| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
Department of * Mathematics and Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts; and Division of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts
Correspondence: Address reprint requests to Professor Michael P. Brenner, Division of Engineering and Applied Sciences, Harvard University, 29 Oxford St., Cambridge, MA 02458. Tel.: 617-495-3336; E-mail: brenner{at}deas.harvard.edu.
 |
ABSTRACT |
|---|
 TOP ABSTRACT INTRODUCTIONEXPERIMENTAL PROCEDURESMATHEMATICAL MODELDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
|---|
|
INTRODUCTION |
|---|
|
TOPABSTRACTINTRODUCTIONEXPERIMENTAL PROCEDURESMATHEMATICAL MODELDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
|---|
; Foty et al., 1994, 1996; Ryan et al., 2001; Steinberg, 1962, 1964; Townes and Holtfreter, 1955; Trinkaus, 1963) and simulations (Glazier and Graner, 1993; Palsson and Othmer, 2000). The surface tension of certain embryonic tissues have been directly measured (Foty et al., 1994, 1996), and as expected, mixtures of cells segregate to minimize the overall surface energy.
If the morphology of a growing tissue is dictated solely by surface energy minimization, then this has implications not only for the position of cells relative to each other but also for the overall shape of the tissues: in the absence of external forces, a tissue-minimizing surface energy should be composed of spherical regions. The goal of the present work is to test this hypothesis within a particularly simple example: the shape of a droplet of a single cell type growing on a nutrient-enriched substrate. As for liquid droplets, the equilibrium shape of such a structure is a spherical cap with a contact angle given by Young's law,
 | (1) |
of the colony at the agar substrate to the surface energies of the liquid and solid. Here, 
CA is the adhesion energy of the cells to each other, CS is the adhesion energy of the cells to the substrate, and SA is the energy per unit area of the substrate. All of these quantities should change when the types of cell and substrate are varied.
Our experiments focus on colonies of Baker's yeast (Saccharomyces cerevisiae), growing on an agar substrate. The advantage of this system is threefold. First, the gene expressing the adhesive protein (FLO11) is known, and thus the cell-cell adhesion CA can be genetically controlled. Second, the adhesivity of the substrate CS can be varied by changing the agar concentration. Third, yeast cells are spherical and have no mechanism for active motility. The experiments demonstrate that, consistent with Young's law (Eq. 1), changing either the agar concentration or the expression of FLO11 modifies the local contact angle of the yeast droplet. Moreover, when the colony is sufficiently small, its shape is a spherical cap, consistent with surface energy minimization. However, above a critical (contact-angle-dependent) volume the spherical structure is unstable, and the colony develops a nonspherical shape. Since these shapes are inconsistent with surface energy minimization, the experiments demonstrate that there must be other forces acting on the tissue. The possible candidates in our experiments are gravity, adhesive gradients, growth stresses, and elastic stresses. We present a mathematical model suggesting that the change in tissue morphology arises from elastic deformations of the colony. The model demonstrates that a spherical elastic droplet on a solid substrate with fixed contact angle is unstable above a critical (contact-angle-dependent) volume, quantitatively consistent with experiments. The model reproduces both the instability threshold and the shape of the yeast droplets near the threshold, consistent with the experiments.
The organization of this article is as follows. In Experimental Procedures, we describe our experimental system and discuss the results. A phase diagram is presented delineating the borderline between spherical shapes (where the colony shapes minimize surface energy), and nonspherical shapes (where other forces are acting). In Mathematical Model, we derive a mathematical model for an elastic droplet on a solid surface, and analyze the stability of the droplet to nonspherical perturbations. The instability threshold is computed and compared with experimental observations. We also present analytic calculations of droplet shapes beyond the transition. Finally, Discussion presents conclusions and directions for future work.
|
EXPERIMENTAL PROCEDURES |
|---|
|
TOPABSTRACTINTRODUCTIONEXPERIMENTAL PROCEDURESMATHEMATICAL MODELDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
|---|
. They noticed that when Baker's yeast (S. cerevisiae) grows in a low glucose medium they adhere to plastic and form biofilms. The ability of the yeast to stick to plastic was traced back to FLO11, a yeast gene encoding a cell surface glycoprotein that allows cells to adhere to agar and to each other. This gene can be turned off (producing the mutant flo11
) or overexpressed (producing the strain sfl1), so that three independent strains of otherwise identical cells with different adhesion strengths exist. Reynolds and Fink found that when wild-type (WT) yeast grows on a low agar concentration (0.3%), it forms a complex structure with reproducible features. Since the cells are nonmotile, the structures that form are entirely the result of the forces that act upon them. The morphologies observed in the Reynolds-Fink experiments are determined by a large number of related effects, including adhesion, nutrient consumption, and water content.
Materials and methods
Yeast strains
We use Baker's yeast S. cerevisiae with different levels of expression of the adhesive protein FLO11 (obtained from the laboratory of Dr. G. Fink, The Whitehead Institute, Cambridge, MA). There are three strains, flo11, wild-type (WT), and sfl1, that express low (zero), normal, and high levels of FLO11, respectively. The strains are characterized by the levels of adhesion as nonsticky, sticky, and super-sticky. The system has many advantages. First, these cells are spherical and nonmotile with an average cell division time of 2 h. Cellular rearrangements are possible through forces the cells exert on each other and on their environment. An aggregate of these cell types has an effective surface energy CA due to adhesive interactions between individual cells. The magnitude of CA is set by the concentration of this cell surface protein, which is genetically controlled.
Preparation of agar substrate and yeast colonies
The growth medium YPD is composed of water, 1% Difco yeast extract, 2% Bacto peptone, and 2% Mallinckrodt dextrose. A desired amount of Bacto agar is added to the growth medium. The mixture is then autoclaved for 20 min at 122°C to dissolve the agar and sterilize the medium. The substrate is prepared by pouring 30 ml of the sterile mixture into a sterile petri dish (Corning, Acton, MA) and allowed to set for 1 h. A sterilized glass plate is placed at the bottom of the petri dish before pouring in the mixture. This makes the transfer of the substrate between the petri dish and the microscope stage more stable and easier. When the plates are set, they are ready for inoculation. Colonies are inoculated by spreading 25 µl of a dilute mixture of yeast cells and liquid YPD. The inoculation procedure ensures that for each plate the number of colonies is small (<20) and spread out. The inoculated plates are placed in a humidified incubator at 28° for a couple of days.
Imaging and data analysis
Once the colonies are visible by eye the imaging process begins with a side-view microscope (Leica Monozoom 7, Leica, Bannockburn, IL) with an attached charge-coupled device camera. This allows the measurement of contact angles that a yeast colony makes with the agar substrate and the two-dimensional shape of the colony as a function of time. For imaging, the glass plate with the agar substrate is cut and removed from the petri dish and then placed on the microscope stage. A dual cold light source (Fiber Lite MI-150, Dolan Jenner, Lawrence, MA) is used to illuminate the colonies from the sides. Time-lapse images of the colonies on the same plate are taken every few hours; between images the plates are placed back in a humidity controlled environment. Even with the glass plate, the transferring of a substrate of agar concentration <1% is not stable. This limits the experiments to agar concentrations of 
1%. We acquire and analyze the images using Metamorph software (Universal Imaging, Downington, PA).
Results
Contact angle
The first set of experiments is designed to measure the contact angle of a yeast droplet for fixed agar concentration, and to determine if it remains constant throughout the growth of the colony, as implied by Eq. 1. Time-lapse images of colonies of the same plate were taken every 2 h. Our initial experiments showed that although the shape of small yeast droplets remains spherical, the contact angle actually increases with time, contradicting Young's law with constant surface energies. We hypothesized that the increase in the angle might arise from the evaporation of water from the colonies and the substrate during the imaging process. We therefore conducted a set of experiments using a number of identically inoculated plates to verify this hypothesis. After the colonies on a given plate are imaged, the plate is discarded to avoid evaporation. Images at later times were taken from a fresh plate from the incubator. These experiments demonstrate that the contact angle remains constant during the entire growth of the colony (Fig. 1). The constancy of the contact angle is obeyed even after the instability (to be discussed subsequently) occurs.
|
strain has the highest contact angle at a given agar concentration, consistent with its higher surface tension. The sticky wild-type (WT) and nonsticky flo11 strains have similar contact angles. Although one might expect the wild-type strain to have a larger contact angle than the mutant flo11 strain owing to the expression of the adhesive protein, this neglects the effect of water on the surface energy of the colony. When the adhesion between cells is sufficiently weak, one would expect the cell-cell adhesion energy CA to be dominated by the surface tension of water; although we have no direct way of measuring CA, we believe this is a consistent interpretation of the data.
|
on varying the agar concentration from 1% to 3%. The contact angle increases with increasing agar concentration. The mechanism through which the yeast colony adheres to the agar is unknown. If the yeast cells adhered to the agar directly, one would expect the contact angle to decrease with increasing agar concentration. This is because higher agar concentrations would give a higher concentration of binding sites between the yeast and agar, consequently changing CS. On the other hand, increasing agar concentration also visibly dehydrates the yeast colony; this effect will also change the cell-cell surface energy CA. Although the molecular mechanisms controlling the contact angle are interesting, the most important conclusion for the present study is that the contact angle remains constant in time, and can be manipulated by changing either the agar concentration or the cell adhesion.
Colony shape
When the colony is sufficiently small, the shape is always a spherical cap, as expected from surface energy minimization. However, we observe that at a critical time during the growth, the spherical shape destabilizes. After the instability, the resulting morphologies include staircase, staircase with centered dimple, and spherical cap with dimple (Fig. 3). A contour plot of time-lapse images of a WT colony (Fig. 4) demonstrates the transition of the colony from spherical to nonspherical. Extensive observations indicate that the character of the instability is determined entirely by the contact angle . For instance, a superadhesive sfl1 colony on 1.5% agar concentration has a similar contact angle to a wild-type colony on 2.1% agar; although the surface tension CA of sfl1 is higher than the wild-type, the Young's law equation (Eq. 1) implies that this is compensated by the higher agar concentration. Despite the differing mechanisms leading to the contact angle, both types of colonies come to a staircase morphology. This suggests that morphology is controlled completely by the adhesion level and agar concentration, which together determine the contact angle of the colony.
|
|
, with no explicit dependence on the type of cells or the agar concentration. To test this hypothesis we measured a phase diagram (Fig. 5) of colony morphologies as a function of colony volume and contact angle. The transition to nonspherical shapes indeed occurs above a contact-angle-dependent critical volume. In Fig. 5, for high contact angles and low colony volumes, the shapes are spherical; for low contact angles and high colony volume, the shapes are nonspherical. The nonspherical regime is divided into three subregimes based on the contact angle: at low angles ( < 40°), the shapes are staircases; at midangles they are staircases with a dimple; at highest angles ( > 70°), they are dimples.
|
strains on four different agar concentrations ranging from 1.8% to 3% and sfl1 strain on 1–1.5%. For each agar concentration, we make eight identical plates. Images are taken from a different plate every 2 h. At the end of the experiment we have 
150 images per agar concentration. The detected edge yexpt(xi), {i = 1...N} is analyzed to obtain contact angles, radii, and areas of the colonies. The edges are then fit to a circular cap yfit(x) using a least-squares method. We then calculate 
2, defined as
 |
is the measurement error per data point. When the colony transitions from a spherical to a nonspherical shape, 2 rapidly increases. We are interested in detecting the early stage of the instability. We define the instability to occur when 2 is in the range between 0.1 and 0.3. Applying this threshold to each set of conditions then yields the phase diagram (Fig. 5).
The fact that the spherical colony shape destabilizes above a critical volume implies that there must be a force other than surface tension affecting the colony shape. The obvious candidates for this additional force are gravity, gradients in adhesivity caused by nonuniform nutrient consumption or waste production in the colony, and elastic stresses. Although gravitational forces should play a role when the colony is sufficiently large, we ruled out gravity by performing experiments on colonies grown while inverted. The colony becomes unstable at exactly the same volume independent of its orientation relative to gravity. We tested for the importance of nutrient consumption or waste production by carrying out experiments where the glucose level in the substrate is varied. Since the expression of adhesive protein is directly controlled by the level of glucose (Reynolds and Fink, 2001), varying the glucose level simulates the effect of nutrient consumption. The experiments showed that the instability occurs at the same critical volume for different levels of glucose, ruling out the possibility of nutrient depletion on causing the instability.
The only remaining candidate is the possibility of developing elastic stresses in the colony. Elastic stresses might be generated by cell growth; however, the very slow cell division timescale (typically 2 h) makes this unlikely. Assuming the colony material is similar to particulate gel, we can compare the growth rate to the stress relaxation rate. Yeast cells divide on average every 90 min; hence, the growth rate is 1/5400 s–1. The shear modulus of a particulate gel of volume fraction 
0.5 is of 103dyn/cm2 and the maximum dynamical viscosity is of 102 dyn/s per cm2 Larson (1999); hence, the stress relaxation rate is 10 s–1, which is much larger than the growth rate. Therefore, the elastic stresses induced by cell growth relax very quickly, and should not affect the colony morphology. Elastic stresses might also arise due to a direct instability of the spherical cap, in which the elastic energy to support a nonspherical shape is less costly than the surface energy for the shape to remain spherical. To explore this possibility, we developed a phenomenological mathematical model of an elastic droplet on a solid surface. The model demonstrates that the spherical cap solution is unstable at a contact-angle-dependent critical volume, reminiscent of the experimental findings.
|
MATHEMATICAL MODEL |
|---|
|
TOPABSTRACTINTRODUCTIONEXPERIMENTAL PROCEDURESMATHEMATICAL MODELDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
|---|
Elastic lubrication theory
We begin by calculating the elastic strain that must exist in the colony for a nonspherical shape to remain in equilibrium. We consider a two-dimensional colony with height z = h(x). The strain field in the colony is 
u(x, z) where u(x, z) is the displacement. For small deviations from a spherical cap (for which there are no elastic strains) we assume the displacement field is measured relative to the spherical cap with identical volume. The yeast colony is incompressible ( · u = 0), owing to the water in the yeast droplet. Displacements in the colony then follow from the equilibrium equations of an elastic droplet,
 | (2) |
 | (3) |
). We remark that the magnitude of G is not the same as the elastic modulus of a single yeast cell (Smith et al., 2000), since the elastic deformations of a yeast colony result in deformation of the network of cells in the colony, instead of the individual cells themselves (Larson, 1999). The bulk elasticity G is therefore much smaller than that of the cells themselves. A typical value of G for particulate gel is 3 x 103 dyn/cm2 for a volume fraction of 0.5 (Larson, 1999).
To compute the strain predicted by Eqs. 2 and 3, we assume that the characteristic length scale of the colony in the horizontal (x) direction, L, is much larger than that in the vertical (z) direction, h. Such a lubrication approximation is common in analyzing thin-film flows in fluid mechanics (Batchelor, 1973). Denoting the components of the displacements u in the x and z directions as ux and uz, respectively, the equilibrium equations are
 | (4) |
Similarly, the incompressibility condition 
xux + zuz = 0 implies 
so that 
Hence when 
we have 
and vertical displacements are unimportant. Similarly, Eq. 4 implies that 
so that we can assume the pressure primarily depends on the horizontal coordinate p = p(x).
With these simplifications the equilibrium equations reduce to a single equation for ux. Henceforth we drop the subscript x and denote the elastic displacement by u. The boundary conditions are that the displacement vanishes on the agar substrate u(z = 0) = 0, and the shear stress at the yeast-air interface vanishes zu(z = h) = 0; finally, the pressure at the yeast-air interface is given by the Gibb's condition (Landau and Lifshitz, 1987),
 | (5) |
Applying the boundary conditions and solving Eq. 4 gives the displacement in terms of h(x),
 | (6) |
A straightforward calculation then gives the total energy of a yeast colony with arbitrary shape h(x),
 | (7) |
Instability of spherical cap
Our goal now is to demonstrate that there are colony shapes with a fixed volume v0 and equilibrium contact angle that can lower their energy by deviating from a spherical cap. We first give a qualitative argument exposing how this instability can arise, and then proceed with a detailed calculation.
Scaling argument
Consider a colony with volume v0 and contact angle . If h is the characteristic thickness of the colony and R is its radius, then v0 hR R2. From Eq. 7, the elastic energy of such a colony is of order 
and the surface energy is of order 
At large enough radius, the surface energy contribution dominates the elastic energy, and thus the colony will deform. These two energies are the same order of magnitude when * (G/R*)1/3 or * ((G/)2
)1/7, where /G is the characteristic length scale representing the competition between surface tension and elasticity. For a typical yeast colony, 10 dyn/cm (Forgacs et al., 1998) and G 3 x 103 dyn/cm2 (Larson, 1999) so the characteristic scale of the instability is 10–2 cm. For volumes v0 > an instability to a nonspherical solution will occur. Note that in this regime, increasing the volume of the colony increases both the elastic and surface energies. However, it is cheaper overall to distort the surface then to spread the colony into the larger area necessary to maintain constant contact angle.
Quantitative argument
The scaling argument can be made quantitative by studying the first variation of Eq. 7. Assuming that h' << 1, we have
 | (8) |
We are interested in solutions to Eq. 8 that are close to a spherical cap. Denote 
as a spherical cap with radius R and volume v0. The radius is related to the contact angle through 
and the pressure enforcing the volume constraint is then 
Taking h = h0 + c
and expanding Eq. 8 to leading order in and integrating twice, we obtain
 | (9) |
= 3G/2. A nonspherical solution exists if there exist nonzero solutions to Eq. 9, satisfying the boundary conditions. The boundary conditions are that the solution is symmetric around the origin '(0) = '''(0) = 0; at the radius R the profile vanishes (x = R) = 0 and the slope obeys the contact angle condition 
(For the spherical cap solution, = 0, so the radius satisfies 
) Finally, since we are considering perturbations to the shape at constant volume, if we fix the volume of the solution to be v0, then the volume associated with must vanish (
). The boundary conditions correspond to five conditions on the solution; Eq. 9 is fourth order, and in addition we have the unknown critical volume 
Hence these conditions are sufficient to uniquely specify the instability.
The most convenient way to find additional solutions is to rescale the horizontal coordinate y = x/R, and introduce v' = . The volume constraint on then implies that v(y = 0) = v(y = 1) = 0. The equation for v is
 | (10) |
= 2R4/3v0 as an eigenvalue. We numerically computed the smallest eigenvalue for which nonzero solutions to this equation exist: = * = 65.12. Hence, we have an explicit formula for the bifurcation curve, tan * = 0.72((/G)2)1/7. Notice that /G is a characteristic length scale. Normalizing the volume by letting V = /(/G)2 gives
 | (11) |
For volume v0 > V other than those given in Eq. 11, the spherical cap solution is unstable. The solid line in Fig. 5 shows the theoretical bifurcation curve. In this comparison we have assumed that = 73 dyn/cm (the surface tension of water) and G = 5 x 103dyn/cm, as described above. The theoretical curve captures the trends of the experiments.
Finally, we note that the shape of the colony close to the bifurcation point also follows from this analysis. The shape of the colony is h(x) = h0(x) + c(x). A weakly nonlinear analysis around the bifurcation point demonstrates that if the volume of the colony increases from 
+ 
v, the solution is 
To leading order in v, the radius of the colony is constant.
Fig. 6 shows the two possible configurations for the colony near the bifurcation point, i.e., the spherical solution and the nonspherical solution. The inset shows the energy of both solutions as a function of distance from the bifurcation point. As advertised, the nonspherical solution has lower energy then the spherical one. Note that this comes about because the spherical solution has a larger radius (and hence higher surface area), to fit the constant contact angle condition.
|
|
DISCUSSION |
|---|
|
TOPABSTRACTINTRODUCTIONEXPERIMENTAL PROCEDURESMATHEMATICAL MODELDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
|---|
< 40°, and when > 70° the colony has a single dimple in the center).
The mathematical model that we have analyzed is phenomenological, in that it assumes that unbalanced capillary forces can be balanced by elastic stresses, although we do not give a microscopic description of how this coupling comes about. As an example, for the deformation of solid bodies, shape modulations of the crystal couple to elastic distortion through surface stresses, a concept that does not exist in our problem. One intriguing mechanism for the coupling is suggested by the striking similarity between our nonspherical morphologies and those that have been discovered in the shapes of a drying droplet of a colloidal suspension (Parisse and Allain, 1996). Although the humidity controlled environment of our experiments should not allow much drying to occur, the growth of the cells in the yeast colony implies that the volume fraction of solid particles in the colony is increasing. Such an increase cannot proceed indefinitely without dewetting the cells in the colony, resulting in elastic stresses. Further work analyzing the precise mechanisms and parameter regimes where such drying stresses could come about is underway.
The demonstration of elastic instability in this simple model of tissue growth points to the possibility of elastic effects in more complex situations. For example, the instability we have identified is the precursor to the complex morphologies discovered by Reynolds and Fink (2001). The precise role of elastic stresses in determining tissue morphologies under more general conditions remains to be seen. The present experience with yeast droplets demonstrates that at least two different materials with different adhesive energies are needed for an elastic instability. The general requirements for elastic stresses to play a role in determining tissue morphology remain to be worked out. It seems possible that the fundamental notion of selective adhesion as a driving force for tissue development needs to be supplemented with elastic effects. If so, there is the fascinating possibility of elastic stresses being regulated during development through, for example, cells modifying their individual stiffness.
|
ACKNOWLEDGEMENTS |
|---|
|
TOPABSTRACTINTRODUCTIONEXPERIMENTAL PROCEDURESMATHEMATICAL MODELDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
|---|
This work was supported by National Institutes of Health grant R01 GM 63618-01, and by the Division of Mathematical Sciences of the National Science Foundation (to M.P.B.).
Submitted on June 12, 2003; accepted for publication January 6, 2004.
|
REFERENCES |
|---|
|
TOPABSTRACTINTRODUCTIONEXPERIMENTAL PROCEDURESMATHEMATICAL MODELDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
|---|
Fink, R., and D. R. McClay. 1985. Three cell recognition changes accompany the ingression of sea urchin primary mesenchyme cells. Dev. Biol. 107:66–74.[Medline]
Forgacs, G., R. Foty, Y. Shafrir, and M. Steinberg. 1998. Viscoelastic properties of living embryonic tissues: a quantitative study. Biophys. J. 74:2227–2234.
Foty, R., G. Forgacs, C. Pfleger, and M. Steinberg. 1994. Liquid properties of embryonic tissues: measurement of interfacial tensions. Phys. Rev. Lett. 72:2298–2301.[Medline]
Foty, R., C. Pfleger, G. Forgacs, and M. Steinberg. 1996. Surface tensions of embryonic tissues predict their mutual envelopment behavior. Development. 122:1611–1620.[Abstract]
Glazier, J., and F. Graner. 1993. Simulation of the differential adhesion driven rearrangement of biological cells. Phys. Rev. E. 47:2128.
Landau, L., and E. Lifshitz. 1986. Theory of Elasticity. Pergamon Press, Oxford, UK.
Landau, L., and E. Lifshitz. 1987. Fluid Mechanics. Pergamon Press, Oxford, UK.
Larson, R. 1999. The Structure and Rheology of Complex Fluids. Oxford University Press, Oxford, UK.
Palsson, E., and H. Othmer. 2000. A model for individual and collective cell movement in dictyostelium discoideum. Proc. Natl. Acad. Sci. USA. 97:10448–10453.
Parisse, F., and C. Allain. 1996. Shape changes of colloidal suspension droplets during drying. J. Phys. II Fr. 6:1111–1119.
Reynolds, T., and G. R. Fink. 2001. Bakers' yeast, a model for fungal biofilm formation. Science. 291:878–881.
Ryan, P., R. A. Foty, J. Kohn, and M. Seinberg. 2001. Tissue spreading on implantable substrates is a competitive outcome of cell-cell versus cell-substratum adhesivity. Proc. Natl. Acad. Sci. USA. 98:4323–4327.
Smith, A., Z. Zhang, C. R. Thomas, K. E. Moxham, and A. P. J. Middleberg. 2000. The mechanical properties of Saccharomyces cerevisiae. Proc. Natl. Acad. Sci. USA. 97:9871–9874.
Steinberg, M. 1962. Mechanism of tissue reconstruction by dissociated cells. II. Time-course of events. Science. 137:762–763.
Steinberg, M. 1964. The problem of adhesive selectivity in cellular interactions. In Cellular Membranes in Development. M. Locke, editor. Academic Press, New York.
Townes, P. L., and J. Holtfreter. 1955. Directed movements and selective adhesion of embryonic amphibian cells. J. Exp. Zool. 128:53–120.
Trinkaus, J. 1963. The cellular basis of fundulus epiboly. Adhesivity of blastula and gastrula cells in culture. Dev. Biol. 7:513–532.[Medline]
This article has been cited by other articles:
 |
|
 |
|
  T. B. Reynolds, A. Jansen, X. Peng, and G. R. Fink Mat Formation in Saccharomyces cerevisiae Requires Nutrient and pH Gradients Eukaryot. Cell, January 1, 2008; 7(1): 122 - 130. [Abstract] [Full Text] [PDF]
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
