## ABSTRACT

The spreading of cells on suitably treated surfaces is briefly discussed and the need to analyse this phenomenon numerically is emphasized. Possible mathematical models for fitting experimental data are classified as statistical, kinetic or empirical and examples of each of these types are given. A possible protocol for analysing cell spreading kinetics and determining goodness of fit and parameter redundancy is presented.

## INTRODUCTION

Studies of cell behaviour *in vitro* have demonstrated that in normal cell types the supporting substratum plays an active part in the determination of morphology (Aplin & Hughes, 1981; Aplin & Foden, 1982), growth and division (Stoker, O’Neill, Berryman & Waxman, 1968; Folkman & Moscona, 1978; Ben-Ze’ev, Farmer & Penman, 1981; Ben-Ze’ev & Raz, 1981), and the pathway of differentiation (Grinnell, 1978; Grinnell & Feld, 1979). Cells settling on a prepared surface may remain rounded for an indefinite period or flatten and spread, and spreading appears to be a prerequisite for DNA synthesis and cell division (Stoker *et al*. 1968; Folkman & Moscona, 1978; Ben-Ze’ev *et al*. 1981; Ben-Ze’ev & Raz, 1981). Studies with permanent fibroblastic and epithelioid cell lines have in general demonstrated a requirement for exogenous ‘promoters’ of spreading added either as a supplement to the medium or as a coating on the growth surface (Grinnell, 1978); primary cells in culture may adhere to a prepared substratum or secrete a ‘carpet’ of adhesive components such as fibronectin, which subsequently allows cell spreading (Grinnell & Feld, 1979). Serum, which is often present as a medium supplement, contains fibronectin (Edsall, Gilbert & Scheraga, 1955), and other adhesion-promoting components (Knox & Griffith, 1980). Cell spreading, which consumes energy (Grinnell, 1978), can occur on a variety of substratum-associated proteins (Aplin & Hughes, 1981). The foregoing attachment process appears to be passive and less specific (Grinnell, 1978).

A variable in adhesion experiments that has been somewhat neglected is that of time. It is evident from the above discussion that the kinetics of cell spreading may depend on the nature of the substratum and cell type in addition to the fundamental molecular processes of membrane and cytoskeletal reorganization. These in turn may be influenced by the position of the cell in the cell cycle, the growth phase of the culture and the method used to obtain a cell suspension (Schor, 1979).

In the present paper we discuss ways of analysing the kinetic behaviour of a population of settled rounded cells, which transforms over a period of time to the spread-out morphology, using the simplifying assumption that each cell can exist in two states: rounded or spread. In the accompanying paper, data collected from fields containing cells that have been defined by an observer as ‘rounded’ or ‘spread’ are analysed as a function of time. In this way it is intended to place the phenomenon of spreading on substratum on a quantitative footing, thus allowing detailed study of the variables that influence cellular morphology.

## MATERIALS AND METHODS

HeLa cells were grown in Minimum Essential Medium with Earle’s salts, non-essential amino acids (1 %), Hepes (20 min) and 10% foetal calf serum. They were brought into suspension using 0·01 % trypsin, 0·004% EDTA in phosphate-buffered saline without divalent cations. Amniotic fluid fibronectin was prepared, and adhesion assays conducted, as described in the accompanying paper.

## RESULTS AND DISCUSSION

### The course of cell spreading

Data obtained using cultured cell lines spreading as a function of time on substrata containing coatings of adhesion-promoting proteins generally take a sigmoid form (Fig. 1 and accompanying paper). Cell types differ, but light and electron microscopic examination of cultured cells (Wang & Goldman, 1978; Heath & Dunn, 1978) suggest that the process may be divided into stages (Grinnell, 1978; Wang & Goldman, 1978). A tentative series of stages is listed in Fig. 2. The cell settles in a rounded conformation (Fig. 2A) and proceeds to explore the surrounding substratum by extending numerous microspikes or small filipodia (Fig. 2B). Some cell types may extend lamellipodia. Once stable attachments have been formed from microspike tips, areas between these are filled in, forming areas of lamellar cytoplasm (Fig. 2c). At this stage the observer using a light microscope recognizes the cell as being spread. Further flattening occurs, however, by advancement of lamellar cytoplasm with ruffling activity and, in some cases, formation of tapering cell processes and focal contacts between cell and substratum (Fig. 2D). It is clear that these different steps in the transformation of morphology may show quite different kinetic behaviour; the rate-limiting step may involve cytoskeletal reorganization (Wang & Goldman, 1978; Heath & Dunn, 1978; Badley, Woods, Carruthers & Rees, 1980).

### An outline of the possible ways to evaluate cell spreading kinetics

Thus the spreading of cells is a complex process involving many factors. Also, any attempt to estimate the rate at which cells spread will be subject to uncertainty as there is no completely objective way to measure the extent to which any individual cell has begun to spread. Nevertheless, certain observations are suggested by our own experiments.

When cells are observed as quickly as possible after settling no cells are seen to be spread.

When cells are applied to suitably coated surfaces and incubated at 37 °C, then eventually all observers will agree that a proportion of the cells show an altered morphology that can justify the term ‘spread’.

Under such conditions as are favourable for cell spreading, all observers agree that a very high proportion of cells, say > 90%, become spread eventually after a sufficient time interval.

The rate at which cells spread is dependent on the nature of the surface to which the cells are applied.

Owing to the subjective element in assessing the extent of cell spreading, there will always be some uncertainty in assessing the proportion of cells spread as a function of time and nature of surface coating. Nevertheless, some quantitative measure of the spreading process is desirable in order that the numerical values of parameters can be estimated, allowing a more rigorous approach to comparing cell spreading kinetics than is possible at the moment.

*F*, is actually spread. Evidently, as a function of time, the value of

*F*is limited by the inequalities: .

The percentage of cells spread is simply 100*F* and this proportion, *F*, is necessarily a function of time and the nature of the surface. For the rest of this paper we shall concentrate on attempts to formulate *F* as a function of time and in the following paper we shall compare the usefulness of certain mathematical models that can be formulated as applied to data obtained under a variety of experimental conditions. We shall investigate three distinct types of model, i.e. statistical models, kinetic models and empirical models.

### Statistical models

*F*from

*F*= 0 at time

*t*= 0 to

*F ≈*1·0 at large time values. From this point of view, the function

*F*(

*t*) is simply the probability that a cell will have achieved the spread state in time

*t*. So we can regard

*F*(

*t*) as a cumulative probability distribution function defined by: ,

where *f*(*t*) is the probability density function. In other words, if we were able to measure the time to achieve the spread state in a large number of cells and plot a histogram of fraction of cells taking time *t* to spread as a function of *t*, then in the limit as the number of cells became infinitely large and the subdivision of *t* became smaller and smaller, the histogram would become a continuous curve, namely *f*(t). We might expect/(t) to beunimodal, the rooto *f*′(t) = 0 representing the average time to spread for the individual cell and we might also expect *f*(*t*) to be roughly symmetrical when the variance would give some idea of the dispersion of spreading times about a mean position. However, there is no reason why the distribution could not be skewed and such possibilities could only be settled by experiment. Let us now turn attention to some possible formulae for *f*(*t*) and, in doing so, we must not lose sight of the fact that, since *F*(*t*) values obtained experimentally are subject to error and since *F*(*t*) profiles are usually simple sigmoid curves there is not likely to be any need for models with more than two or three parameters.

### The truncated normal distribution

*LI*and

*o*. However, there is one objection to this; the normal distribution is symmetrical about

*t = μ*and allows negative

*t*values. Since this is physically impossible we suggest a special form of the normal distribution, truncated so as to exclude negative

*t*values. According to this model the probability density,

*fT*(

*t*), function would then take the form: — 0 otherwise,

where the definite integral in the denominator replaces the usual √ (2π). The interpretation of *μ*T and *σ* T is then simply that the time to spread is normally distributed with mean time *p* r and standard deviation σ T but *t* values are confined to *t* ⩾ 0 making the distribution unsymmetrical.

### The symmetrical normal distribution

*t = μ—t*

_{0}is just as likely as a time

*t = μ+t*

_{0}. A way round this is to define a distribution that declares

*t*> 2

*u*as representing all cells spread and takes

*f*(0) =

*f*(2

*μ*) = 0. The resulting symmetrical distribution is derived from the normal one and has the probability density function,

*f*

_{s}(t), defined by: = 0 otherwise.

Now *μ*_{s} has the interpretation of the mean time to spread and *σ*_{s} is a measure of the dispersion of values around the mean. This distribution can be thought of as arising from the normal one by raising the horizontal axis and redefining the time scale to run from *t* = 0 to *t* = 2*μ*.

### The log normal distribution

and the interpretation of *μ*_{L} and σ_{L} is straightforward. We actually perform the experiment in real time *t* but if we think of the ‘biological time scale’ as being Int, then in the Int space *μL* and σ L, are the mean and standard deviation of Int. There is no problem with t < 0 but the distribution is, of course, skew.

### Kinetic models

*F*(

*t*) with no probabilistic interpretation. Now there are clearly several ways in which this could be done. We know that there is, first of all, a phase during which the cells settle under gravity. However, this phase is very short in experiments in shallow dishes and can safely be neglected (unpublished data; accompanying paper). Morphological changes then begin (Figs 1, 2). A possible analogy with chemical kinetics could presume that there is a stochastic transfer constant,

*k*say, and that the rate of transformation from rounded to spread could be taken as proportional to the number of rounded cells remaining. Assuming (Fig. 2) that the observer differentiates only two states as in : then

*F*(

*t*) would be defined by: and it only remains to enquire more closely into the meaning of

*k*. It is proposed (Fig. 2) that at the contact surface cells send out processes to explore the nature of the surface. Occasionally a stable contact forms and then the cell transforms via a rate process into the altered morphology. Obviously the rate at which this happens is roughly proportional to the number of cells remaining in the unspread state and so

*k*constitutes an overall measure of the proportion finding satisfactory contact and the rate of transformation.

*k*

_{1}and

*k*

_{2}as in: where now

*ki*would measure the rate of forming contact sites and

*kz*would estimate the rate of conversion into the spread state. The equation for

*F*(

*t*) would be: and this allows a sigmoid

*F*(

*t*) profile unlike equation (4).

In defence of using such kinetic models with irreversible steps, we simply indicate that in the time course and experimental situation envisaged the reverse transformation from spread to rounded morphology is not appreciable.

### Empirical equations

*F*(

*t*), since very sweeping assumptions have been made in developing them. However, the statistical models can be strongly defended on the grounds that the

*p*and avalúes obtained certainly do have the significance described whatever underlying molecular interpretation there is. It is perhaps not so easy to defend the interpretation of the

*k*

_{i}values obtained using the kinetic equations, since exponential models are quite versatile and the sort of

*F*(

*t*) profiles obtained experimentally would be well fitted even if the interpretation of the meaning of the

*k*

_{i}was in error. The only way to settle this would be to measure the rates of transformation of individual cells via such primed or transition states as must exist and this has not yet been satisfactorily achieved. In any case, we might expect large inter-cell variations in the

*k*

_{i}values leading, in the end, to a probabilistic type of interpretation. For purposes of comparison, we could try a model such as the Hill equation in the form: suggested by the sigmoid nature of many of the observed F(

*t*) profiles. Equation (6) has an interpretation only in a very limited sense, i.e. when

*K*is positive and

*n*a positive integer, for then it models the totally cooperative binding of

*n*ligand molecules to a receptor with no intermediate species (Bardsley & Waight, 1978). The only value in such a model is that we can thereby obtain an independent measure of how well experimental data can be fitted by arbitrary models with the same number of disposable constants. Parameter values, as new surface treatments are studied or new media are used, can then be consulted to estimate the significance of any changes resulting. This will be developed in the accompanying paper.

### Scaling parameters

*A*final point stems from the fact that all the expressions for

*F*(

*t*) given so far have the asymptotic property that: In other words, after a sufficient period of time has elapsed, 100 % of the cells become spread. In actual experiments it is unusual for all the cells to be viable and maximum values of 90—95 % of total cells being spread eventually are more likely. To accommodate this we could introduce a scaling factor, λsay, that would normally be of the order of 0·9 and this could be included as a parameter to be estimated. In other words, the experimental data could be fitted by G(t) defined as: This, of course, raises the problem of how many parameters are needed to fit the data. Clearly, any improvement in goodness of fit resulting from the introduction of such an extra parameter would have to be assessed against the increase in number of degrees of freedom in the model. To discuss this we now formulate a possible experimental approach.

### A possible experimental approach

*t*=

*t*

_{1},

*t = t*

_{2},…,

*t = t*

_{n}, a number of microscope fields could be counted in replicate, by independent observers if possible, and in this way estimates of

*F*(

*t*) together with standard errors could be determined. If we call these estimates E(

*t*

_{1}), E(

*t*

_{2}),. .., E(

*t*

_{n}) then they could be individually weighted by the standard errors, SEi, SE2, …, SE", and the sum of squares of weighted residuals,

*Q*, formed according to: By non-linear regression analysis,

*Q*could then be minimized and the best fit parameters obtained along with confidence limits on the parameter magnitudes.

Then comes the problem of estimating goodness of fit to experimental data using different models. Where rival models have the same number of parameters, then the best model is simply that with the lowest *Q* value. However, to decide whether any significant improvement comes with introducing extra parameters as, for instance, a choice of whether to introduce the scaling factor Å referred to previously then we could compute the likelihood ratio test statistic (Lindgren, 1976).

## CONCLUSION

In this paper we have argued that there exists a need to provide quantitative measures of cell spreading so that comparisons of different media, cell types and surface coatings can be made numerically. We have shown that simple models with only two or three parameters can be defined in several ways and we have classified alternative approaches as statistical, kinetic or empirical, arguing that in the first two cases a definite interpretation can be given to the numerical estimates of the parameters of the model. Ways of designing experiments and discriminating between alternative models have been discussed and the use of the F-test statistic recommended for assessing the need for models with increasing degrees of freedom. Parameters obtained in this way could then be studied as functions of other experimental variables such as temperature, concentration of solutes or density of surface coating. In the following paper we shall demonstrate how these theoretical considerations can be put to practical use.

## ACKNOWLEDGEMENTS

This work was supported in part by a grant from the Medical Research Council to J.D.A.

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