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.
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.
The percentage of cells spread is simply 100F 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.
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
where the definite integral in the denominator replaces the usual √ (2π). The interpretation of μT 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
Now μs has the interpretation of the mean time to spread and σs 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 = 2μ.
The log normal distribution
and the interpretation of μL and σL 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.
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.
A possible experimental approach
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).
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.
This work was supported in part by a grant from the Medical Research Council to J.D.A.