Lewis Et Al. 1977

J. theor. Biol. (1977) 65, 579-590

Thresholds in Development

J. LEWIS, J. M. W. SLACK AND L. WOLPERT

Department of Biology as Applied to Medicine, The Middlesex Hospital Medical School, London W1P 6DB

(Received 21 May 1976, and in revised form 15 August 1976)

The interpretation of gradients in positional information is considered in terms of thresholds in cell responses, giving rise to cell states which are discrete and persistent. Equilibrium models based on co-operative binding of control molecules do not show true thresholds of discontinuity, though with a very high degree of co-operativity they could mimic them; in any case they do not provide the cells with any memory of a transient signal. A simple kinetic model based upon positive feedback can account both for memory and for discontinuities in the pattern of cell states. The model is an example of a bistable control circuit, and transitions from one state to another may be brought about not only by morphogenetic signals, but also by disturbances in the parameters determining the kinetics of the system. This might explain some aspects of transdetermination in insects.

An attempt is made to analyse the precision with which a spatial gradient of a diffusible morphogen could be interpreted by a kinetic threshold mechanism, in terms of the length of the field, the steepness of the concentration gradient, and the intrinsic random variability of cells. It is concluded that it would be possible to specify as many as 30 distinct cell states in a positional field 1 mm long with a concentration span of 103. Mechanisms for reducing the positional error are considered.

1. Introduction

Many pattern-forming processes in development may be viewed in terms of a mechanism involving positional information. Two main steps are envisaged: the cells have their position specified with respect to certain boundary regions, and then they interpret this positional information by an appropriate choice of cell state. The choice of cell state may correspond to a change in determination or initiate a course of cytodifferentiation. Much more attention has been given to how positional information may be supplied than to the process of interpretation. A classic mechanism for specifying position depends on a gradient in some morphogen acting as a positional signal, as in the insect epidermis (Lawrence, Crick & Munro, 1972), the insect

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580 J. LEWIS ET AL.

egg (Sander, 1975), hydra (Wolpert, Clarke & Hornbruch, 1972) and the chick limb (Tickle, Summerbell & Wolpert, 1975). Possible molecular mechanisms for setting up such gradients have been considered by Gierer & Meinhardt (1972). A gradient in a positional signal is not the only way of providing positional information, as is shown by the phase-shift model of Goodwin & Cohen (1969) and the progress zone model of Summerbell, Lewis & Wolpert (1973). Nevertheless, it seems reasonable to assume that ultimately the positional value of the cell will be governed by the concentration of one or more chemical compounds.

The first step in the interpretation of positional information is to choose amongst a set of different states, discrete choices being associated with thresholds in the response of the cells: for concentrations just above a threshold, the cells will adopt one state, and for concentrations just below it, another state. If the positional signal is graded, the spatial boundaries between cells in different states will correspond to critical concentrations of the signal substance, as first pointed out by Dalcq & Pasteels (1937). Spemann, indeed, evidently saw here a serious objection to the gradient concept of Child (1941) when he wrote: “Another difficulty seems to lie in the fact that the gradient . . . must be conceived of as continuous, whereas the series of formations whose differentiation would be determined by that gradient . . . is absolutely discontinuous” (Spemann, 1938).

We have to meet this objection, if we are to believe that a smooth concentration gradient of a morphogen can provide positional information which the cells can then interpret to give the spatial pattern of cellular differentiation (Wolpert, 1971). The rather abstruse analysis provided by catastrophe theory (Thorn, 1975) leaves some important biological questions unanswered. We need to know how many discrete distinctions could plausibly be specified by such a gradient, and how reliably, given the natural variability of cells. Crucial as the problem is, it has received virtually no attention. We wish here to suggest some simple mechanisms, and to discuss the fidelity with which they could generate patterns.

Thresholds are intimately related to another feature of cell determination: its persistence. In many cases the differences set up in a positional field remain long after the field-like properties have disappeared together with the positional signal. The cells must remember the effect of the signal (Wolpert & Lewis, 1975). The threshold and the memory are, we believe, two aspects of a single process: both depend on a positive feedback loop in the intracellular control system.

The detailed chemistry of thresholds in embryonic development is unknown. Elsewhere in biology, however, certain threshold phenomena have been analysed thoroughly, and the role of positive feedback has been

THRESHOLDS IN DEVELOPMENT 581

made plain; as, for example, in the triggering of the nerve action potential (Hodgkin, 1967) and in the induction of the lac permease in E. coli (Kennedy, 1970). In other cases, such as the very abrupt and striking response of certain enzyme activities in plants to the amount of phytochrome activated by light (Mohr & Oelze-Karow, 1976), something of the chemistry is known, but the mechanism is as yet only half-elucidated.

2. Equilibrium Control and Kinetic Control

Thresholds are often dismissed as being easily accounted for by co-operative binding of control molecules to an allosteric enzyme. The threshold, on this conception, consists simply in a steep shift in the level of an equilibrium between different forms of the enzyme as the concentration of the control molecule changes (Fig. 1). The threshold is thus viewed essentially as an equilibrium phenomenon. We wish to contrast this with a view of thresholds which is essentially kinetic.

-2

Morphogenetic

is

B -3-

-I

I I

--L----y---- Territory 1 Intermediate Territory 2

FIG. 1. A simple equilibrium model for a “threshold”. The concentration gradient of S spans two decades across five cells. The lower curve depicts the degree of saturation of an allosteric enzyme obeying the Hill equation with K = lo-la, PZ = 4. The intermediate response of the cells is arbitrarily drawn to occur between YI = O-2 and Yz = 0.8.

582 J. LEWIS ET AL.

Although the cell as a whole is in no sense at thermodynamrc equilibrium, it is possible to isolate sets of metabolites whose relationships are very near those of equilibrium (Newsholme & Start, 1973). One may thus consider the effect of variations in a signal substance on a subsystem where all the other substances assume their equilibrium concentrations passively. Consider, for example, the case where the observed response of the cell isdirectlyproportional to the degree of saturation of a receptor enzyme, E, by the morphogen S. Suppose that E binds IZ molecules of S with very high co-operativity. Under these conditions the equilibrium degree of saturation, Y, will show a sigmoid dependence on S with a Hill coefficient approaching y1 (Fig. 1):

E-I- nS z$ ES,, with dissociation constant K

y - IWnl L-S]”

The equilibrium dependence of Y upon S is always continuous. One might admittedly mimic the effect of a true threshold of discontinuity, if the dependence of Y on the signal concentration S were so steep that there was a large difference between Yin adjacent cells in the graded signal field. But some rather extreme conditions would have to be met. Suppose, for example, that Y changed from 10% to 90% of maximal. With a Hill coefficient of 4 this would require a difference of S of three-fold between adjacent cells. At this rate, the concentration of S would change by 31° fold across ten cell diameters and by 3 loo fold across 100 cell diameters. Such very steep concentration gradients do not seem plausible (see below). Thus any equilibrium chemical control of this sort which could produce a steep enough response to mimic a threshold would need a receptor with a number and a co-operativity of binding sites much greater than those of most known allosteric enzymes. But even then, a crucial requirement for cell determination would be missing: the cells would have no memory of the effect of the signal. They would all revert to the same state after the signal gradient was gone. Admittedly, true discontinuities may occur in equilibrium systems under- going phase transitions, and may be associated with hysteresis, as in the supercooling of water or the orientation of domains in a magnet. But phase transitions depend on the establishment of long-range order in very large (strictly speaking, in infinite) assemblies of molecules, and are an unlikely means of producing the subtle and multifarious thresholds occurring in animal cells. The memory of a change of determination that is heritable from one cell generation to the next must be a non-equilibrium phenomenon.

Let us therefore consider the cell as a kinetic system, in which chemicals react far from equilibrium at rates depending on their concentrations. Such a system can jump discontinuously from one sort of steady state to another


as external control parameters are changed (Kacser, 1960; Prigogine & Nicolis, 1971; Edelstein, 1972; Thomley, 1972). We illustrate this sort of threshold phenomenon by a simple and biochemically plausible model, whose basic formulation was suggested to us by Graeme Mitchison. The model has interesting properties which are not intuitively obvious, and which it shares with a much more general class of kinetic systems.

We consider the activation of a gene G by a signal substance S. Tran- scription of the gene G is assumed to be promoted in a linear fashion by S, and in a sigmoidal fashion by its own product g, giving a positive feedback; g is in addition degraded at a rate proportional to its concentration. If we eliminate various intermediate chemical variables from the system of equations describing transcription, translation and metabolism, the behaviour of this subsystem when subject to slowly changing conditions may be summed up by a single equation for g. To be precise, let the net rate of change of g be

z=k,S+ kg2 ----y - hg k+g

where the k’s are constants. For a given value of S, there are steady states at the values of g such that dg/dt = 0 (Fig. 2). A steady state is stable if any small departure from it is self-correcting: that is, if dg/dt is negative for slightly higher values of g, and positive for slightly lower values of g. The stable steady states for a given S can be read directly from the graph of dgldt as a function of g.

Figure 2 shows the graphs for three different concentrations of the signal substance S. When S is low, there are two stable steady states, at go(S) and gr(S). When S is high, there is only one, at gl(S). The critical intermediate signal concentration SC divides the regime where there are two possible stable steady states from the regime where there is only one. Suppose that a cell starts with the gene G inactive, that is with g = 0, and is exposed to a signal concentration that increases gradually from zero. The concentration of the gene product g will stay close to the lower stable value go(S) until S reaches the threshold SC. When S becomes greater than SC, the first downward dip of the dgldt curve fails to meet to the zero line, and there ceases to be a steady state in the neighbourhood of the point g&S’,) to which the cell has been brought. The concentration of g will thereupon rise autonomously until it reaches the upper stable value gl(S). Thus there is a discontinuous transition from one steady state to another as S is increased past its threshold value SC. In principle, an infinitesimal increment from S, -dS to SC +dS will bring this transition about. Our kinetic system stands here in contrast with the equilibrium systems, where no such discontinuity occurs.

J. LEWIS ET AL.

FIG. 2. The rate of change, dg/dt, of the concentration of the gene product as a function of the instantaneous concentration g, for three different tied concentrations of the signal substance S. The curves are calculated assuming

The arrows on the curves point in the direction of increasing g where dg/dt > 0, and of decreasing g where dg/dt < 0. A steady state occurs where a curve cuts the g-axis; it is stable if the arrows converge on it, and unstable if the arrows diverge from it. When S = 0, there are stable steady states at the points A [where g = go(O) = 0] and D [where g = gI(0)] and there is an unstable state at C. When S equals the critical value S,, there is one stable steady state at E [where g = gI(&)] and a singular unstable state at B [where g = g&S,)]. When S = 2S,, there is only one steady state, at F [where g = gl(2SJ], and it is stable.

The kinetic system, furthermore, has a memory. If the signal concentration S, having been raised above threshold, is then brought down again to zero, the system will not return to the lower stable state go(O); it will instead be left in the upper stable state gl(0). The gene will, in effect, have been turned on permanently. Much of the argument can be carried over into a far more general case. Discrete alternative stable modes are a natural feature of chemical kinetic systems, and the discontinuous transitions from one mode to another can be triggered by transient control signals.

Our kinetic system is an example of the type of bistable control circuit postulated by Kauffman (1975) to explain determination and transdetermination in the imaginal discs of Drosophila. It shows clearly why transitions between alternative steady states may occur more readily in one direction than in the other: in our particular model, the upper stable steady state g1 is stable for example even against large fluctuations in S, whereas the lower stable steady state go is not. Random disturbances of other concentrations, affecting the values of kl, k,, k, and k4, could likewise cause transdetermination preferentially in one direction. A fluctuation in the concentration


of a small molecule, which might pass from cell to cell through gap junctions, could cause transdetermination of a group of cells simultaneously, as is observed (Gehring, 1972).

A related feature of the system is that the cells in an embryonic field may continue to be sensitive, say, to an increase in the concentration of a signal substance, even after they have ceased to be affected by a reduction. One might thus understand why, for example a graft of ectopic polarizing tissue in a chick limb bud can cause an almost complete reduplication of digits, whereas excision of polarizing tissue at the same stage causes only slight deficiencies (Tickle, Summerbell & Wolpert, 1975).

3. Positional Signalling and Positional Precision

To give the body its spatial pattern, developmental signals must specify the positions where cells cross thresholds. Most developmental fields are about 1 mm or 100 cell-diameters in maximum linear dimension, and specification takes place over a period of hours. As Crick (1970) has pointed out, this makes diffusion a plausible means to set up a positional signal, and there is some experimental evidence from Hydra (Wolpert, Clarke & Hornbruch, 1972), insects (Sander, 1975; Lawrence, Crick & Munro, 1972) and the vertebrate limb (Tickle, Summerbell & Wolpert, 1975) which is consistent with this. The local concentration of a substance diffusing across the field may serve as the signal S which controls the choice of cell state. Thresholds in the response to S would appear as a spatial pattern of discontinuities demarcating different cell states. There is no reason in principle why there should not be many thresholds in the response to a single signal, depending on different receptor circuits, so that many distinctions are simultaneously determined in the one field.

There are, however, some limitations imposed by the random variability of cells. Given our threshold mechanism, we can relate the random errors in the placing of thresholds in the body to the random errors in the control of chemical concentrations inside cells. The errors in the placing of thresholds will in turn set a limit to the number of thresholds that can be drawn in a reliable spatial sequence across one developmental field.

There are some rather scanty data bearing on these points. The lengths of most parts of the body in higher animals seem to be determined with a precision of about 3% or better (Wolpert, 1972; Maynard-Smith, 1960). The random variability of intracellular concentrations will be discussed more fully below: it may be much greater, perhaps of the order of 20% for many substances. The number of distinct states typically specified per embryonic field is hard to establish. There is some suggestion that all the

586 J. LEWIS ET AL.

14 segments of the insect body may be rendered distinct by a single axial field acting round about the time of blastoderm formation (Sander, 1975); and it is possible that the number of territories specified during primary induction along the main body axis in vertebrates may be of the same order of magnitude. On the other hand, the subdivision of Drosophila imaginal discs into compartments, for example, seems to proceed simply by binary distinctions (Garcia-Bellido, 1975; Crick & Lawrence, 1975).

Concentration gradients could be established in various ways. Diffusion is perhaps the simplest conceptually and we shall assume it in this discussion. But most of our general conclusions do not depend on this specific assump- tion. Consider, then, a field a hundred cell-diameters long, in which a signal gradient is set up by diffusion from a source at one end. Could the gradient serve to specify, say, ten distinct regions? Could it determine lengths to a precision of 3 %? Let us assume that the signal substance is kept at a constant concentration S, at the source, and that elsewhere it is degraded uniformly, at a rate proportional to its concentration. The steady-state distribution will then be of exponential form. The concentration S at a distance x from the source will be given by

or equivalentIy S = So emax

x = d In So/S (2)

where a is a constant depending on the ditIusion constant and the rate of degradation.

If a threshold occurred invariably at S = S,, there would be a transition in the spatial pattern precisely at

x, = ; In s,/s,.

If the cells are variable, however, with their individual thresholds ranging from S, - AS, to S,+ AS,, then the position of the transition will be variable by &Ax,, where

AX c = ; AS&

We can express the positional error as a fraction of the length L of the field, using equation (1) :

Thus the positional error can be reduced in two ways: by reducing the error


in the setting of the threshold, A&/S,, and by increasing the concentration span of the gradient, So{&.

How big a concentration span is plausible? The upper extreme of concentration which one might plausibly expect for a small molecule in the soluble pool is about 10T2 M. If for the lower extreme we take ten molecules per cell, i.e. about lo-lo M, we have a maximum concentration span of 10’. Consideration of the flux to be drawn from each source cell, however, and of the homeostasis of sources, sets a rather more stringent limit of 10’ for the concentration span.

The positional error Ax& would then be less than the error in the threshold AS/S, by a factor of In lo’, that is, about 12. For a concentration span of only 10, the positional error would be only five times larger than for a concentration span of 105.

The value of A&/S, is the crucial unknown. It depends on the random variability of chemical concentrations in cells, and we have found it remark- ably difficult to obtain information on this basic point; it is also hard to distinguish experimentally between intrinsic variability in single cells, and random errors in the techniques of measurement. A cytochemical study of mouse fibroblasts (Killander & Zetterberg, 1965) showed that the RNA content had a coefficient of variation of 22% and the dry mass of 11% even for sister cells at the same stage of the division cycle, with a DNA content variable by no more than 2%. Measurements of chloride concentration in squid giant axons (Keynes, 1963) of resting potential in frog muscle fibres (Adrian, 1956) and of pH in crab muscle fibres (Aickin & Thomas, 1975) have standard deviations corresponding to coefficients of variation in ionic concentrations of about 8 %, 12% and 30% respectively; these can be regarded as upper limits on the variation intrinsic to the cells.

From rather inadequate evidence such as this we would guess that a variability of 20% would be typical of many intracellular concentrations. Several different substances must take part in the threshold reactions, and the error in each may be expected to contribute to the error in the definition of the threshold concentration of S. In the same way, errors in many different metabolite concentrations will contribute to the error in the concentration of any one given metabolite. The cell’s system of homeostatic controls, however, presumably keeps the error at almost every point of the reaction network within standard limits, avoiding disastrous accumulations of error. In the first instance let us therefore suppose that the threshold setting error AS,/s, is of the same order as most concentration errors in the cell, say 20%. Then we can find the positional error from equation (2): for concentration spans of 106, lo3 and 10, it will be roughly 1*5x, 3% and 9x, respectively. Thus our arguments lead to an estimate which is perfectly

588 J. LEWIS ET ,jL.

compatible with the observations of positional errors of about 3 %. As many as 30 distinct states could be specified in reliable serial order by one positional signal, although its gradient would need to be rather steep.

4. Modification of the Error

The size of the threshold setting error AS,/,!& might in any case be reduced in various ways. The concentrations of substances involved in the threshold reactions might be controlled with special accuracy. If DNA is involved, as seems likely, its concentration may be standardized by confining the action of the signal to a particular phase of the cell cycle, such as G,. Fluctuations in the concentrations of small molecules may be reduced by their exchange through low-resistance junctions between the cells (Loewenstein, 1973; Furshpan & Potter, 1968; Slack & Warner, 1975). The signal substance might act through a reaction of high order in S, so that a 20% error in the level of the immediate reaction product required for threshold would correspond to a much smaller percentage error in the required threshold level of S itself. But we recognize that we are here glossing over complex problems raised by variability and fluctuations in control systems.

Because of the errors in the thresholds of the individual cells, the boundary between tissues composed of cells in different states may not be sharp. There may be a transition region, of width Ax,, where cells above and below their individual thresholds are mixed together. The position of the centre of the transition region will be defined by an average over many cells, and will have an accuracy better than Ax,. The effect of cell variability can be much reduced if the cells are free to move a little. The cells in the transition region may then sort out if, for example, those in the same state of determination stick together more strongly than those in opposite states (Steinberg, 1976). Lawrence (1975) has suggested such a mechanism for maintaining the sharp and precisely placed boundaries at compartment borders in Drosophila.

5. General Discussion

The kinetic model for thresholds given above shows how cells may change their state discontinuously at particular concentrations of a signal substance, and how this change of cell state may persist when the signal is withdrawn. Although our argument has been presented specifically in terms of chemical concentration gradients, thresholds must be an essential feature of any theory of development, if it is to explain the origin of qualitative differences between cells. This holds true irrespective of whether the differences arise from cytoplasmic differences in the egg, or from signals from other cells.


If a signal can supply positional information only over a small field, and yet is to determine a gross feature of a large animal, it must do its work when the embryo is still small; it must assign different states of determination to the cells, and they must remember the assignments when the embryo has grown big, and the signal is no longer there. Mesoderm must remain meso- dermal, thoracic somites must help to make a thorax, and the leg bud must develop into a leg. Thus in the pattern of differentiation of the body, all but the finest details must depend on temporary signals that set up lasting distinctions; and discontinuities of cell character are characteristic of that process. Such perpetuated differences represent discrete alternative steady modes of the dynamical system of reacting chemicals in the cell, and cannot in general be continuously graded. On the other hand, where a morphogenetic signal is permanently present there is no need for such discontinuities : a maintained graded signal could maintain a smooth gradation of cell character and an equilibrium model for interpretation might be appropriate, as perhaps in hydra. A distinction should be drawn between two different ways in which a smooth gradation between two types of tissue might occur. In the one there could be continuous variation in the character of the constituent cells; in the other, the tissues might be composed of discrete types of cell, mixed in smoothly variable proportions.

The analysis above has shown that a concentration gradient across a small field could easily define the position of a discontinuity of cell state with a precision of 3 y0 of the length of the field. In a field of 100 cell diameters it should be possible to specify as many as 30 distinct cell states in a reliable sequence. A basic factor determining the precision of such a pattern is the random variability of the individual cells. This is probably an important source of variability between organisms which is neither genetic nor environmental.

This work is supported by the Medical Research Council.

REFERENCES ADRIAN, R. H. (1956). J. Physiol. 133,631. &CKIN, 0. C. & THOMAS, R. C. (1975). J. Physiol. 252,803. CHILD, C. M. (1941). Patterns and Problems in Development. Chicago: University of

Chicago Press. CRICK, F. H. C. (1970). Nature, Land. 225,420. CRICK, F. H. C. & LAWRENCE, P. A. (1975). Science, IV. Y. 189, 340. DALCQ A. & PASTEELS, J. (1937). Arch. B&l. 48,669. EDELSTEIN, B. B. (1972). L theor. Biol. 37,221. FURSWAN, E. J. & POTTER, D. D. (1968). Curr. Top. Devel. Biol. 3, 95. GARCIA-BELLIDO, A. (1975). In Cell Patterning, Ciba Foundation Symp. 29, 161. Amster-

dam : Associated Scientific Publishers.

590 J. LEWIS ET AL.

GEHRING, W. (1972). In Biology of imaginal Disks (H. Ursprung 8z R. Nothiger, eds), p. 35. Berlin: Springer.

GIERER, A. & MEINHARDT, H. (1972). Kybernetik X2,30. GOODWIN, B. & COHEN, M. H. (1969). J. theor. Biol. 25,49. HODGKIN, A. L. (1964). l’?ze Conduction of the Nervous Impulse. Liverpool: University Press. KAC~ER, H. (1960). Symp. Sot. exp. Biol. 14,13. KAUFFMAN, S. (1975). In Cell Patterning, Ciba Foundation Symp. 29, 201. Amsterdam:

Associated Scientific Publishers. KENNEDY, E. P. (1970). In The Lactose Operon (J. Beckwith & D. Zipser, eds), p. 49.

Cold Spring Harbor Lab. KEYNES, R. D. (1963). J. Physiol. 169,690. KILLANDER, D. & ZETTERBERG, A. (1965). Expl. Cell Res. 38,272. LAWRENCE, P. A. (1975). In Cell Patterning, Ciba Foundation Symp. 29, 2. Amsterdam:

Associated Scientific Publishers. LAWRENCE, P. A., CRICK, F. H. C. & MUNRO, M. (1972). J. cell Sci. 11,815. L~EWENSTEIN, W. R. (1973). Fedn. Proc. Fedn. Am. Sots. exp. Biol. 32, 60. MAYNARD-SMITH, J. (1960). Proc. R. Sot. B. 152,397. MOHR, H. & OELZE-KAROW, H. (1976). In Light and Plant Development (H. Smith, ed.).

Kent : Butterworth. NEWSHOLMIX, E. A. & START, C. (1973). Regulation in Metabolism. London: John Wiley &

Sons Ltd. PRIGOGINE, I. & Nrco~xs, G. (1971). Q. Rev. Biophys. 4, 107. SANDER, K. (1975). In CelZ Patterning, Ciba Foundation Symp. 29, 241. Amsterdam:

Associated Scientific Publishers. SLACK, C. & WARNER, A. E. (1975). J. Physiol. 248,97. SPEMANN, H. (1938). In Embryonic Development and Induction, p. 331. Yale: Yale Uni-

versity Press. STEINBERG, M. S. (1970). J. expl. Zool. 173,395. SUMMERBELL, D., LEWIS, J. H. & WOLPERT, L. (1973). Nature, Lond. 244,492. THOM, R. (1975). Structural Stability and Morphogenesis. Reading, Mass. : Benjamin. THORNLEY, J. H. M. (1972). An. Bot. 36, 861. TICKLE, C., SUMMERBELL, D. & WOLPERT, L. (1975). Nature, Lond. 254, 199. WOLPERT, L. (1971). Curr. Top. devel. Biol. 6, 183. WOLPERT, L., CLARKE, M. R. B. & HORNBRUCH, A. (1972). Nature, Lond. 239,101. WOLPERT, L. & LEWIS, J. H. (1975). Fedn. Proc. Fedn. Am. Sots. exp. Biol. 34, 14.

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Lewis Et Al. 1977