fitting envelope over the real shape may therefore provide a surprisingly good estimate of the true contact probabilities.

I have referred to "end" contact by fibers as contributing to contact probabilities. These end contacts appear as the vV terms in Eqs. (19) and (22). I have shown elsewhere" that genuine fiber ends, nodes, and the curvature of fibers all contribute to the apparent value of v in these terms. I argued that the magnitude of this term in a reticulate polysaccharide gel such as agarose is likely to be small, as seems to be found by e~per iment .~ , '~ The vV term in Eqs. (19) and (22) may therefore be ignored to a first approximation in the case of agarose gels. In the case of polyacrylamide gels these terms may be quite large, partly because of the curvature of polyacrylamide molecules and because the application of Eq. (19) is difficult.

There are some further restrictions on the applicability of Eqs. (19) and (22). Strictly speaking, they apply only to the accommodation of the first particle in the reticulum. When subsequent particles are inserted at random there is a finite probability that they will impinge upon a particle that has already been accommodated. The accommodation probability therefore decreases as the reticulum becomes saturated with particles. If the accommodation probability for protein molecules in gel chromatography is significantly reduced because of overloading, molecules penetrating the partially saturated gel will behave as though they are larger than their true size and will move through the column faster than they would at infinite dilution. This accounts for the broadening of elution bands in overloaded gel chromatography experi- ments. The geometrical analysis of this phenomenon is beyond the scope of this paper.

Equations (19) and (22) are correct as presented here only if the orientations of fibers and particles are uncorrelated. For this to be so i t is necessary that either the fiber orientations, or the particle orientations, or both, are isotropic. If not, then the mean cross section of the particle (Ap) is not equal to (A,)/4. If in these circumstances a correct evaluation of 4A were substituted into Eqs. (19) and (22), those equations could still be used to evaluate contact and accommodation probabilities. An example of such a reworking is given below in the section on spheroids. This problem does not arise if the particles are spheres, since the concept of the orientation of a sphere is meaningless.

ACCOMMODATION PROBABILITIES FOR SPHEROIDS

An interesting example of a nonspherical solid for which accommodation probabilities can be calculated is the spheroid. This is the solid generated by

246 WEST

rotating an ellipse about either its major or its minor axis. The shapes of molecules of biological interest are not usually true spheroids, but a suitable spheroid is a much better approximation to the shape of many protein molecules than is a sphere. The properties of spheroids are amenable to algebraic manipulation, and will therefore serve to illustrate the interpreta- tion of the equations deduced above. This example has also been worked by Giddings et al.,13 whose solution, however, omitted the vV term.

Prolate Spheroids

If an ellipse is rotated about its major axis the solid generated is a prolate spheroid. Let the major radius of the ellipse (the radius along the major axis) be a and the minor radius b. If the axial ratio a / b is 1 the spheroid is a sphere. As the axial ratio increases the prolate spheroid becomes cigar shaped. The shape of a spheroid is defined by its axial ratio, its size by its axial ratio, and the magnitude of either a or b.

The properties of the prolate spheroid required for substitution into Eqs. (19) and (22) can all be expressed in terms of a and b, viz.,

volume of a spheroid:

4nab2 v = - 3

surface area of a prolate spheroid:

{FZF (23b) where E =

a ab . sin-'€

E

It is therefore sensible to chose either a or b as the reference radius, and to express the other principal radius in terms of it. If we choose a as the reference radius and define 12 = b / a, then Eq. (23a) becomes

4nk2a3 4nqva3 v = ___ -- - 3 3

so that qv = k2. Equation (23b) becomes

A, = 4na2 [ - k 2 + k . s i n - ' ( d z ) ] = 4 2 q A

2 2 . J l T

so that qA is the term in square brackets. Equation (19) becomes

4?rvk2a3 3 - (2m~g ,a + 4nvk2a2) . exp -

aP - - d a

This is the differential with respect to the major radius. The differential with


respect to the cube root of the volume is

d P a 213 d P d(V1I3) = jb) . da

This differential enables us to compare the probability of voids isometric to a set of particles of equal volume but of different shapes.

The accommodation probability is

4nvk2D3

a 3 . dD (27) lP(2nLqAD + 4nvk2D2) * exp -

which is equivalent to Eq. (22).

Oblate Spheroids

If an ellipse is rotated about its minor axis the solid generated is an oblate spheroid. If the axial ratio a / b is 1 then the spheroid is a sphere. As the axial ratio decreases the spheroid becomes discus shaped. We define k = b / a as before. For an oblate spheroid b > a and therefore 4- is imaginary. We must therefore reevaluate qA. The surface area of an oblate spheroid is

which may be rewritten

[ k 2 ln(k(d + 1))] 4na2 - + = 47ra2q,

2 E’

where qA is the term in brackets as before. With these replacements for q v and qA, Eq. (25) may be used for oblate spheroids.

Extreme Cases

When a = b the spheroid is a sphere, q A = qv = 1 and Eqs. (19) and (22) are identical to those found by Ogston for a sphere [Ref. 3, Eqs. (12) and (13)]. If b is finite and a is reduced to zero the oblate spheroid becomes a double-sided flat disk whose surface area is 2nb2 and whose volume is zero. Then Eq. (22) is equal to exp - (nLb2/2). Waki et a1.l’ proved that the mean number of fibers transecting a unit plane inscribed in a random network of fibers is L/2, from which it follows directly that the probability that no fibers transect a disk whose radius is b is also exp - (nL b2/2). As the axial ratio of a spheroid approaches infinity its surface area approaches a2ab. If a remains constant while b + 0, the spheroid becomes a straight rod of finite length and infinitesimal thickness. Then as a / b + 00, As +. 0, V +. 0, and

248 WEST

exp - (LAs/4 + uV) -+ 1. This is the correct accommodation probability for a rod of finite length and infinitesimal thickness within a finite random network of infinitesimally thick fibers. If a / b + 00 while either ab or ab2 remains constant, a + 00 and the accommodation probabilities approach exp - (s’ab) and zero, respectively. Equation (22) is therefore shown to yield the correct result when a = b and when a / b is taken to either of its limits.

If a spheroid of volume V is placed a t random in a reticulum for which L, v, and the fiber radius r have defined values, then the accommodation probability for the spheroid [Eq. (27) ] , the probability of its accommodation with at least one contact [from Eq. ( Z O ) ] , and the volume fraction of isometric voids [dP/d(V’I3), Eq. (26)] all vary with the axial ratio of the particle. In Fig. 4 the axial ratio has been varied while the volume remains constant, and these three functions have been plotted vs the axial ratio. The accommodation probability has its maximum value for the sphere, for which the surface/volume ratio is minimal. The plots of the probability of accommodation with a t least one surface contact have been plotted using two formulae for /L+ . C, . dJ/. For curve Pp the integral was evaluated as L * (maximum perimeter), corresponding to the fibers all being orientated at right angles to the major axes of the particles-“broadside” contacts. For curve Pm the integral was evaluated as L . (mean perimeter), corresponding to random orientation of the fibers with respect to the particles. Curve Pp always lies above curve Pm, except a t the sphere. The minimal curve of this set is not shown. It corresponds to “end on” contacts for prolate spheroids, when the major axes of the particles and fibers are all parallel, and “edge on” contacts for oblate spheroids. It has its maximum value a t the sphere, where it coincides with Pp and Pm curves. In the case of prolate spheroids the perimeter presented for contacts is 2 s b = Baku, and Eq. (19) becomes

P, = (1 - exp - (2sLka + vAS)) * r . exp - (7 LA, + uV)

The Pa curve, based on Eq. (19), is presented for contrast: its very different shape is due to the different geometry of the shell within which the surface contacts are counted.

It may be that in biological tissues in which it is physiologically necessary to maximize the contacts by a network of veins or capillaries upon a matrix of cells (for instance, to optimize irrigation with nutrients) the optimum arrange- ment is for the capillaries to be orientated preferentially in one plane and for the cells to be prolate spheroids (or polyhdra circumscribing prolate spheroids) whose major axes are preferentially orientated perpendicularly to that plane. However, i t would also be necessary to consider the contact probabilities for cells subsequent to the first, and the packing fraction of differently shaped cells, before coming to this conclusion. Moreover, there are likely to be other physiological considerations, such as optimizing the surface area/volume ratio, which influence the shape of cells. Both these questions are beyond the scope of this paper.

I thank Dr. A. D. Gilbert and Dr. A. M. Davie, both of the Department of Mathematics, University of Edinburgh; for a critical reading of the manuscript and advice with the algebra. The


figures were photographed by Mr. D. Stuart at the MRC Clinical and Population Cytogenetics Unit, Edinburgh. I am a staff member of the United Kingdom Medical Research Council.

References 1. Leatherby, M. R. & Young, D. A. (1981) J . Chem. SOC. Faraday Trans. 1 77, 1953-1966. 2. Dea, I. C. M., McKinnon, A. A. & Rees, D. A. (1972) J . MoZ. Biol. 68, 153-172. 3. Ogston, A. G. (1958) Trans. Faraday SOC. 54, 1754-1757. 4. Sewer, P. (1983) Electrophoresis 4, 375-382. 5. Onsager, L. (1949) Ann. N Y Acad. Scz. 51, 627-659. 6. Zimm, B. H. (1946) J . Chem. Phys. 14, 164-179. 7. Isihara, A. (1950) J . Chem. Phys. 18, 1446-1449. 8. Isihara, A. (1951) J . Chem. Phys. 19, 1142-1147. 9. Laurent, T. C. (1967) Biochzm. Biophys. Acta 136, 199-205.

10. West, R. M. (1987) Biopolymers 26, 343-350. 11. Waki, S., Harvey, J. D. & Bellamy, A. R. (1982) Biopolymers 21, 1909-1926. 12. Rodbard, D. & Chrambach, A. (1970) Proc. Natl . Acad. Scz. USA 65, 970-977. 13. Giddings, J. C., Kucera, E., Russell, C. P. & Myers, M. N. (1968) J . Phys. Chem. 72,

14. Chalkley, H. W., Cornfield, J. & Park, H. (1949) Science 110, 195-297. 15. Cornfield, J. & Chalkley, H. W. (1950) J . Wash. Acad. Scz. 41, 226-229. 16. Siegel, L. M. & Monty, K. J. (1966) Biochim. Biophys. Acta 112, 346-362.

4397-4408.

Received April 24, 1987 Accepted August 5, 1987

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The structure of agarose gels, and the accommodation within them of irregularly shaped particles