J DVc
where D distance2 time is the diffusion coefficient of cells in brain tissue. The theoretical models, referred to above, considered the brain tissue to be homogeneous so the diffusion and growth rates of the tumour cells are taken to be constant throughout the brain. This is not the case, of course, when considering tumour invasion into white matter from grey. With constant diffusion the governing equation 11.1 with 11.2 is then As we shall see this model gives reasonable agreement with the CT...
Background and Experimental Results
There is an obvious case for studying bacteria. For example, bacteria are responsible for a large number of diseases and they are responsible for most of the recycling that takes place. Their use in other areas is clearly going to increase as our understanding of their complex biology becomes clearer. Here we are interested in how a global pattern in bacterial populations can arise from local interactions. Under a variety of experimental conditions numerous strains of bacteria aggregate to form...
Info Stg
where a, b, a, p and K are positive parameters. If we include activator inhibition in the activator-inhibitor system in the first of these we have, for f and g, f u,v a bu ku2 , g u,v u2 v, 2.9 where k is a measure of the inhibition see also Section 6.7 in Chapter 6 in Volume I. Murray 1982 discussed each of these systems in detail and drew conclusions as to their relative merits as pattern generators he presented a systematic analytical method for studying any two-species reaction diffusion...
Info Bdu
D we again use the Fisher-Kolmogoroff approximation D v2 4p from 11.36 as was also used by Burgess et al. 1997 . We again associate the diffusion coefficients in white and grey matter by Dw vW 4p and Dg vg 4p with the experimentally observed linear velocities vw and vg, respectively. To deduce estimates for the diffusion coefficients we need to determine the growth rate of the tumour cells. Alvord and Shaw 1991 cite doubling times of one week to one month for gliomas in vivo. The resulting...
Numerical Solution for the Epidermal Wound Repair Model
Sherratt and Murray 1990, 1991 numerically solved the model system of equations 9.12 and 9.13 together with 9.14 - 9.18 in a radially symmetric geometry and the results were compared with data from a variety of experimental sources both in vitro and in vivo. The results are given in Figure 9.6 together with the quantitative results from experimental studies on re-epithelialisation. For example, in Van den Brenk's 1956 Figure 9.6. The decrease in wound radius with time for the normal healing of...
Quadriradii Define
V Cpu mnpp - p . 6.65 The matrix source term is probably necessary to induce branching. From the above sections we know that this system, given by 6.63 - 6.65 , can generate a spectrum of complex spatial patterns if the parameters are in the appropriate ranges. After nondimensionalising the system using 6.21 , linearising about the relevant steady state u 0, n p 1 and looking for solutions proportional to ea 1 l k x we get refer to Section 6.3 the dispersion relation, which is similar to 6.31...
Evolution and Morphogenetic Rules in Cartilage Formation in the Vertebrate Limb
In Section 6.6 in the last chapter we showed how a mechanical model could generate the cartilage patterns in the vertebrate limb. There we proposed a simple set of general morphogenetic construction rules for how the major features of limb cartilage patterns are established. Here we use these results and draw on comparative studies of limb morphology and experimental embryological studies of the developing limb to support our general theory which is essentially mechanism-independent of limb...
Info Ldy
for 1 lt a lt 1. It is the functional form of an arc of circle with centre 1 2 1 2a , 1 2 1 2a and radius 1 2 1 1a2 1 2. As a 1 we get a cusp shape while for a 0 we get a diamond shape. For positive a we get ovate shapes. Figure 9.11 shows how the shapes change with a. The solutions for four different shapes are illustrated in Figure 9.12. Such solutions let us predict, from our model, the variation in healing time with various aspects of initial wound geometry. In one study Sherratt and Murray...
A Pattern Formation Mechanism for Butterfly Wing Patterns
The variety of different patterns, as well as their spectacular colouring, on butterfly and moth wings is astonishing. Figures 3.12 a and b show but two examples see also Figure 3.22 below. There are close to a million different types of butterflies and moths.4 The study of butterfly wing colours and patterns has a long history, often carried out by gifted amateur scientists, particularly in the 19th century. In the 20th century there was a burgeoning of scientific activity. A review of the...
I
Figure 4.24. a A computed solution giving a longitudinal strip the arrow denotes increasing cell density. Parameter values for equations 4.30 are D 0.25, N s 1, r 389.6, a 1782. b The snake Thamnophis sauritus sauritus generally exhibits longitudinal striping. c Longitudinally striped California king snake Lampropeltis getulus californiae. From Murray and Myerscough 1991 Figure 4.24. a A computed solution giving a longitudinal strip the arrow denotes increasing cell density. Parameter values...
WolfDeer PredatorPrey Model
We must now include the deer as a dynamic variable. With the explicit inclusion of a deer population, we can be more specific about wolf foraging which we modelled earlier by random diffusion. Here we represent movement associated with foraging by a response of the wolves directly to the deer density. In the simplest form, the prey-taxis describes a local response of the wolves to a 'deer gradient.' In other words, wolves move towards regions of higher deer density which assumes that there is a...
Info Pha
Figure 11.11. continued See the comment in parentheses in Figure 11.10. Parameter estimates are obtained from data on mid to high grade astrocytomas glioblastoma multiforme . High grade astrocytomas account for 50 of all astrocytomas, which in turn make up 50 of all primary brain tumours Alvord and Shaw 1991 . We use the Alvord and Shaw 1991 estimate of one week to 12 months as the doubling times for gliomas. So, for cell proliferation of a high grade astrocytoma, we take a growth rate of p...
Info Rbb
1 u' u r uu' r 1 u' u r uu' r The boundary conditions, in place of 9.52 and 9.53 , are now Eu' T u' u r 1 P u' u r uu' r 1 0 at r 1 9.57 When Sherratt 1991 solved this equation with these boundary conditions he again found, as we would now expect, a critical pcrit above which solutions could not be found. He also found that the solutions for a range of parameter values exhibited the intense aggregation of actin at the wound edge. However, it was not possible to obtain the actin cable formation...
CellChemotaxis Model Mechanism
The model we consider here involves actual cell movement. Pattern formation models which directly involve cells are potentially more amenable to related experimental investigation. There is also some experimental justification from the evidence on pige-ment cell density variation observed in histological sections which we described above. Also, Le Douarin 1982 speculated that chemotaxis may be a factor in the migration of pigment cells into the skin. Heuristically we can see how chemotaxis...
Formation of Microvilli
Micrographs of the cellular surface frequently show populations of microvilli, which are foldings on the cell membrane, arrayed in a regular hexagonal packing as shown in the photograph in Figure 6.26. Oster et al. 1985a proposed a modification of the model in Section 6.8 to explain these patterns and it is their model we now discuss. Figure 6.26. a Micrograph of the hexagonal array on a cellular surface after the microvilli foldings have been sheared off. The photograph has been marked to...
Modelling Hair Patterns in a Whorl in Acetabularia
The green marine alga Acetabularia, a giant unicellular organism see the beautiful photograph in Figure 3.23 is a fascinating plant which constitutes a link in the marine food chain Bonotto 1985 . The feature of particular interest to us here is its highly efficient self-regenerative properties which allow for laboratory controlled regulation Figure 3.23. The marine algae Acetabularia ryukyuensis. Photograph courtesy of Dr. I. Shihira-Ishikawa Figure 3.23. The marine algae Acetabularia...
Biology of Tooth Initiation
Vertebrate teeth vary in size and shape yet all pass through similar stages of development. In the vertebrate jaw, there are two primary cell layers the epithelium, which is arranged in sheets, and the underlying mesenchyme, a conglomerate of motile cells, connective tissue and collagen. Figure 4.10 schematically shows the early events in their initiation. The first sign of developing structure of the tooth organ is the tooth pri-mordium. The tooth primordium first becomes evident in the...
Shamanism and Rock Art
In Section 12.3 we described a model mechanism, involving excitatory and inhibitory neurons an activator and inhibitor model for generating visual hallucination patterns some of these images and their corresponding shapes in the visual cortex are illustrated in Figure 12.9. Visual perception does not depend on light, of course, as is easily demonstrated by shutting your eyes and relaxing. More complex patterns are obtained, with your eyes closed, if you press a finger into the corner of each...
ef cshYu Px xudx 11462
The function v x can take one of two forms as shown in Figures 14.9 a and b with either one or two maxima in both cases there is a maximum value for some x gt xu assuming that xu lt xv . Note that the distribution for pack 1 remains symmetric about the den location. A single maximum for pack 2 is ensured if which suggests that there is a critical relative strength of adhesion between packs beyond which packs which have the greater response to foreign RLU marking could be forced to split their...
Whhw
Figure 12.28. a Example of a basic pattern evolving into cult drawings found in San rock paintings second, third and fifth in Southern Africa and the Shonean Coso rock painting fourth in California. Redrawn from Lewis-Williams and Dowson 1988 . b Examples of basic crenalated phosphene designs and related cave motifs. Here i is a migraine pattern, ii is an hallucinogenic pattern from Tukano shamanistic art in Colombia, iii is a rock painting from Montevideo, Baja, California, iv is Hohokam rock...
Spatial Patterning of Teeth Primordia in the Alligator Background and Relevance
We have already given several reasons in Volume I for studying the crocodilia in general when discussing their remarkably long survivorship. In the previous sections in this chapter we saw how the study of stripe patterns on the alligator increased our understanding of the biology. In the case of teeth there are numerous reasons, other than pedagogical ones, for studying the development of dentition in A. mississippiensis. The development of teeth primordia in the vertebrate jaw of the...
X Uxx Tonx Sx Suxx u
The dispersion relation for the growth rate X for each mode k is then given by X Dok2 0 ikx iDoY k3 0 X ikX Dok2 k x Sto yt to 1 Parameter ranges and wavenumbers k which result in X k gt 0 give rise to growing spatial patterns. From 10.71 we see that if k2 is small but nonzero, X lt 0 and so small wavenumbers, which correspond to long wavelengths, are stable. If k2 1 then Dok x sto YT to X STo 2 To ' So, if the parameters x, to, y , and 8 are in appropriate domains it is possible for X to be...
Geographic Spread and Control of Epidemics
The geographic spread of epidemics is less well understood and much less well studied than the temporal development and control of diseases and epidemics. The usefulness of realistic models for the geotemporal development of epidemics be they infectious disease, drug abuse fads or rumours or misinformation, is clear. The key question is how to include and quantify spatial effects. In this chapter we describe a diffusion model for the geographic spread of a general epidemic which we then apply...
Info Pjt
Applying the continuity conditions 1.45 and 1.46 the following series of equalities must hold for i 0,1,2, .If the expressions inside the square roots become negative we have to use the identities tan iz i tanh z, arctan iz i arctanh z. 1.69 We are, of course, interested in the sign of the smallest eigenvalue, A A0, which satisfies the above equality. It is easy to show that A0 is negative if and only if the expressions 1 - Ye A0 and G2 - Ye A0 appearing under the square roots have opposite...
Info Rrt
i x, t 1 ef t eikx, v x, t t eg t J2 where 0 lt s 1 and the k are the wavenumbers associated with the Fourier series of the random initial conditions. To approximate the actual experimental situation, where the initial concentration of chemoattractant is exactly zero, we set g 0 0. For illustration we choose f 0 1. We look for spatially varying solutions superimposed on the temporally growing solution. Since we are looking for solutions on a finite domain with zero flux boundary conditions we...
Hickerson 1965 Sioux
Figure 14.12. The intertribal buffer zone between the Chippewa and Sioux settlements in Wisconsin and Minnesota from about 1750 to the mid-1880's. Redrawn from Hickerson 1965 and reproduced with permission of the American Association for the Advancement of Science copyright holder Figure 14.12. The intertribal buffer zone between the Chippewa and Sioux settlements in Wisconsin and Minnesota from about 1750 to the mid-1880's. Redrawn from Hickerson 1965 and reproduced with permission of the...
Chippewa and Sioux Intertribal Conflict c17501850
There is a well-documented human application of the general mechanistic theory we have proposed in this chapter which tends to justify intertribal warfare as a traditional means of survival. Morgan 1887 suggested that buffer zones or disputed areas between tribes was a universal feature of tribal societies.3 These buffer zones between accepted tribal territories were not generally occupied by members of either tribe and tended to be entered only by hunting groups of considerable strength 15 to...
Show That The A B Turing Parameter Space For Diffusion-driven
where b and d are positive constants. Which is the activator and which the inhibitor Determine the positive steady states and show, by an examination of the eigenvalues in a linear stability analysis of the diffusionless situation, that the reaction kinetics cannot exhibit oscillatory solutions if b lt 1. Determine the conditions for the steady state to be driven unstable by diffusion. Show that the parameter domain for diffusion-driven instability is given by 0 lt b lt 1, db gt 3 2V2 and...
Info Tpi
d S n -1 1 4y kx2 1'2 0 tm -4 k- which on substitution in 3.16 gives Now, from the kinetics mechanism 3.12 , Smax Sth is the level which effects a switch from g 0 to g g3 in Figure 3.14 a . Substituting in 3.8 we can calculate the distance xth from the vein where g g3 and hence, in our model, the domain of a specific pigmentation. Thus xth is the solution of An alternative form of the equation for xth is In dimensional terms, from 3.11 , the critical distance xth cm is thus given in terms of...
D Rju
If we now use 3.25 and the experimental points from Figure 3.21, we can determine D, k and C from a best fit analysis. From the point of view of experimental manipulation it is difficult to predict any variation in the degradation constant K since we do not know what the morphogen is. There is, however, some information as to how diffusion coefficients vary with temperature. Thus, the parameter whose value we can deduce, and which we can potentially use at this stage, is the diffusion...
Pigmentation Pattern Formation on Snakes
Snakes order squamata reptiles and amphibians in general, in fact are numerous and highly diverse in their morphology and physiology. Snakes and lizards exhibit a particularly rich variety of patterns many of which are specific to snakes. The fascinating, and visually beautiful, book by Greene 2000 is a good place to start. He discusses their evolution, diversity, conservation, biology, venoms, social behaviour and so on. Another very good book, by Klauber 1998 , is more specific and is...
Info Czt
1. Multi-Species Waves and Practical Applications In Volume 1 we saw that if we allowed spatial dispersal in the single reactant or species, travelling wavefront solutions were possible. Such solutions effected a smooth transition between two steady states of the space independent system. For example, in the case of the Fisher-Kolmogoroff equation 13.4 , Volume I, wavefront solutions joined the steady state u 0 to the one at u 1 as shown in the evolution to a propagating wave in Figure 13.1,...