## ABSTRACT

Unfertilized eggs of the sea-urchin

*(Psammechinus miliaris)*were left for known but varying times in contact with homologous sperm suspensions containing known numbers of spermatozoa. Counts were made of the numbers of fertilized and unfertilized eggs at times ranging from o to 300 sec. after mixing.If spermatozoa are considered as particles moving in random directions, the frequency of sperm-egg collisions can be calculated if the density and mean speed of the sperm suspension are known.

The information in (1) and (2) enables an estimate to be made of the probability of a successful sperm-egg collision.

The estimated probability of a successful collision,

*p*, varies with sperm density,*d*. At the lowest density used, 7·44 ×10^{4}/ml.,*p*was found to be 0·226. At the highest density, 9·62 ×10^{6}/ml.,*p*was about 0·011. The inverse relationship between*p*and*d*may be due to sperm-sperm interactions of a physical nature.The presence of jelly round the eggs increases

*p*. This disposes of the possibility, raised in the past, that egg jelly may have an adverse effect on the fertilizing capacity of homologous spermatozoa under normal conditions of fertilization.The technique of subjecting eggs to a pre-determined number of collisions facilitates investigation of the conduction time of the block to polyspermy. Preliminary experiments suggest that the conduction time may be of the order of seconds rather than fractions of a second.

Sources of error arising from the ‘kinetic ‘treatment of sperm suspensions are discussed in detail. One source of error concerns the alleged chemotaxis of spermatozoa towards eggs and egg secretions. No chemotaxis was observed.

## INTRODUCTION

The change in cortical structure which passes over the egg surface of *Psammechinus miliaris* in about 20 sec. at 18°C. can only be the block to polyspermy if the probability of a successful sperm-egg collision is lower than one. For during the passage of the cortical change over the egg surface, previous calculations (Rothschild & Swann, 1949) indicate that there will be some fifteen potentially successful sperm-egg collisions after the cortical change has started in an egg immersed in a sperm suspension of density 10^{6}/ml. An obvious extension of these experiments was to investigate the effect of polyspermy-inducers on the conduction velocity of the cortical change and on the speeds of the spermatozoa. A decrease in the former or an increase in the latter would increase the probability of polyspermy. As, however, nicotine, an excellent polyspermy-inducer in the sea-urchin egg, had neither of these effects, the question again arose of the probability of a sperm-egg collision being low and of nicotine exerting its effect by increasing this probability (Rothschild & Swann, 1950). The methods used to investigate the questions of the low probability of a successful sperm-egg collision and of the relationship, if any, between the cortical change and the block to polyspermy were at that time indirect. Both these questions have now been examined in a more direct way than before. A brief note on the relationship between the conduction velocities of the block to polyspermy and of the cortical change has been published (Rothschild & Swann, 1951), but will be discussed in more detail in this paper. The probability of a successful sperm-egg collision was studied by immersing unfertilized eggs in sperm suspensions of known density for known but varying periods of time, ranging from o to 300 sec., and later counting the proportions of fertilized and unfertilized eggs. Interpretation of the results in terms of probabilities involves the hypothesis that a suspension of sea-urchin spermatozoa can be considered analytically as an assemblage of gas molecules, moving in random directions. Apart from the difficulty of verifying this hypothesis experimentally, there are obvious differences between a suspension of spermatozoa and an assemblage of gas molecules. The most serious of these concerns the statement, often made but not substantiated, that a substance diffuses out of unfertilized eggs and has a chemotactic effect on spermatozoa of the same species.

Although chemotaxis of plant spermatozoa towards secretions of plant eggs or archegonia undoubtedly occurs, the position is far less clear in the gametes of animals. Possible explanations, other than chemotaxis in the true sense, of the accumulation of spermatozoa in the neighbourhood of egg secretions and eggs, have been discussed in detail by Rothschild (1951), while we have noted that the dark ground ‘tracks ‘of sea-urchin spermatozoa (Rothschild & Swann, 1949) do not bend or point preferentially towards eggs. Moreover, neither in the case of *Echinus esculentus* nor of *P. miliaris* have we been able to confirm Dakin & Fordham’s experiments (1924), in which sea-urchin spermatozoa accumulated to a marked degree in capillaries containing egg water, though Lillie’s activation^{*} test for the presence of egg water (1919) was positive. Another possible source of error in the kinetic treatment concerns the effect of egg jelly on spermatozoa. Though many experiments on fertilization are done on eggs without jelly, the jelly may of course be present in natural fertilization. The influence of the jelly has therefore been investigated, parallel experiments being done on eggs with and without jelly. A convenient method of establishing whether jelly is, or is not, present round eggs is to place a sample of the experimental suspension in sea water containing Janus Green B, which is strongly taken up at the surface of the jelly (see Harvey, 1941).

A further difficulty in the interpretation of the experiments centres round the well-known fact that the fertilizing power of a spermatozoon quickly declines with time (Lillie, 1915). Therefore, although eggs were on occasions left in contact with spermatozoa for as long as 300 sec., this was done with the knowledge that interpretation of data obtained after eggs had been in contact with sperm suspensions for minutes rather than seconds might involve new variables which would certainly obscure estimates of the true probability of a successful collision. An arbitrary upper limit of 45 sec. was therefore selected for the time of contact between eggs and spermatozoa. This question is discussed in more detail later. Though of uncertain value for probability estimates, the information gained in the range 45 − 300 sec. is of interest from another point of view, the decline in the fertilizing capacity of a spermatozoon.

Another source of error is introduced if the sperm suspensions are too dense. The disturbing factor might be called sperm-sperm interactions, and although on *a priori* grounds it might have been expected that in a dense suspension, spermatozoa would ‘interfere ‘with each other, the existence of such interactions at high sperm densities is strikingly demonstrated in the experiments described in this paper. This fact has interesting implications in other spheres of gametological research.

Although the experiments required no apparatus apart from beakers, pipettes and a microscope, the actual procedure was rather complicated, and is therefore described in detail in the next section. A serious, and to a certain extent unexpected, difficulty was experienced in finding a method of suddenly killing or inactivating the spermatozoa, without preventing the eggs developing or exhibiting the normal signs of having been fertilized, such as fertilization membranes, cleavage, or the well-known differences between the cytolysis of fertilized and unfertilized eggs. The following reagents were tried and found to be unsatisfactory. CuCl_{2}2H_{2}O: this inhibits fertilization but only after the eggs have been in sea water containing it for much longer than the duration of these experiments. Acidified sea water: this method of inhibiting fertilization has been used successfully by Tyler & Schultz (1932) and Tyler & Scheer (1937). Using the eggs and spermatozoa of *P. miliaris*, Tyler’s results could not be reproduced, no doubt because of species differences. HgCl_{2}: as is well known, this substance is highly toxic to spermatozoa; but it also has an irreversible toxic effect on eggs. The same applied to ceric sulphate and aluminium hydroxide, both of which are toxic to sea-urchin spermatozoa. It was found impossible to arrange the dosage of these substances so that though the spermatozoa were inactivated, the eggs were left in an identifiably fertilized condition. As will be seen, the experimental procedure made it essential that after the desired number of sperm-egg collisions, the spermatozoa should be incapable of recovering from the ‘killing’ treatment. A suitable concentration of hypotonic sea water (45% sea water in distilled water), however, was found to kill the spermatozoa and to prevent fertilization; but it had no harmful effect on the eggs.

## EXPERIMENTAL PROCEDURE

This is conveniently explained with the aid of Table 1, which represents one set (no. 4) of operations in a series, which may contain as many as ten sets, carried out at approximately the same time. A case where the eggs were in contact with a sperm suspension of known density for 5 sec. has been selected. The times on the right in

Table 1 show when pipettes or beakers were emptied into other beakers. For example, the eggs in beaker 3 were inseminated from the pipette at *t = 0* and after 5 sec. were transferred to beaker *2*. The hypotonic sea water in beaker 2 consisted of 35 ml. sea water and 55 ml. distilled water, so that after addition of beaker 3, the contents consisted of 45 % sea water in distilled water. The sea water in beaker 1 was made hypertonic with NaCl, the hypertonicity being adjusted so that after addition of the contents of beaker 2, the final ‘sea water’, in which the eggs were to develop, had the same tonicity as sea water. The success of this experimental procedure depended on the following considerations. (1) As one experiment involved a number of separate operations, each of the type shown in Table 1, consideration had to be given to the possibility that the mean speeds of the different sperm suspensions might vary because of delays during the experiment. This difficulty was obviated by carrying out the inseminations of the experimental (not the control) suspensions at 15 sec. intervals, in which time the speeds of the spermatozoa in the different suspensions did not decline significantly. This necessitated two people carrying out the operations on a rather complicated time schedule, because after insemination of eggs in beaker 3 in Table 1, for example, these eggs had to be transferred at specified time intervals to two different solutions. Without careful timing there would therefore have been a tendency for the third operation in series 4 to conflict with other operations in different series. At the same time, all the *t =* o operations had to be carried out as quickly as possible, for the reasons mentioned above. (2) Spermatozoa in beaker 3 must be instantly inactivated on being put into beaker 2, i.e. there must be no possibility of fertilization in beaker 2. Alternatively, the probability of fertilization in beaker 2 must be determined. This contingency can be dealt with by a control in which eggs and spermatozoa are added simultaneously to beaker 2, run at the same time as the main experiment. (3) Spermatozoa which appear to have been killed by the hypotonic sea water in beaker 2 may recover after transfer to beaker 1. This possibility can also be dealt with by a control experiment in which unfertilized eggs are added to beaker 1, spermatozoa to beaker 2, and the contents of beaker 2 to beaker 1, after 120 sec. (4) Eggs may be parthenogenetically activated by the hypotonic-hypertonic sea-water treatment. This eventuality necessitates a further control in which all operations are normal but spermatozoa are omitted from the series. (5) Another control is needed to determine the percentage fertilization at the same sperm and egg density as in the experimental series, in a sample from the same egg suspension, under normal conditions.

Before each operation in each series, beakers were very gently agitated by hand, in an irregular way. The experiments, on eggs of *P. miliaris* with and without jelly, were done at room temperature which varied from 17 to 18·5 ° C. during the season.

Sperm counts were made absorptiometrically and, as the new model of the Spekker was used, the procedure was slightly different from that previously described. In the new model there is a linear drum scale and ‘zero ‘is 1·30 instead of 2 ·00 as in the earlier model. Instructions to cover such differences have been published (Rothschild, 1950).

## RESULTS

Eggs and spermatozoa were allowed to interact for a series of known but different times; later, counts were made of the number of fertilized and unfertilized eggs in samples corresponding to each interaction time. The resultant information, which is depicted graphically in Fig. 1 a, b, enables an estimate to be made of the probability of a successful sperm-egg collision, *p*,^{*} or of the number of sperm-egg collisions that on the average will be needed for fertilization to take place. The method of making these estimates is given in an Appendix at the end of the paper. The results of all experiments, at different sperm densities, are given in Table 2, from which it will be seen that the probability of a successful sperm-egg collision varies with the number of spermatozoa in the sea water round the eggs. At a sperm density of about 10^{7}/ml., by no means a dense suspension, the probability of a successful collision is 1·5 in 100. This sperm density, which comes into the category of ‘very dilute’ (Rothschild, 1951), would be too low for use in a standard manometric experiment.

### Effect of jelly on p

The technique of subjecting eggs to a known number of sperme-gg collisions is probably the best method of quantitatively investigating the effect of varying environmental conditions on the fertilization reaction. One of the most interesting of these environmental conditions concerns the jelly which in nature surrounds the eggs, but which is removed in most fertilization experiments. The effect of jelly on *p* is shown in Table 3, from which it will be seen that its presence increases the probability of a successful collision. In the case of Exp. 1 in this table, reference to Table 2 shows that even if a scaling factor were introduced for the decline in *p* with increasing sperm density, which is trivial over such a small range, the removal of the jelly still reduces the probability of fertilization. This question does not arise in the other experiments in this series, as the sperm density was the same in the presence and absence of jelly.

### Block to polyspermy

The experiments described in this paper led logically on to an examination of the conduction velocity of the block to polyspermy. Although a brief note on these experiments, which were of a preliminary nature, has been published (Rothschild & Swann, 1951), it is convenient to recapitulate the results as they are so closely bound up with the others reported in this paper. The results of one experiment are given in Table 4. The experimental procedure was to run two experiments at the same time: in Exp. 1 the spermatozoa were killed in the usual way after a known time, 25 sec.; in Exp. 2, at the time when the spermatozoa were killed in Exp. 1, more spermatozoa were added, the sperm density being increased by a factor of 100. From results described ear her in this paper, it is known what proportion of eggs will have been fertilized in a particular time at a particular sperm density and therefore how many blocks to polyspermy will have started during that time. If, therefore, instead of killing the spermatozoa at *t* = 25, the intensity of the sperm bombardment is increased, there should be a negligible incidence of polyspermy unless a considerable number of sperm-egg collisions occur during the propagation of the block to polyspermy.

Table 4 shows that there are some three times as many polyspermie eggs in Exp. 2 as there were unfertilized eggs in Exp. 1. In other words, nearly half of the eggs which were fertilized in 25 sec. had not finished propagating their block to polyspermy in that time, and became polyspermie because of the new sperm bombardment they received after the 25 sec. period was terminated.

## DISCUSSION

### Sources of error:

Some of the sources of error, associated with treating a suspension of spermatozoa as an assemblage of particles or gas molecules moving in random directions, have been mentioned earlier (pp. 403, 404). A further source of error may be introduced by the assumption that the mean speed of a sperm suspension, , is 150 *µ*/sec. This requires two comments. First, it is a lower figure than that previously given (180*µ*/sec., Rothschild & Swaim, 1949). Dark ground sperm tracks taken during the course of these experiments indicated that the lower figure was the more accurate, at any rate during the season in question.^{*} Secondly, the figure 150 *µ*/sec. is assumed not to vary with sperm density. The variation of a spermatozoon’s speed according to the density of the suspension is a complicated subject. Earlier views that sperm speeds were inversely proportional to density were mainly based on measurement of O_{2} uptake which is a poor index of motility (Rothschild, 1948). In any case, no significant differences between sperm speeds in dense and dilute suspensions were observed using the dark ground track system (Rothschild & Swann, 1950), an observation which was confirmed at various sperm densities this year. The existing evidence therefore supports the view that sperm speeds do not vary greatly with density. Superficially, the fact that the probability of a successful collision declines with increasing sperm density might be held to lend support to the view that sperm speeds are lower in dense suspension. This, however, is an unlikely explanation of the decline in *p* at the higher densities because the sperm speeds would have to be so low to account for the observed probabilities, which differ by a factor of 20 when sperm densities differ by a factor of 100. The variation in *p* may be due to sperm-sperm interactions of a mechanical nature.

### Sources of error: slope of the log_{10}u, t plot

The arbitrary selection of the period 0−45 sec. (Fig. 1b) for the time during which it is safe to assume that the speed or fertilizing capacity of a spermatozoon remains constant may introduce errors. In future experiments, for example on the influence of changes in environmental conditions on the fertilization reaction, it would be safer to restrict observations to a relatively short period, as was done in the experiments on the effect of jelly on *p*. The statistical analysis supports this contention. In all experiments the values of *u* for *t* large, 45−300 sec., were lower than would have been expected on the basis of the slope of the log_{10}*u, t* plot up to *t* = 45. This deviation from linearity, which is very clear in Fig. *1b*, is a measure of the decline in the fertilizing capacity of the spermatozoa, either through a decrease in speed, or through the loss of some substance from the spermatozoa to the medium, with concomitant impairment of fertilizing capacity.

### Sources of error: n

Estimates of n, the sperm density, may introduce a rather serious source of error, not because the actual counts are subject to large error—the fiducial limits are of course known (Rothschild, 1950); but because there is at present no way of counting the proportions of live and dead spermatozoa in a suspension. This problem is well known in the field of mammalian sperm physiology, in which it has been solved (Hancock, 1951).

### Egg jelly

One possible explanation of the beneficial action of egg jelly is that it acts as a sperm trap (Rothschild & Swann, 1949). When eggs are inseminated with fairly dense sperm suspensions, more spermatozoa are observed in the neighbourhood of an egg than in an equivalent volume of nearby sea water. This happens because spermatozoa which collide accidentally with the jelly tend to stick to it, while, on the average, the same number of spermatozoa enter and leave an ordinary volume element of sea water. It does not follow that because an abnormal number of spermatozoa stick to the surface of the jelly, an abnormal number of spermatozoa will collide with the egg surface. The surface of the jelly presents a distinct barrier to the passage of spermatozoa, as can be seen by comparing the number of spermatozoa within the jelly, which is quite fluid, with the number on the jelly surface.^{*} The claim has sometimes been made that spermatozoa sustain a *loss* in fertilizing capacity after being in contact with egg jelly in sea water. There are reasons for querying the occurrence of this phenomenon under natural conditions (Rothschild, 1951) and these experiments show that even if it occurs, it has no bearing on the fertilization reaction in nature. Two other possible explanations of the effect of jelly in increasing *p* are worthy of mention. First, jelly may increase the mean speed of the suspensions, thus increasing the egg-bombardment frequency. There is no evidence for or against this explanation. Secondly, the jelly may have some orienting effect on the spermatozoa, increasing the probability that they collide with the egg surface at the right angle (Tyler, 1948). There is no evidence for or against this explanation, though the possibility that egg jelly has a function of this type has attracted the attention of a number of workers. Perhaps the most likely explanation is that egg jelly has a beneficial effect on spermatozoa in preventing loss of fertilizing capacity, which is equivalent to increasing the sperm density.

### The block to polyspermy

The experiments on this subject suffer from the disadvantage that a sperm density of 3 × 10^{8}/ml. is too high to obtain a morphologically satisfactory polyspermie fertilization reaction. At such high sperm densities there is a tendency for eggs to cytolyse, which may be due to an excessive number of spermatozoa entering the eggs; but it may also be connected with a high concentration of the haemolytic substance Androgamone III in the neighbourhood of the eggs. When counting eggs at the end of the experiment, a decision, which on occasions is difficult, has to be made as to whether an egg should be considered as cytolysed or polyspermie. As under normal conditions of insemination there is a negligible percentage of cytolysed eggs, it might be held that all eggs which are cytolysed under conditions of insemination with abnormally high sperm densities are cytolysed because of polyspermy. If this problem were obviated by reducing the sperm density resulting from the second insemination, the number of polyspermie eggs would be reduced and the results of the experiment would be correspondingly less clear cut. When these experiments are repeated more systematically, a series at lower second insemination sperm densities will be carried out.

In order to obtain some quantitative information regarding the conduction velocity of the block to polyspermy, as revealed by this experiment, the incidence of polyspermie eggs may be examined on the assumption that the block to polyspermy takes 1 or 20 sec. to pass completely over the egg surface. If eggs and spermatozoa are left in contact with each other for 24 sec., at a sperm density of 3 × 10^{6}/ml., the percentage of fertilized eggs at the end of that time will be about 85. One second later, that is 25 sec. after the beginning of the experiment, none of these fertilized eggs will be available for polyspermy. The maximum percentage of polyspermie eggs, if the conduction time of the block to polyspermy is 1 sec., is therefore 15. This is incompatible with the results in Table 4, in which the percentage of polyspermie eggs was 44, following the addition of more spermatozoa at *t* = 25. If, however, the conduction time of the block to polyspermy is 20 sec., which is also the conduction. time of the cortical change, the position is different. Five seconds after the beginning of the experiment, 45 % of the eggs will still be unfertilized. These eggs will not have finished propagating their blocks to polyspermy by the time the additional spermatozoa are added at *t* = 25, and consequently a little less than 45% of the eggs in the suspension will be available for polyspermy. The observed percentage of polyspermie eggs was 44. When this experiment has been systematically repeated, it may be possible to make a more accurate estimate of the conduction time of the block to polyspermy. Although this experiment is inconsistent with the concept of a high-speed block to polyspermy, with or without a high probability of a successful collision, interpretation on the basis of a 20 sec. block to polyspermy introduces difficulties which cannot be resolved without further experiments. Suppose that in a particular suspension, 80% of the eggs are fertilized in 20 sec. This means that each of these eggs has sustained an effective collision in that 20 sec. In the next 20 sec. a number of eggs in this 80 % will receive further effective collisions. The number of such collisions will be of the order of 0·7 at the sperm density in question. But the proportion of polyspermie eggs at this sperm density is far lower than this. The implication of this argument is that though it has been proved that the conduction time of the block to polyspermy is longer than has hitherto been suspected (i.e. more than i sec.), and that there is an associated low probability of a successful collision, the cortical change cannot be equated to the block to polyspermy without making additional hypotheses which at present are not experimentally justified.

### General

As might be expected, the development of a new method of investigating the fertilization reaction tends to raise new questions rather than answer old ones. On the basis of previous work, it was predicted that the probability of a successful sperm-egg collision was significantly less than one: this prediction has now been verified; but new problems, such as the decline in the probability of a successful collision with increasing sperm density, have arisen. Possible explanations of this phenomenon have been put forward, but further experiments are required before they can be confirmed or rejected. The principle of subjecting eggs to known concentrations of spermatozoa for known periods of time suggests a series of experiments involving the effects of variations in the environment on the fertilization reaction. For example, a number of agents are known to extend the fertilizing capacity of spermatozoa. But it is not known how this extension is achieved; nor can such effects in general be put on a quantitative basis. Equally, it should now be possible quantitatively to examine the effects of ageing, both of spermatozoa and of eggs, on the fertilization reaction. If such experiments were carried out in the same detail as those described in this paper, they would be extremely time-consuming. But, as is shown in the Appendix, estimates of *p* can be made when eggs have been subjected to known concentrations of spermatozoa for only two times, for example 10 and 30 sec.

## APPENDIX

### Estimation of the probability of a successful sperm-egg collision

(1)

*Provisional estimate of p*. Determination of the probability of a successful spermegg collision involves a knowledge of the number of collisions in unit time,*Z*, and of the observed proportions of fertilized and unfertilized eggs,*f*and*u*, in a suspension which has been in contact for time*t*with spermatozoa at density*n*.Consider a typical egg in the egg population. On the average, one spermatozoon will collide with this egg every τ (= 1/Z) sec. This is equivalent to a series of successive trials, at times τ, 2 τ, 3 τ,…, there being a constant probability

*p*of a success being associated with each trial. In this case a trial is a collision and a success is a successful collision. The probability*P*_{τ}of it taking exactly*r*trials to achieve a successful collision is equal to the probability that the first*(r—*1) trials will be unsuccessful and the*r*th trial successful, i.e.where

*q*= 1 −*p*.By the Addition Theorem, the probability of a successful trial (a collision followed by fertilization), at any one of the times τ, 2 τ, 3 τ,…,*S τ*, is given by where α =—(log*q*)/α and log*q*is negative. In terms of a sample from the population ofEquation (3·1) can be written in the form so that if log*u*is plotted against*t*, the slope of this line, a, is given by (log*q*)/τ. As a can be measured, log*q*, which equals—τ α, can be evaluated:*p*, the probability of a successful collision, equals 1—*q*. Calculations relevant to one particular experiment are given in Table 5, in which the number of fertilized and unfertilized eggs found in egg suspensions which had been in contact for different times with sperm suspensions of known density is shown. Immediately below this is given the proportion of unfertilized eggs corresponding to the actual numbers, but corrected for the incidence of fertilization (1%), which occurred when eggs and spermatozoa were in contact with each other for o sec. No correction was necessary for the incidence of unfertilizable eggs in this example, and in other cases these two corrections never amounted to more than 3 %. The calculations for the provisional estimate of*p*follow. The slope of the log_{10}*u, t*plot up to 45 sec. (Fig. 1 b) was determined by minimizing the sums of squares.*Z*, the number of collisions in unit time, is a function of the egg radius*a*, the sperm density*n*, and the mean speed of the sperm suspension . The relationship has been given and discussed in previous papers (Rothschild & Swann, 1949).- (2)
*Accurate estimate, p*. If, at time*t*_{1,}*f*1 fertilized and*u*_{1}unfertilized eggs have been counted in a sample of*N*_{1}eggs, the probability of obtaining such a result in random sampling is given by, where*Q*_{1}= 1—*P*_{t}*=*the true proportion of unfertilized eggs after time*t*. Since the results obtained in one sample do not affect the results in other samples, the probability of obtaining the combined data isThe data is made up of

*k*samples at times*t*_{i},*i*= 1..*k. L*is a function both of the data and of the parameter a, contained in the probabilities*Q*_{i}. When*L*is regarded as a function only of the data, it gives the probability of observing each possible combination of the data; but when considered as a function of a, the data being given, it is known as the Likelihood of each possible value of α (Fisher, 1922). The operation of maximizing the Likelihood enables a choice to be made of the a which is most appropriate to the data. In many cases the logarithm of the Likelihood, rather than the Likelihood itself, is used for finding the maximum. This is permissible since log*L*is a steadily increasing function of*L*. For*L*or log*L*to be a maximumThe expression—

*∂*/*∂*α log*L*is called the Score and finding the Maximum Likelihood estimate is equivalent to setting the Score equal to zero. This method of analysing the data is shown in Table 6. The plot of log (*u*/*N*) against*t*was found to give an approximately straight line (Fig. 1 b, p. 5) with slope 0·0224. This provides a provisional estimate of α, log 10 × 0·0224 =0·0516-Examination of the total Score, − 67·975, shows that this provisional estimate of a is too low. Further trials and linear interpolation show that if α = 0·052775, the total Score becomes +0·115, which is sufficiently near zero for our purposes. A difference*Δ K*from a zero Score causes a systematic error Δ α from the true estimate of α. Δ α /S(α) = Δ*K*/√ [*I*(α)]. If this latter quantity is numerically less than 0·1, Δ α is small compared with the sampling error of α. As*p*=1–exp*(− α τ), p =*0·115. - (3)
*Precision of p*. The sampling variance of the Maximum Likelihood estimate is given by where ∞ (λ) = the average value of λ, and*I(p)*—the amount of information associated with the estimate. Similarly, it can be shown that*I(p)*and*I*(α) being related by the equationFrom these relationships it is found (approximately) that the variance of

*p*isThe standard error of the estimate of

*p*isThere is therefore a fiducial probability of 0·95 that

*p*, the estimate of which is 0·115, lies between 0·132 and 0·098. (4)

*Initial hypothesis*. The underlying hypothesis is that after the addition of spermatozoa, the proportion of unfertilized eggs declines exponentially with time.*K/√ [I(α*,)] is distributed as a normal variate with zero mean and unit variance for each sample; consequently Σ{*K*^{2}/*I*(α) is distributed as*x*^{2}, the number of degrees of freedom being three, not four, since*p*has been used in the determination of these quantities. The values of K^{2}/*I*(α) and*√*{*K*^{2}*/I(α)*are given in Table 6, the latter being 1·748. Such a value of x^{2}on three degrees of freedom is attained or exceeded with a probability of slightly less than 0·70, which shows that there are insufficient grounds for rejecting the original hypothesis.(5)

*‘Best’ interaction times*. Examination of Table 6 shows that*I(α)/N*, the amount of information per egg, is markedly lower at 5 sec. than at the other times during which the eggs and spermatozoa were allowed to interact, and is a maximum at 30 sec. The importance of being able to assess the ‘best ‘period for interaction, in reducing the time-consuming nature of these experiments, needs no emphasis.

## SUMMARY

We are greatly indebted to Prof. R. A. Fisher, F.R.S., for patient and unstinting advice.

## REFERENCES

*J. exp. Biol*

*Philot. Tram. A*

*Nature, Land*

*Arbacia punctulata*egg

*Biol. Bull. Woodt Hole*

*Arbacia*

*Proc. Nat. Acad. Sci. Wath*

*Problems of Fertilization*

*Echinut etculentut*spermatozoa

*J. exp. Biol*

*J. exp. Biol*

*Biol. Rev*

*J. exp. Biol*

*J. exp. Biol*

*Exp. Cell. Ret*

*Phytiol. Rev*

*Urechit caupo*

*J. exp. Zool*

*J. exp. Zool*

^{*}

Activation here means the *initiation* of sperm activity, not an increase in activity, in sperm suspensions made motionless by immersion in isotonic NaCl.

^{*}

Except in a few self-evident cases, *p* is an *ettimate* of the actual probability, often written .

^{*}

We are indebted to Mrs M. M. Swann for taking the dark ground sperm track photographs.

^{*}

The argument that the jelly merely increases the effective egg radius, and therefore the bombardment frequency, is open to similar objections.