Nov. 20, l!X5,i
NOTES
A Note on the Differential Equation of Simple Enzyme Kinetics BY MANUEL F. MORALES~ AND DAVID E.GOLDMAXI
Experimental measurement of these two decay constants provides a basis for estimating individual velocity constants. For example, the sum of ml and m2is - k1 multiplied by a readily measured facEo So. I n the spgcial case that tor, viz., l/K Eo is negligible relative to either 1/K or So in exPression 7, then one m goes t o zero (exponential term degenerates to a constant), and the other m approaches -kiSopL/p6. This latter case has already been treated by GUtfreund and rough ton^^ although their rationale has not yet been published. The logical consistency of this or any other approximation (see below) can be checked by ascertaining that throughout the relevant time interval, the neglected terms, as estimated from the approximate expression, do indeed remain negligibly small. As we have noted already, there is only one maximum in p‘, ;.e., only one instant a t which 9‘’ = 0; however, under many conditions p“ may remain small (and (p”/pL)Eo negligible relative to s‘ or p‘ in the differential of eq. 3) for a considerable time before and after the point of inflection. The “steady state approximation” consists in assuming that for the range in question 9‘‘ is small enough so that i t may be neglected in eq. 2. The resulting equation is awkwardly integrable to givep(t), or easilyintegrable if one transforms to the variable, s ( l / R )In s s = -p&t Cl (8) where c1is a constant of integration which must be evaluated for the range of approximation. Equation 8 is directly usable if s is measured. If p is measured one must further assume that (P’/Pk)Eo is negligible relative to either S O r p (in equation 3) in Order to Obtain a usable equation
RECEIVED JUNE22, 1955 The classical model of enzyme action2 assumes that enzyme, E (in total concentration, Eo), reacts with substrate, S (in total concentration, SO),to form “complex,” ES, which then breaks down irreversibly to regenerate E and produce products, P, ;.e. k1
E + S ~ E S k- 1 kz ES-E+P ( 11 and = ~ ( t )we , may write the
Letting [p]= p ( t ) [s] exact differential equation for p in system (1) as P“ p’ Eo
m,
= (1
- $) - (1 -
2) ($ +
s)
(2)
whereP6 = k2EO is the maximum possible ration:) rate Of PrEduction Of Pi’ (KSoI (1 K S o )1 pL, and K k1/(k-1 k2). Material balance requires that
+
+
p j
+ +
s P (P’/P&)Eo = So (3) where i t will be recognized that [ES] = (p’/pk)Eo. I t is evident that both p 6 and p6 are both “natural” constants of-the system, and i t will also be observed that K and pL (hence pi’) are usually considered to be obtainable from conventional enzyme kinetic experiments. The object of this note is to sketch approximately the theoretical time course of p and p‘, and to examine the conditions under which these approximations can be fitted to data for the purpose of inferring kinetic constants, e.g., k1, k-1, etc. If a t t = 0, p = 0 and s = SO, i t is evident from physical considerations that P’(0) = and that, P , 0 ’ O, and P’ is again P’ must therefore have attained a maximum P:, at Some Of t. At this P” = 0, so from eq. 2 we have
-
2)(k + 2 .-) The r.h.s. of equation 4 is positive, i t follows that (1
- :’)
=
(4)
(1 -
P‘ Eo
-
SO
6069
+ +
+
+
( I / R )In
(So
-p) -p
=
-pit
+ Cz
(9)
where Cz is also a constant of integration to be evaluated for the range of approximation. Equations 8 and 9 can be compared with experimental data to check the validity of the assumptions which they contain, even though such comparisons do not yield information other than that presumed to be obtainable in conventional enzyme kinetics. Indeed, if in equation 9 one assumes p/So small relative to unity, and therefore retains only the linear term in expanding the logarithm, the result can be put in the form
p’ may equal, but may never exceed, Ps‘. The only case in which pi pi’ is when p,’ p k , ;.e., KSo 03 ; in all other cases p ; falls short of p:. p = p:t + Constant (10) During the Phase” Of the reaction P and which is the traditional equation used in enzyme p’ are both small, and i t is therefore permissible to kinetic studies, wherein the measured rate (sometheir products and squares in eq. 2 , which -.).
-.).
what loosely called the ‘‘initialrate”) is supposed to be pi. P During the terminal phase of reaction one may kiSopA again linearize equation 2 by appealing to the fact The solution of eq. 5 is that both p’ -t 0 and (So - p) 0. Straightaway m2emll - mlem2t this permits one to drop the term containing p f 2as p/so - 1 = (6) a factor, and obtain ml - m? where ml and m2are given by P” = kipL(So - $1 - k l { l / R Eo (So - p ) ) P ’ (11) -(1/R EO + SO) {(1/x Ec + So)* - ~ P & / ~ I I ’ / ~Equation 11 is still not integrable unless or until i t 2/ki can also be assumed that (SO -- p) has become neg(7) ligible relative to either 1/K or Eo. From this ( 1 ) In accordance with Navy regulations, it is noted that the opinions point on the equation is a second-order equation expressed in this article are those of the authors and do not necessarrly with constant coefficients. Its solution contains reflect the view3 of the N a v y Department or the Naval Service at then becomes - =P “I -
+
-
*
+ +
+
large ( 2 ) .I
Michaelis and M. L. Menten, Biochem. Z
, 49,333 (1913).
(3) Cited by F. J. W. Roughton in (1954).
D m . Faraday
Soc.. 17, 116
6070
KOTES
Vol. 77
two exponential terms whose decay constants are given by
with a-dextrin. Long chain fatty acids, alcohols and esters fall into this category. Their chemical characteristics are not pronounced so that it is difficult to find indicators of a sensitivity adequate for use in paper chromatograms. Although various sugIn certain favorable cases these transients could gestions have been made to solve these diffic~lties,~ also be employed in deducing velocity constants. paper chromatography does not hold its due place For example, if k L 1 were very large compared to among analytical tools in the lipid field. (kz klEo), one decay constant would tend to a-Dextrin can be used successfully on chromatozero, and the other to - k - l , while if k2 were very grams of fatty alcohols, acids, methyl or ethyl eslarge compared to ( k - l klEo), one decay con- ters and monoglycerides. Diglycerides and triglycstant would tend to - klEo, and the other to - kz. erides do not respond to this indicator method, lx-ith modern, high speed computing devices i t is since they do not react readily with a-dextrin. quite feasible to find by trial the unique set of val- However, they can be detected by splitting them ues for the parameters needed to reproduce a given into their components after chromatographic sepexperimental concentration time curve for any com- aration. The necessary hydrolysis is carried out ponent of the reaction system. This is easily done in situ enzymatically by means of commercial panwith an analog computer where the programming creatin. I n this way it is possible to make visible is direct and simple. However, the usefulness of the separated di- and triglycerides on the paper. the procedure may be limited in situations where Two other indicators were found to be applicable there are wide differences in order of magnitude be- to certain types of lipids. Iodine vapors have been tween the various parameters, e.g., Eo and So. used for detecting a variety of substances4 but not U. S. NAVAL MEDICAL RESEARCH INSTITUTE for higher fatty acids. I n the lipid series we found BETHESDA,MARYLAND i t specific for spotting unsaturated components. For instance, oleic acid and its derivatives appear as brown spots on chromatograms when exposed to Indicators for the Paper Chromatography of Lipids‘ iodine vapors. The limitation of the method is BY HELMUT K. MANGOLD, BEVERLY G. LAMPAND HERMANNgiven by the fact that symmetrical oleodipahitin SCHLENK cannot be located unambiguously in actual chromatograms. A more specificreagent is lead tetraaceRECEIVED JUNE 24, 1955 tate which is applicable to the chromatography of The inclusion reaction has received considerable monoglycerides. The reagent hydrolyzes in air to attention in recent years. One of the basic princi- form brown lead dioxide ; with a-monoglycerides it ples resulting from research in this field is that undergoes the normal glycol splitting reaction and shape rather than chemical character of a molecule yields colorless lead compounds.j Consequently, determines whether or not i t can be included by a monoglycerides appear as white spots on a brown certain host. The complexing reaction can be background in chromatograms sprayed with lead carried out in minute amounts such as are used in tetraacetate solution. For the detection of di- and practical paper chromatography. This is easily triglycerides i t can be used after splitting them by demonstrated by spraying a filter paper bearing a the above-mentioned enzymatic hydrolysis. spot of octadecane with a-cyclodextrin and subseComplexing is the most universal method and is quently exposing i t t o iodine vapors. The area certainly applicable to many other compounds becovered with hydrocarbon remains white, while the sides the lipids. remainder of the paper turns bluish purple. This Table I shows R f values obtained by chromatooccurs because of different reactivities of the free graphing model mixtures and individual lipids for and of the complexed a-dextrin. Free a-dextrin identification, Different solvent systems were reacts with iodine to form a deeply colored inclusion used in ascending technique with silicone impregcomplex very similar to that resulting from starch nated paper, i.e., a reversed phase chromatography. with iodine. I n fact, the two reactions are closely Whatman No. I paper was modified according to related2 and the same phenomenon can be observed procedures reported in the literature. We found with starch on a spotted paper. Comparative examination proved a-dextrin to be the preferable impregnation with silicone6 to be the most versatile indicator in actual Chromatograms. When the a- preparation for our purpose. Since silicone is not dextrin is already occupied by a guest molecule, in washed out by the developing solvents as hydrocarthe above case the octadecane, it is no longer avail- bons are from impregnated papers, the same paper able for the iodine reaction which produces color. can be used for reversed phase chromatography in Alccordingly,the spot of hydrocarbon stays white. the second dimension. Uniform coating is easily This holds for other compounds having low chemi( 3 ) J. Asselineau, Bull. SOL. chim. France, 884 (1952) (review); cal reactivity besides hydrocarbons ; for example, H. P. Kaufmann and W. H. Kitsch, Fette und Seifen, 56, 154 (1954); esters, ethers or other types of molecules that react H. P. Kaufmann and H. G. Kohlmeyer, ibid.. 67, 231 (1955); y .
+
+
(1) Supported by research grants from t h e U. S. Atomic Energy Commission, t h e Sational Institutes of Health (P.H.S. 4226, of t h e Public Health Service). and by the Hormel Foundation. Presented a t t h e 127th meeting of the American Chemical Society, Cincinnati, Ohio, April, 1955. Hormel Institute Publication S o . 133. (2) F. Cramer, “Einschlussverbindungen,” Springer, Berlin, 1954, p. 71 ; K . E. Rundle, J. F. Forster and R. R. Baldwin, THISJ O U R N A L , 66, 2116 (1044); R. E. Rundle, ibid., 89, 1769 (l0?7).
Inouye, M . Noda and 0. Hirayama, J . A m . Oil Chemists’ Soc., 32, 132 (1955); B. D. Ashley and U. Westphal, Arch. Biochcm. Biophrs., 66, 1 (1955); J. W. Dieckert and R. Reiser, Science, 120, 678 (1954). (4) G. B. Marini-Bettolo-Marconi and S . Guarino, Expeuienlia, 6, 309 (1950); G. B. Marini-Bettolo, Estraffod a i Rendiconti dell’ I S f i f Z l o Superiove d i S a n i f a , XVII, 471 (1954). (5) J. G. Buchanan, C. A. Dekker and A. G. Long, J . C h m . SOC., 3162 (1950). (6) T. H . Kritchevsky and A. Tiselius, Science, 114, 299 (1921).
