Principles of General Physiology

This is the law of mass action in its simplest form. The history of this law, which is the foundation both of chemical statics and dynamics, will be referred to below, under the head of equilibrium. If C is the concentration at the time t, then - ' is the velocity of change during so short a time that the rate does not alter, and, according to the law of mass action, where k is a constant, varying with each individual case and known as the velocity constant.

In order to use this equation for any practical purpose, it must, of course, be integrated, so that the change during a measurable time can be investigated. The reader will note that we have another case of the " compound interest law," and the simplest form of the integral is where Cj and C., are the concentrations of the molecule undergoing change, ^ and t.-, are the times after the commencement of the reaction at which the concentrations are found to be Cx and C2. This is known as a unimolecular reaction.

It is obvious that, in practice, any quality of the substance concerned which is a mathematical function of its concentration, such as electrical conductivity or optical rotation, can be used to represent C, provided that it is known what function of the concentration this quality is. Let us take a step further and suppose that the reaction is one in which two molecules combine together to form one or more different ones. This is a bimolecular reaction, since the change of concentration of two molecules must be taken into account. What is the law here ? Take for a moment the kinetic point of view and suppose that we increase the concentration of the one reacting component, A, and leave the other, B, intact ; the rate is clearly proportional to (A), using brackets to express concentration, since the number of times that a molecule of B meets with one of A is proportional to the number of A to be met with in a given space. Suppose that we increase (B), leaving (A) alone, then the number of times collision takes place is proportional to (B) ; therefore, if both are changed, the velocity of the reaction is proportional to the product of the concentrations of the two molecules, or

In practice, most cases of bimolecular reactions can be simplified for integration, since the concentration of A and B can be made to change equally. In the saponification of esters, say ethyl acetate, by sodium hydroxide, the equation is : — where a and b are the initial concentrations of ester and alkali, and x is the amount of sodium acetate produced in the time t. If equivalent quantities of ester and alkali are taken, this equation becomes Hydrolysis of Gaiie-Sugar by Acids. — This process can clearly be expressed as a unimolecular reaction, since it is* completely accounted for, as regards its rate, as the change of concentration of the one kind of molecule, cane-sugar. Applying the formula to it, it is found that the constant k is the same at all stages of the reaction. Indeed, as mentioned before, the determination of this velocity constant has been used as a method of determining the hydrogen ion concentration of various acid solutions.

Application to Enzymes. — Suppose that cane-sugar is hydrolysed by the enzyme invertase, instead of by acid. What sort of values of k do we obtain ? The value of making measurements of this kind in the case of enzymes is that we thereby obtain indications as to what to look for as causes of any divergence found, and also, by the regularity of the time course of the divergence, we are able to estimate the accuracy of the method of experiment adopted.

In all cases where we are investigating the action of an enzyme on a single substrate, as in the majority of hydrolytic reactions, we might expect to find that the unimolecular equation is followed. In point of fact, in the case of invertase, if we calculate the velocity constant by the unimolecular formula, we find that it steadily rises as the reaction proceeds ; in a particular case, from 0 '00058 to 0'00097 (Victor Henri, 1903, p. 55). Taking other enzymic actions, we find, on the contrary, almost invariably, a decrease in the value of k. E. F. Armstrong (1904, 1, p. 506) found it to fall from 0'0640 to 0'0129 in the case of lactase.

We may now proceed to find what possible factors might have the effect of diminishing the rate more than it should be, by mass action, on account of the mere diminution in the number of molecules of the substrate, leaving, for the present, the exceptional case of invertase. There are two things to be kept in mind with regard to this question. The first is that there is every reason to suppose that all the reactions with which we are dealing are reversible, and that there are two opposite reactions proceeding simultaneously, so that the net result observed in any experiment is the difference between the rates of these two reactions. Suppose that our reaction is the

hydrolysis of a substrate into simpler molecules. As the equilibrium position is approached, the opposing synthetic reaction becomes more and more marked by mass action of the increasing products of hydrolysis, and, at the equilibrium position, becomes equal in rate to the hydrolytic one. It is unnecessary to remind the reader that what is spoken of here is the actual rate, that is, the total amount of change in a given time, not the velocity constant, which is, of course, independent of the active masses.

Under the conditions in which most enzyme experiments are made in vitro, the equilibrium position is so near that of complete hydrolysis, owing to the excess of water present, that the synthetic reaction is too small to exercise any very perceptible influence, so that. .other causes for the slowing of hydrolysis must be sought for. When the conditions are such that the synthetic reaction is considerable, as in the cases of lipase and emulsin quoted at the beginning of this chapter, the reversibility of the reaction plays an important part. Fig. 80 (page 300) shows how the hydrolytic reaction in the case of lipase is favoured by the presence of water.

The second thing to be considered is that the rate of a catalysed reaction is dependent on the concentration of the catalyst. If, then, anything happens during the course of the reaction which diminishes the amount of enzyme present, either actually, by destruction, or effectively, by paralysing its activity, the result will be a progressive diminution in the rate of the reaction. We note that this disappearance of enzyme may be irreversible, when there is actual destruction, or reversible, when it is merely removed from the sphere of action, either by temporary paralysis, caused by the products of the reaction, or by adsorption on the surface of some substance present in the system, as is the case with many of the so-called " anti-enzymes."

It is well to mention here that the particular cases which are used in the following pages for illustrative purposes are not to be taken by the reader as the only ones of the kind known. Numerous others will be found in my monograph (1913, 2). We will take first the question of the actual destruction of enzyme. A solution of any enzyme is found to lose its activity if kept. This is, in great part, due to the complex colloidal state of these substances. The rate of this loss of activity varies very much according to the individual case, and is accelerated by rise of temperature. The time course of the process was investigated by Tammann (1892, 2) in the case of emulsin. In the presence of substrate the rate of destruction is much decreased (Bayliss and Starling, 1903; Vernon, 1904), although not entirely prevented. This protective action of substrate has been ascribed to chemical combination with the enzyme. Without denying that this may sometimes occur, I have found that mere adsorption (Bayliss, 1911, 1) by charcoal is protective in the case of trypsin, and there is no proof that this may not be the general explanation.

With very low concentration of enzyme, even in the presence of substrate, activity has been found to disappear before completion of the reaction (Tammann, 1892, 2, and Bayliss, 1913, 1 , p. 248); this fact has led to certain erroneous statements with regard to "false, equilibrium." On the other hand, experiments made for the purpose (Bayliss, 1904) have shown that, in the case of trypsin, there is no detectable loss during a few hours, although the velocity constant has diminished considerably. The spontaneous destruction of enzyme is, therefore, not the sole cause of the decrease in activity.

Reversible Inactivation. — It is found by experiment that the addition of certain substances, amongst which are to be found the products of the reaction itself in a large number of enzymic reactions, has a great effect on the activity of enzymes. This action is brought about in several different ways : — (1) Enzymes are colloids, and therefore liable to aggregation or precipitation by a variety of agents. Emulsin is precipitated by benzaldehyde, so that, even in the small quantities produced in the course of the hydrolysis of amygdalin, it is probable that a certain degree of aggregation and diminution of active surface is produced. We have seen (page 301) that enzymes act by their surfaces, and further evidence will be given presently. (2) Most enzymes are extraordinarily sensitive to changes of hydrogen ion concentration. This is a common property of the colloidal state and points

to the intervention of electrical charge. The slight diminution in hydrogen ion concentration caused by the addition of blood serum is sufficient to bring about great retardation in the action of emulsin (Bayliss, 1912, 2). Trypsiri is inactive except in slightly alkaline reaction, and there is a particular narrow concentration of hydrogen ion in which it, as indeed enzymes in general, are most active. Now it has been shown by Brailsford Robertson and Schmidt (1908-9) that, during the course of a digestion by trypsin, there is a progressive increase of hydrogen ion concentration, or diminution of hydroxyl ion, due to the production of amino-acids, which, combining with the free alkali originally added, diminish the alkalinity, and thus retard the action of the enzyme more and more as the reaction goes on. It is found, indeed, that the addition of amino-acids has a powerful effect in slowing trypsin digestion.

An observation that has probably been made by many workers with trypsin is that, supposing that one starts a digestion of caseinogen with trypsin, adding only the optimal amount of alkali, after a week or two in the incubator one finds that the digest is distinctly acid to litmus, and the rate of action can then be made in increase by addition of more alkali. A reaction, then, can be caused to proceed more rapidly by removal of the products, as was pointed out by Kronecker (1874), and this result is due, not only to reversibility, but also to the fact that the products are more or less toxic to the enzyme itself.

As would be expected, the sensibility of enzymes to various substances is much great IT than that met with in ordinary chemical reactions. It is held by some investigators that there is a special "affinity" of each enzyme for certain products of its action, in that the rate of change is affected more by these than by other related substances. Thus, E. F. Armstrong (1904, 2) found that fructose retards the action of invertase more than the corresponding concentration of glucose does, and the statement is made that invertase is "controlled" by fructose. The fact, however, that fructose also " controls " the action of maltase more than glucose does (Philoche, 1908, p. 243), although it is not a constituent of the system concerned, suggests that the relationship is not one of a chemical nature between the constitution of the enzyme and of its substrate. The excessive sensibility of enzymes to acidity and to some inorganic salts warns us that great caution must be exercised in the interpretation of results of this kind. Bourquelot (1913, 2, p. 3) finds that the hydrolysis of arbutin is retarded by hydroquinone, but not that of salicin. Hence the action is not on the enzyme itself, and increase of the reverse reaction is suggested.

The acceleration of rate, in the course of the action of invertase, is probably due to the production of some substance in small amount which increases the activity of the enzyme. Acid does this, and it has been stated that an acid is produced in small quantities during the action of inveitase, and it may perhaps be Isevulinic acid. So far as I am aware, no measurements of the hydrogen ion concentration during the reaction have been made. There is a certain similarity between this production by an enzyme of substances which affect its own activity and the process called by Ostwald " Autocatalysis " (1902, II. (2), pp. 263-266). In the first illustration at the beginning of the present chapter we saw that the spontaneous hydrolysis of methyl acetate in water is greatly accelerated by the addition of acid. Now in the hydrolysis itself, free acetic acid is formed, which must act as a catalyst, although not a powerful one. Moreover, it increases in amount by its own activity, so that, if we determine the velocity constant at different times, we find that it increases at a greater and greater rate.

In an experiment of this kind which I performed, at the beginning of the reaction the velocity constant was 49 x 10~7, in nineteen days it had risen to 593 x 10~7, and in forty-two days to l,498xlO~7. In half-normal hydrochloric acid, it was initially 1,600 tinii's that in water and equilibrium was reached in about six hours, so that, if there were no autocatalysis in water the attainment of equilibrium would have taken 6x 1,600 hours, or 400 days. This fact may assist the reader to realise how slow the spontaneous reaction is, and how ini|>ossililr it is for the equilibrium position to be unaltered by acid unless the catalyst accelerated both the hydrolytic and synthetic reactions.

Perhaps a few more details will be useful with regard to the phenomenon of autocatalysis. When the curve of a reaction of this kind is plotted with time as abscissae and actual rate of change as ordinates, it is found to have an S shape. The rate is slow at first, becomes quicker and then slows again. This course is typical of autocatalysis and is, obviously, due to the deficiency of catalyst at the beginning and deficiency of substrate at the end ; the latter fact causes diminution in rate by mass action. The rate of the reaction, as measured by the amount of ester hydrolysed in unit time, must not be confused with the velocity constant, which increases steadily throughout.

The difference between true autocatalysis and the effect on enzymes described above is that, in the former process, the actual quantity of the catalyst is altered, positively or negatively, whereas in the latter the enzyme causes the production of substances which act upon itself in a similar positive or negative way, the effect increasing more and more as the reaction progresses, so that the change of concentration of the catalyst is not actual but only effective.

We may now consider the effect of different concentrations of enzyme as added intentionally at the beginning of the reaction. A practical point of some importance may appropriately be mentioned here. As Bredig points out (1902, p. 187), in comparing the results of the action of enzymes under different conditions or concentrations, we ought to compare the reactions at the same stage, since, in this way only, can we be certain of having the same proportion of substrate and products, and, moreover, if the reaction goes in stages, we should otherwise obtain very false information. What we must compare, then, are the times taken to effect equal changes, not the changes produced in the same time.

In practice this is most conveniently done by taking series of measurements and plotting them as curves. If amounts of change are made ordinates and time abscissae, a horizontal line drawn to cut all the curves at the stage desired will give the time values required. What is always found, except when there is very little enzyme in proportion to the substrate, or vice versa, is an obvious disproportion between the amount of enzyme and its effect. This may be seen in the following table taken from an experiment of my own with trypsin and caseinogen (1904). The first column gives the relative amounts of the enzyme added to the same volume of substrate. The second column gives the times taken by each to produce the same amount of change, measured by the electrical conductivity, that is, the increase of the concentration of carboxyl groups (see page 219 above). The third column gives the mean rate in each case, namely, the reciprocal of the time taken (multiplied by 1,000 to avoid long fractions). The fourth column gives a measure of the activity of the enzyme as obtained by division of the actual rate by the amount of enzyme present, or, in other words, it represents the activity of equal amounts of enzyme when present in different concentrations, and may be called " specific activity."

It is obvious that the smaller quantities of enzyme are considerably more effective in proportion to their concentrations than the larger ones are. Taking the numerical values of enzyme concentrations 2 and 4, for example, we find that the value of 4, instead of being double that of 2, is less than this. Let us suppose that, instead of being multiplied by 2, it is multiplied by some root of 2, 2"*, and let us see what values are to be given to x to satisfy the various data. As a first approximation, try x = 2, then the value of enzyme concentration, 4, should be that of concentration, 2, multiplied by 2~2, that is, 12-4 x 1'4 = 17'4, instead of the experimental value of 18 '2, a fairly satisfactory agreement. This is in fact the rule known as the square root law of Schiitz and Borissov, but we see that it is mereiy an approximation. If we take different stages of the same experiment, we find, in fact, that the value of the exponent increases nearly to 2 towards the middle of the reaction, but may be very nearly unity at the beginning. In this latter case

there is linear proportionality. It is also different where the enzyme concentrations are very far removed from one another, or of high values, being smaller if we compare concentrations of 64, 128, and 256, in arbitrary units, with those of 1, 2, and 4 in the same units. Details of these experiments will be found in a paper by myself (1911, 1, pp. 9094). The reader will probably notice at once that these results are precisely what we should expect if the velocity were controlled by adsorption of substrate by enzyme ; it was, in fact, this relationship which first led me to suggest the hypothesis of adsorption as applying to the case (1906, p. 224).

It is then impossible to formulate a general law, correlating the concentration of enzyme with its activity, and capable of giving numerical results, except one of considerable complexity. Each case must be investigated for itself until we know more of the changes taking place in the colloidal state of the enzyme during the course of the reaction. This state is, no doubt, the cause of the variations of the adsorption exponent which we have met with ; it may have any value between one and two, values above two are rare.

There are three classes of substances which may next be considered, since they affect the rate at which the enzyme acts. 1. Electrolytes. — Pepsin and trypsin are inactive except in acid or alkaline solutions respectively. Amylase requires neutral salts, which have also a beneficial effect, as a rule, on enzymes generally. The question will be referred to again later. 2. Co-enzymes. — Bertrand (1897) found that the oxidase of Japanese lacquer is ineffective without the presence of manganese and called this substancg the " co-enzyme " of laccase. These oxidation systems will be considered in Chapter XX.

Magnus (1904) discovered that the lipase of the liver loses its activity when dialysed, but recovers it when bile salts are added. It seems probable that this action is exerted on the enzyme itself, since it applies to the action on soluble esters as well as to that on fats, which might be supposed to be better emulsified by bile salts. These latter, having so great a power of lowering surface tension, are no doubt able to bring about a greater colloidal dispersion of the enzyme, thus increasing its active surface.

Another interesting case of a co-enzyme is that of alcoholic fermentation. Yeast juice contains an enzyme, or rather enzyme-system, "zymase," which brings about the formation of alcohol and carbon dioxide from sugar. Harden and Young (1906) showed that such juice, filtered through Martin's gelatine filter, which keeps back the colloids, was separated into two constituents, neither of which was active by itself, but became so on mixing again. Since inorganic phosphates increase the activity of yeast juice, it was thought that they might be the co-enzyme, but experiments showed that these alone were incapable of restoring activity to thecolloidal matter left on the filter, and that it was necessary to add boiled yeast juice in addition. Both substances are indeed required.

3. Anti-enzymes. — When a foreign protein is injected subcutaneously into an animal, some kind of a neutralising substance is produced. This is known as the " anti-body," while the injected substance producing it is the " anti-gen." There is as yet no satisfactory proof that any substance other than a protein can act as an antigen. The antibodies are of various kinds, sometimes they precipitate the antigen and are known as precipitins, sometimes they act in neutralising its toxic properties in some other way and are called " anti-toxins," sometimes they cause agglutination of bacteria, " agglutinins," and so on.

Now statements have been made that when enzymes are used as antigens, anti-enzymes, true antibodies in the above sense, that of Ehrlich, are formed. It is to be noted that a true anti-enzyme must be specific and act only on the particular enzyme which caused its production, its antigen. It is therefore incorrect to describe any substance which retards the action of an enzyme as an anti-enzyme ; otherwise, alkali would have the right to be called " antipepsin."