J. theor. Biol. (1975) SO, 45-58
Allosteric and Related Phenomena: An Analysis of Sigmoid and Non-hyperbolic Functions ROBERT E. CHILDS AND WILLIAM G. BARDSLBY
Department of Obstetrics and Gynaecology, University of Manchester, St Mary’s Hospital, Manchester Ml3 OJH, England (Received 25 March 1974) The graphs of rational polynomial functions are of considerable importance in enzyme kinetics but a full analysis of the turning points and inflexions is only possible for 2 : 2 functions due to the complexity of the first and second derivatives for 3 : 3 and higher degree functions. This paper describes a simple method which can be applied to rational functions of
any degree in order to discover the relationstip
between the coefficients
necessary for the curve to have an initial sigmoid inkxion and a Gnal inflexion (which implies a maximum in the graph). This technique is then
applied to the current allosteric models. In addition, a full analysis of the 2 : 2 function with a maximum is given together with a method of separa-
ting this type of hehaviour (partial substrate inhibition) function (dead-end substrate inhibition).
from the 1 : 2
1. htroauction
Steady state processes in biological systems invariably consist of steps which are individually zero, first or second order with respect to one variable when all others are held constant. For instance, transport processes involve complexing between membrane components and diffusing entities, responses of pharmacological preparations to agonists or blocking agents are mediated via interactions between receptors and drugs, ligand binding to proteins, elimination of drugs from the body and binding of substrates, products, inhibitors and other effecters to enzymes all follow this type of bebaviour. Since combinations of polynomials and rational polynomials can always be written as rational polynomials, it follows that these systems can all be represented by functions of the type a,+alx+a2x2+. . .a,~” y = j30+/9~x+~2x2+. . .&pm i.e. by pure algebraic expressions with integer exponents. In all the cases 4s
46
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referred to, x would be the concentration of substance under consideration and y would be the appropriate steady state or equilibrium response measured experimentally. The purpose of this paper is to explore the types of polynomials encountered and to investigate analytically the graphical methods that can give information concerning the numerator and denominator coefficients and degree. In all cases, the physical situation dictates that the function cannot be negative and the denominator can have no real roots for positive values of x. Also, with the exception of enzyme inhibition, the degree of the numerator cannot exceed that of the denominator. 2. Transport Processes It is well known that the steady state rate of membrane transport can often be approximated by a 1 : 1 function when all experimental conditions are held constant and only the concentration of the material being transported is varied (Stein, 1967; Neal, 1972). True first order transport implies unlimited membrane capacity and is therefore not found in this or any other biological system except at very low concentrations when rational functions with LY,,= 0 and /?a finite become approximately first order. 3. Drug-Receptor
Interactions
Responses of pharmacological preparations are usually described by 1 : 1 functions (Ariens, 1964; Paton, 1970). Empirical equations such as y = Yc”(c”+K”)-’ have also been employed (Parker & Wand, 1971), but these are merely approximate curve fitting attempts that have no theoretical justification, since they often lead experimentally to non-integer powers of n. If we consider the response (R) of a typical preparation (e.g. contraction frequency, tension developed, pressure etc.), there will usually be a finite value of the parameter being measured (say R,) even when the drug concentration is zero, i.e. with P,(x) having no constant term. It seems to the present authors better to analyse the response of pharmacological preparations using y = (R-R,) as a dependent variable according to methods used in enzyme kinetic studies rather than by the log plots and empirical curve fitting procedures currently used which prevent maximum utilization of experimental data.
ALLOSTERIC
AND
RELATED
PHENOMENA
47
4. Ligand Binding to Proteins Where only one ligand binding site is present, the saturation function (J’) will be 1 : 1 with a0 = 0 but when several binding sites are present we have P = PnWlQmW where n = m and P has no constant term, and n is the number of binding sites. If these sites are independent and equivalent, then common factors will cancel and where allosteric effects are found (Monod, Wyman BEChangeux, 1965; Koshland, Nemethy BEFilmer, 1966), the functions will be at least 2 : 2 and the saturation function may be sigmoid (Botts, 1958; Ferdinand, 1966). 5. Pkarmacokinetica It has now been recognized for some time that the rate of elimination of drugs and metabolites is described more accurately by hyperbolic functions than first order processes (Lundquist & Wolthers, 1958; Wagner, 1973). Since the disappearance will be effected byseveral competing pathways (e.g. uptake by cells or plasma proteins, metabolism, conjugation, excretion, etc.), it is likely that the 1 : 1 function is a poor approximation and more thorough experimental work would indicate higher degree polynomials (with u. = 0). 6. EnzymeSubstrate
Interactions
The steady state velocity (u) of an enzyme catalysed reaction is a function of the concentration of substrates (A, B, C etc.) and products (P, Q, R etc.) but when all substrates and products are held constant and only A is altered we have where P has no constant term and where the coefficients are necessarily positive functions of the other substrate and product concentrations. The degree of the numerator will be determined by the number of times substrate adds in the catalytic sequence and also by the nature of that sequence, but the degree of the denominator will be greater than that of the numerator if the substrate also forms dead-end complexes with the enzyme (dead-end substrate inhibition). Rate equations will be increased in degree if isoenzymes are present or if the enzyme has several active sites provided that the kinetic constants for these sites differ; thus two 1 : 1 functions generate a 2 : 2 function etc. There is, in principle, no difficulty in calculating rate equations for any mechanism (King BE Altman, 1956; Volkenstein & Goldstein, 1966; Cha,
48
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1968; Seshagri, 1972), but often it is helpful to find the degree of the functions independently as an aid to mechanistic interpretation. Polynomial functions were first introduced for this purpose by Cleland (1963), but there still remains some confusion about the type of information made available to the kineticist by graphical methods. This is particularly the case with allosteric effects where sigmoid and other complex types of v/A curves and non-linear double reciprocal plots are often taken to indicate homotropic effects, although several mechanisms can give this type of behaviour (Rabin, 1967; Atkinson, 1966; Sweeny & Fisher, 1968; Harper, 1971, 1973; Ainsworth, 1968; Ainsle, Shill & Neet, 1972). Double reciprocal plots will be dominated by the lowest degree term in the numerator, i.e., if ~1~= 0, parabola if a, = a, = 0, cubic and so on, but, if a, is finite, then no matter what the degree of the numerator and denominator, the double reciprocal plot will asymptotically approach a straight line as A-’ + co. The slopes of the asymptotic lines and the actual value of the function when A- ’ = 0 will be poly nomial functions in the other variables but the equations will now be inverted and undefined at the origin, i.e. they will be functions of the form: slope or intercept = f(B) =
po+81B+~2P+.
. .B,Brn
a B +cr B2 + 1
2
cI B” -*-
.
n
7. Enzyme Inhibitor Interactions
The effects of inhibitors on slopes and intercepts of double reciprocal plots have been fully discussed (Cleland, 1963), and it is now a routine exercise to replot slopes and intercepts as functions of inhibitor concentration Z according to WI = f’@)[l + PnU)IQ,(OI where P has no constant term. 8. Some General Properties of Rational Polynomials
We consider rational polynomials as given by equation (1) where n < m and ai; pi > 0 all i (i.e. positive coefficients only). Then the following properties hold: (1) If a0 = 0 the curve passes through the origin (otherwise y = ao/po when x = 0). (2) If n < m then lim y = 0. x+m (3) If IZ = m then lim y = cl,/&,. x-a3
(4) The graph can have no vertical asymptotes for x 2 0.
ALLOSTBRIC
AND RELATED
PHENOMENA
49
(9 A graph of Cv- aoMo) ( i.e. no constant in the numerator)
passes through the origin and so all rational polynomials can be reduced to one of this type and most of the later discussion concerns this type. This can result in the numerator having zero or negative coefficients and the function becoming negative. (6) The graph of y(x) may have, under some circumstances, an inflexion of positive slope giving an initially sigmoid shape to the curve (Fig. 1). The conditions for such an inflexion are found as follows from the derivative: Y’W -Y’(O) = 280 %z s; -alBoB1-aeBo82+a0B:)~ for x -4 1. In all cases we are to consider, we are interested only in those where the gradient is positive (or zero) at x = 0. All future discussion will make this assumption.
Y
X
Fro. 1. Conditions under which we obtain sigmoid inilcxions and/or maxima in equation (1) (a0 =O). (a) a~8~-a~80h-aogoB+ao~ c 0. @I e&-add%-e&h + a& > 0. (4 w%-~ - a,~ A 2 0. (d) an&l - ~-18. c 0. W always.
From the above expression then, we see that near the origin the gradient is increasing with increasing x if azbf-alAbA -aJdb+adC > 0. (2) Knowing that the curve must eventually “turn over” to approach the asymptote a./?;’ (if n = m) or the x-axis (if n < m), we conclude that the gradient must at some point start decreasing. Thus we get a point of infiexion to an initially sigmoid curve (Fig. 1) if equation (2) holds. (7) The graph of y(x) may have several maxima and a useful result in the later analysis is the slope of y(x) as x --, co for the case n = ni. 4 T.B.
50
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We tind :+t 2 = a,B~1x-2(Bn-1/Bn-tln-11a.>. Thus we get a decrease in y to a finite value as x + co, if and no decrease in y as x + co, if a&,- 1 2 j?, a,- i (see Fig. 1). The double reciprocuZ form of equation (1) is as follows: (y-‘)
= /%+Bm-lwl)+~ a,(x-l)m-n-t..
* * +BoW’)” . +a,(x-‘)m
’
From (i) (ii) (iii)
this we may note the following properties. y-l --) co as x -’ --t 0 for m > n. y-’ -+ /?./a, as x-l + 0 for m = n. If a0 = 0 then y-’ approaches a straight line asymptotically as x-l -+ co (PO # 0). If a,, = a, = 0 then y-l approaches a parabola asymptoticahy as x-i -+ co and so on. (iv) The same inequalities as in section (7) for y versus x arise in the slope of the double reciprocal plot at x-l = 0 for n = m, i.e.
W- ‘1 =
B,a,‘(B,-l/B,-a~-l/a~) at x-l = 0. d(x- ‘) Thus a negative initial slope in the double reciprocal plot is associated with decreasing y as x -P co (for the case n = m).
9. Special Cases of 2 : 2 Functions Since for a 2 : 2 function the first derivative is a quadratic and the second a cubic, there are only four possible shapes for this function, i.e. smooth curve with no maximum but with or without a sigmoid inflexion and a curve with a maximum but with or without a sigmoid inflexion. Botts (1958) has discussed the conditions on the coefficients for these cases and suggested that more complex functions such as stair-step curves with horizontal sections may be possible with 3 : 3 functions where the first derivative is a quartic and second derivative a sixth degree polynomial. Teipel & Koshland (1969) have shown, by numerical examples, that complex curves can arise with 4 : 4 functions. We now present a more detailed study of the 2 : 2 function and the special cases encountered experimentally, paying particular attention to the case of dead-end and.pa.rtial substrate inhibition.
ALLOSTBRIC
AND
RELATED
51
PHENOMENA
CASE 1: a, = 0 AND a&,
> /%/I%
Case I is for partial substrate inhibition. (Without take /?,, = 1.) y = (a~X+a~X2)/(1+~~X+~~X2>.
loss of generality
we (3)
We find turning points (if any) where
x=xo=/9z(ala;‘-j?ij?;‘)
where
(4)
G = B2(al a; ’ -0.5B1 /?; 1)2 + (1 - 0.258: /3; ‘). Consideration
of the inequality
0 < G < 1 leads to the inequality alaZ’
< j&B;’
and hence no positive turning points; whereas G > 1 implies a1 a;’ and leads to just one positive turning point. Let al a2- ’ = /I1 fl; ‘8,
8 > 1 for maximum
turning point
> j?J?;’
(5)
and B2 = j?: 4, Then substitution
4 > 0 for initially
sigmoid curves.
(6)
of equation (4) into (3) gives (for 0 > 1):
1
(e-1)2
= p.a2.j3;’
(say), p > 1.
Y(X = xo) = 2 See Fig. 2(a). Rearrangement
’ + (2++e-i)-2p(++e2-e)+
1
(7)
gives the following relation between 0,+ and p : 4 _ (e-p)2 4P(P-1)
(8)
and so ~de--P) xo = 2/?&-l)
p2 = -A--
&p - 1)’
Thus from an experimental plot of y versus X, we can measure x0, p and a2/jJ2 which enables us to calculate /I2 and hence a2 and to find 4 as a function of 8. From a double reciprocal plot we can measure the slope (So) of the
52
R.
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BARDSLEY
straight line asymptote as X-I + co [Fig. 2(b)] so giving PO
8=
/.&
=
B:xxP-l)
x0 - 2P~ow32);
MO
@a2
and then a1 = Bla2(e/S2). Zn$exions
Points of inflexion occur where /MA a2-alB2)x3+382a2xz+382al x+0& a1 --a21 = 0 which on substitution for 0 and #Jdefined above may be written y3-3ay-a-
By means of the trigonometric
Bl l+- 24 = 0 (9) p2 ( e-1 ) method we can solve equation (9) to give:
x = ~ A4 1+ 2(e2+4-e)‘.cos0 n ~2w) { J& nI where
8, = *cos--~
e+2+1
I 2n7-c.
2&(e2++-e)+
>
3 ’
n = 1,2,3.
Note that we require e+2+i 2J&e2
< 1
+ 4 - e)*
for all three roots to be real, i.e. 4, > 4. By this method al, a2, PI, /I2 can be determined and points of inflexion verified. CASE II:
a0 = 0; a1 = 0
In an effort to explain sigmoid v/A curves in enzyme kinetics, rate equations without linear terms have been derived (Sweeny & Fisher, 1968). Enzyme mechanisms where a substrate adds twice in a sequence uninterrupted by product release give rate equations of this type. This case gives a double reciprocal plot which is parabolic. The curve y(x) is aZway.ssigmoid and has no turning point in the first quadrant. This case is easily distinguished from that of a sigmoid curve where al is finite since in this latter case, the double reciprocal plot will reach a linear asymptote which actually has a negative y-l intercept. In fact, this seems easier to spot experimentally than a sigmoid inflexion.
ALLOSTBRIC
AND
RELATED
53
PHENOMENA
CAsEm:a~=0;a,=O Here we are dealing with a case of “dead-end substrate inhibition” and find y(x) can neuer be sigmoid but that there is always a turning point. Experimentally, it is often difficult to distinguish between dead-end substrate inhibition where y + 0 as x + cc and the case of a full 2 : 2 function (or 3 : 3 etc.) where we get partial substrate inhibition, particularly if the horizontal asymptote for large x is very small. A useful discriminatory plot for distinguishing between these two cases is that of y -’ versus x; since if CI~ = 0 [Fig. 2(c)], then there can be no intlexion in the positive quadrant for this plot whereas for the partial substrate
---_
(12, P;’
X
(b)
(cl I:2 / --
Y-'
y
/I 2:2 -A
/
X
ho. 2. (a), (b) 2 : 2 case showing experimentally obtuinable parameters of text. (c) Plot of y-l/x for 1 : 2 and 2 : 2 functions showing dit%rence x+03.
used in analysis in behaviour as
54
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inhibition case (Fig. 2), we find that there must be exactly one inflexion as shown by the following analysis : a2
a2
#
0,
PI a2
= 2{af+3a,
a2x+3a3x2+a2(/4 (a,x+a2x2)3
For this second case (partial substrate inhibition) of the second derivative can be written: x3--x2-bx-c
= 0;
a2-al
/12)x3)
.
the cubic in the numerator
a, b, c > 0,
and since there must be at least one inflexion in the positive quadrant, then it factorizes as follows : (x-rl)(x2+px+c/r,)
rl > 0.
= 0;
The quadratic factor here can be shown to have no positive root. CASE
IV:
I&,
=
0;
p2
=
0
This situation occurs in mixed dead-end and partial inhibition now inhibitor concentration. Consideration of the function
where x is
a,x+a2x2
= ~(x+ala;‘-l);‘-(ala;‘-8;‘)/(l+B1x)) Y = 1+&x 1 and its derivatives shows that there can be no sigmoid inflexion. CASE
v:
a0 = 0; PI = 0
This gives a sigmoid inflexion always together with a maximum. CASE
VI:
a,, = aI = PI = 0
This gives a sigmoid inflexion but no maximum. 10. Some Allosteric Models In all of the models discussed in the folIowing section, the coefficients are such that there are no turning points but in most cases a necessary constraint is found before sigmoid curves are possible. (A)
HILL
EQUATION
Kx” Y =1+Kx”
ALLOSTBRIC
AND
RELATED
PHENOMENA
55
This has an inflexion where x = “J(n- 1)/K@+ 1) and has no turning point but steadily approaches y = 1. The experimental value of n is thought to bc a measure of the allosteric linkage (Wyman, 1%7). For the case n = 2, for example, the Hill equation is really an approximation to equation (3) in the case where there is no turning point (i.e. ai #?z < cQi) and in this case a2 = B2 as it is modelling a saturation function. A similar analysis to the following will apply in the case where there is a V,,, instead. Experimental workers plot lo&/l -JJ) versus log x expecting to obtain a straight line of slope n if the data actually obeys the Hill equation. If they are actually plotting values which satisfy qx+a2x2
Y = 1+pix+o12x2
where u2 = B2
i.e. a1+cr2x log(-J!:hX+hl 1-Y > (1 +a -4x
>
then it is reasonable to suppose that the “straight” line chosen is the tangent to the curve at the point of inllexion and thus n is the gradient of this tangent. The inflexion to equation (11) occurs at x = [ai/a2& -cl,)]* and the slope of the tangent there is: -44 -al> a2+a1 (Bi-d+2(~1 a2>+(Bi -I)*’ We see that if a1 = & = 0 then the slope is 2 and the function is that of the Hill equation. Similar considerations could be applied for cases where n > 2. a2
l+
(B)
ADAIR
MODBL
FOR
HABMOGLOBIN
The saturation curve given by Adair [1925, equation (4)] is: Y=
0*25KI x +0’5K2 X2+0*75K3 l+KIx+K2x2+K3x3+K4x4
X3
+ K,
X4
from which we see by equation (2) that a sigmoid inflexion occurs if 2K2 > Kf. (C)
PAULING
MODEL
Pauling (1935) gives two saturation equations [equation
PaperI. K’p + (2a + 1)K’2p2 + 3a2K’3p3 -~ct~K’~p~
(0 Y = 1 + 4K’p
+ (4a + 2)K’“p2 + 4c?K”p3 + r~~K’~p~
(3) and (4) in his
56
R.
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CHILD.7
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which is sigmoid if a > 4 and (ii) y =
K’p + 3aK”p’
+ 3a3K13p3 +a6K’4p4
1 + 4K’p +6aK2p2 + 4a3K’3p3 + a6K’4p4 which is sigmoid if a > 4. (D)
MONOD
MODEL
In Monod et al. (1965) y = Lca(l +ca)“-‘+a(1 F L(1 +ca)“+(l
+ay-l +a)
which is sigmoid if: (n - l)(Lc’ + l)(L + 1) - n(Lc + 1)2 > 0. A number of interesting cases here are: c = 1: in which case YF is never sigmoid. L = 1; in which case yF is sigmoid if n(c- l)2 -2(c2 + 1) > 0. n = 4; in which case Yp is sigmoid if 3(Lc2 + l)(L+ 1) > 4&c+ 1)2. (E)
KOSHLAND
MODELS
In the tetrahedral model [Koshland et al., 1966, equation (13)], sigmoid curves arise if KBB > 4 K&. In the square model [equation (18)] if KBB > 3 KiB. In the linear model [equation (24)] if K
BB
> KABW:B
+ 3?4~3 + 1)
2+KAB
In the concerted model [equation
*
(30)], there can be sigmoid curves if:
KfC < 3.
(F)
In Volkenstein
VOLKENSTEIN
AND
GOLDSTEIN
MODEL
& Goldstein (1966) v = 2E K
1
k2K2S+k4S2 K1 K2+2K2 S+S2 > (
_ k2+k-!. kI
K
’
=
b+k-3
’
k,
giving a2-a1B1
= Ki
k4 K2
- -2kz G
ALLOSTBRIC
AND
i.e. k3k4(k2 +k-,) > 2k, kz(k4+k-,) a maximum is possible if k2 > 2k,.
RELATED
57
PHENOMENA
for sigvnoid inflexion. Note also that
11. An Inductive Proof for Sigmoid Carves An inductive proof that sigmoid curves cannot result from II independent 1 : 1 Michael&Menten isoenzymes or one enzyme with n independent active active sites each of 1 : 1 mechanism is as follows. Assume that a typical site or isoenzyme has a rate equation of the form: +AK;‘) -1 "r = V&%4(1 then the sum of n such terms will be of the form
with (~1~-.x1 /3J8 > 0 for sigmoid intlexions. that such curves are non-sigmoid. Assume
We now prove by induction
(a2-w%)s = - ,tl J-F;‘. Then, from:
we see that (a2
-alBl)n+l
=
(a2-alBlX-K+iG+21
?I+1 = - z1 v,q2. _ Nowforn
= 1: (a2
andforn
-a1 /II) = - VI K;’
= 2:
- wV2 = -(V,Iq2+V2K3. Thus the proof is complete. In the further case of 2 : 2 functions (nonsigmoid) it can similarly be shown that resulting sums of curves must be nonsigmoid. Similar analysis by induction can be used to prove that the addition of n 1 : 1 functions can never lead to an n : n function having a maximum. (a2
REFERENCES G. S. (1925). J. b&d Chcm. 63,529. G. R., SHILL, J. P. & NEET, K. E. (1972). J. bid. AINSWORTH, S. (1968). J. tbeor. B&d. 19, 1. ANENS, E. J. (1964). Molecular Phanmcdgy, vol. 1. New ADAIR, AIMLE,
Chem. 247,7088. York:
Academic
Press.
58
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E.
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AND
W.
G.
BARDSLEY
ATKINSON, D. E. (1966). Rev. Biochem. 35, 85. BOTTS, D. J. (1958). Trans. Faraday Sot. 54, 593. CHA, S. (1968). J. biol. Chem. 243, 820. CLELAND, W. W. (1963). Biochim. biophys. Acta 67, 104, 173, 188. FERDINAND, W. (1966). B&hem. J. 98,278. HARPER, E. T. (1971). J. theor. Biol. 32,405. HARPER, E. T. (1973). J. theor. Biol. 39, 91. HILL, A. V. (1910). J. Physiol., Lord. 40, 190. KING, E. L. & ALTMAN, C. (1956). J.phys. Chem. 60, 1375. KOSHLAND, D. E., NEMETHY, G. & FILMER, D. (1966). Biochemistry, N. Y. 5, 365. LIJNDQUIST, F. & WOLTHERS, H. (1958). Acta pharmac. tox. 14,265. MONOD, J., WYMAN, J. & CHANGEUX, J. (1965). J. molec. Biol. 12, 88. NEAL, J. L. (1972). J. theor. Biol. 35, 113. PARKER, R. B. & WAND, D. R. (1971). J. Pharmac. 177, 1. PATON, W. D. (1970). In Molecular Properties of Drug Receptors (R. Porter, M. J. O’Conner & A. Churchill, eds). London. PAIJLING, L. (1935). Proc. natn. Acad. Sci. U.S.A. 21, 186. RABIN, B. R. (1967). Biochem. J. 102,22C. SESHAGRI. N. (1972). J. theor. Biol. 34. 469. STEIN, w. D.‘(196?). The Movement’of Molecules Across Cell Membranes. New York: Academic Press. SWEENY, J. R. & FISHER, J. R. (1968). Biochemistry, N. Y. 7, 561. TEIPEL, J. & KOSHLAND, D. E. (1969). Biochemistry, N. Y. 8, 4656. VOLKENSTEIN, M. V. & GOLDSTEIN, B. N. (1966). Biochim. biophys. Acta 115,471. WAGNER, J. G. (1973). J. Pharmacokinetics Biopharmaceutics 1, 103. WYMAN, J. (1967). J. Am. them. Sot. 89,2202.
Allosteric and Related Phenomena: An Analysis of Sigmoid and Non-hyperbolic Functions ROBERT E. CHILDS AND WILLIAM G. BARDSLBY
Department of Obstetrics and Gynaecology, University of Manchester, St Mary’s Hospital, Manchester Ml3 OJH, England (Received 25 March 1974) The graphs of rational polynomial functions are of considerable importance in enzyme kinetics but a full analysis of the turning points and inflexions is only possible for 2 : 2 functions due to the complexity of the first and second derivatives for 3 : 3 and higher degree functions. This paper describes a simple method which can be applied to rational functions of
any degree in order to discover the relationstip
between the coefficients
necessary for the curve to have an initial sigmoid inkxion and a Gnal inflexion (which implies a maximum in the graph). This technique is then
applied to the current allosteric models. In addition, a full analysis of the 2 : 2 function with a maximum is given together with a method of separa-
ting this type of hehaviour (partial substrate inhibition) function (dead-end substrate inhibition).
from the 1 : 2
1. htroauction
Steady state processes in biological systems invariably consist of steps which are individually zero, first or second order with respect to one variable when all others are held constant. For instance, transport processes involve complexing between membrane components and diffusing entities, responses of pharmacological preparations to agonists or blocking agents are mediated via interactions between receptors and drugs, ligand binding to proteins, elimination of drugs from the body and binding of substrates, products, inhibitors and other effecters to enzymes all follow this type of bebaviour. Since combinations of polynomials and rational polynomials can always be written as rational polynomials, it follows that these systems can all be represented by functions of the type a,+alx+a2x2+. . .a,~” y = j30+/9~x+~2x2+. . .&pm i.e. by pure algebraic expressions with integer exponents. In all the cases 4s
46
R.
E.
CHILDS
AND
W.
G.
BARDSLEY
referred to, x would be the concentration of substance under consideration and y would be the appropriate steady state or equilibrium response measured experimentally. The purpose of this paper is to explore the types of polynomials encountered and to investigate analytically the graphical methods that can give information concerning the numerator and denominator coefficients and degree. In all cases, the physical situation dictates that the function cannot be negative and the denominator can have no real roots for positive values of x. Also, with the exception of enzyme inhibition, the degree of the numerator cannot exceed that of the denominator. 2. Transport Processes It is well known that the steady state rate of membrane transport can often be approximated by a 1 : 1 function when all experimental conditions are held constant and only the concentration of the material being transported is varied (Stein, 1967; Neal, 1972). True first order transport implies unlimited membrane capacity and is therefore not found in this or any other biological system except at very low concentrations when rational functions with LY,,= 0 and /?a finite become approximately first order. 3. Drug-Receptor
Interactions
Responses of pharmacological preparations are usually described by 1 : 1 functions (Ariens, 1964; Paton, 1970). Empirical equations such as y = Yc”(c”+K”)-’ have also been employed (Parker & Wand, 1971), but these are merely approximate curve fitting attempts that have no theoretical justification, since they often lead experimentally to non-integer powers of n. If we consider the response (R) of a typical preparation (e.g. contraction frequency, tension developed, pressure etc.), there will usually be a finite value of the parameter being measured (say R,) even when the drug concentration is zero, i.e. with P,(x) having no constant term. It seems to the present authors better to analyse the response of pharmacological preparations using y = (R-R,) as a dependent variable according to methods used in enzyme kinetic studies rather than by the log plots and empirical curve fitting procedures currently used which prevent maximum utilization of experimental data.
ALLOSTERIC
AND
RELATED
PHENOMENA
47
4. Ligand Binding to Proteins Where only one ligand binding site is present, the saturation function (J’) will be 1 : 1 with a0 = 0 but when several binding sites are present we have P = PnWlQmW where n = m and P has no constant term, and n is the number of binding sites. If these sites are independent and equivalent, then common factors will cancel and where allosteric effects are found (Monod, Wyman BEChangeux, 1965; Koshland, Nemethy BEFilmer, 1966), the functions will be at least 2 : 2 and the saturation function may be sigmoid (Botts, 1958; Ferdinand, 1966). 5. Pkarmacokinetica It has now been recognized for some time that the rate of elimination of drugs and metabolites is described more accurately by hyperbolic functions than first order processes (Lundquist & Wolthers, 1958; Wagner, 1973). Since the disappearance will be effected byseveral competing pathways (e.g. uptake by cells or plasma proteins, metabolism, conjugation, excretion, etc.), it is likely that the 1 : 1 function is a poor approximation and more thorough experimental work would indicate higher degree polynomials (with u. = 0). 6. EnzymeSubstrate
Interactions
The steady state velocity (u) of an enzyme catalysed reaction is a function of the concentration of substrates (A, B, C etc.) and products (P, Q, R etc.) but when all substrates and products are held constant and only A is altered we have where P has no constant term and where the coefficients are necessarily positive functions of the other substrate and product concentrations. The degree of the numerator will be determined by the number of times substrate adds in the catalytic sequence and also by the nature of that sequence, but the degree of the denominator will be greater than that of the numerator if the substrate also forms dead-end complexes with the enzyme (dead-end substrate inhibition). Rate equations will be increased in degree if isoenzymes are present or if the enzyme has several active sites provided that the kinetic constants for these sites differ; thus two 1 : 1 functions generate a 2 : 2 function etc. There is, in principle, no difficulty in calculating rate equations for any mechanism (King BE Altman, 1956; Volkenstein & Goldstein, 1966; Cha,
48
R.
E.
CHILD.9
AND
W.
G.
BARDSLEY
1968; Seshagri, 1972), but often it is helpful to find the degree of the functions independently as an aid to mechanistic interpretation. Polynomial functions were first introduced for this purpose by Cleland (1963), but there still remains some confusion about the type of information made available to the kineticist by graphical methods. This is particularly the case with allosteric effects where sigmoid and other complex types of v/A curves and non-linear double reciprocal plots are often taken to indicate homotropic effects, although several mechanisms can give this type of behaviour (Rabin, 1967; Atkinson, 1966; Sweeny & Fisher, 1968; Harper, 1971, 1973; Ainsworth, 1968; Ainsle, Shill & Neet, 1972). Double reciprocal plots will be dominated by the lowest degree term in the numerator, i.e., if ~1~= 0, parabola if a, = a, = 0, cubic and so on, but, if a, is finite, then no matter what the degree of the numerator and denominator, the double reciprocal plot will asymptotically approach a straight line as A-’ + co. The slopes of the asymptotic lines and the actual value of the function when A- ’ = 0 will be poly nomial functions in the other variables but the equations will now be inverted and undefined at the origin, i.e. they will be functions of the form: slope or intercept = f(B) =
po+81B+~2P+.
. .B,Brn
a B +cr B2 + 1
2
cI B” -*-
.
n
7. Enzyme Inhibitor Interactions
The effects of inhibitors on slopes and intercepts of double reciprocal plots have been fully discussed (Cleland, 1963), and it is now a routine exercise to replot slopes and intercepts as functions of inhibitor concentration Z according to WI = f’@)[l + PnU)IQ,(OI where P has no constant term. 8. Some General Properties of Rational Polynomials
We consider rational polynomials as given by equation (1) where n < m and ai; pi > 0 all i (i.e. positive coefficients only). Then the following properties hold: (1) If a0 = 0 the curve passes through the origin (otherwise y = ao/po when x = 0). (2) If n < m then lim y = 0. x+m (3) If IZ = m then lim y = cl,/&,. x-a3
(4) The graph can have no vertical asymptotes for x 2 0.
ALLOSTBRIC
AND RELATED
PHENOMENA
49
(9 A graph of Cv- aoMo) ( i.e. no constant in the numerator)
passes through the origin and so all rational polynomials can be reduced to one of this type and most of the later discussion concerns this type. This can result in the numerator having zero or negative coefficients and the function becoming negative. (6) The graph of y(x) may have, under some circumstances, an inflexion of positive slope giving an initially sigmoid shape to the curve (Fig. 1). The conditions for such an inflexion are found as follows from the derivative: Y’W -Y’(O) = 280 %z s; -alBoB1-aeBo82+a0B:)~ for x -4 1. In all cases we are to consider, we are interested only in those where the gradient is positive (or zero) at x = 0. All future discussion will make this assumption.
Y
X
Fro. 1. Conditions under which we obtain sigmoid inilcxions and/or maxima in equation (1) (a0 =O). (a) a~8~-a~80h-aogoB+ao~ c 0. @I e&-add%-e&h + a& > 0. (4 w%-~ - a,~ A 2 0. (d) an&l - ~-18. c 0. W always.
From the above expression then, we see that near the origin the gradient is increasing with increasing x if azbf-alAbA -aJdb+adC > 0. (2) Knowing that the curve must eventually “turn over” to approach the asymptote a./?;’ (if n = m) or the x-axis (if n < m), we conclude that the gradient must at some point start decreasing. Thus we get a point of infiexion to an initially sigmoid curve (Fig. 1) if equation (2) holds. (7) The graph of y(x) may have several maxima and a useful result in the later analysis is the slope of y(x) as x --, co for the case n = ni. 4 T.B.
50
R.
E.
CHILDS
AND
W.
G.
BARDSLEY
We tind :+t 2 = a,B~1x-2(Bn-1/Bn-tln-11a.>. Thus we get a decrease in y to a finite value as x + co, if and no decrease in y as x + co, if a&,- 1 2 j?, a,- i (see Fig. 1). The double reciprocuZ form of equation (1) is as follows: (y-‘)
= /%+Bm-lwl)+~ a,(x-l)m-n-t..
* * +BoW’)” . +a,(x-‘)m
’
From (i) (ii) (iii)
this we may note the following properties. y-l --) co as x -’ --t 0 for m > n. y-’ -+ /?./a, as x-l + 0 for m = n. If a0 = 0 then y-’ approaches a straight line asymptotically as x-l -+ co (PO # 0). If a,, = a, = 0 then y-l approaches a parabola asymptoticahy as x-i -+ co and so on. (iv) The same inequalities as in section (7) for y versus x arise in the slope of the double reciprocal plot at x-l = 0 for n = m, i.e.
W- ‘1 =
B,a,‘(B,-l/B,-a~-l/a~) at x-l = 0. d(x- ‘) Thus a negative initial slope in the double reciprocal plot is associated with decreasing y as x -P co (for the case n = m).
9. Special Cases of 2 : 2 Functions Since for a 2 : 2 function the first derivative is a quadratic and the second a cubic, there are only four possible shapes for this function, i.e. smooth curve with no maximum but with or without a sigmoid inflexion and a curve with a maximum but with or without a sigmoid inflexion. Botts (1958) has discussed the conditions on the coefficients for these cases and suggested that more complex functions such as stair-step curves with horizontal sections may be possible with 3 : 3 functions where the first derivative is a quartic and second derivative a sixth degree polynomial. Teipel & Koshland (1969) have shown, by numerical examples, that complex curves can arise with 4 : 4 functions. We now present a more detailed study of the 2 : 2 function and the special cases encountered experimentally, paying particular attention to the case of dead-end and.pa.rtial substrate inhibition.
ALLOSTBRIC
AND
RELATED
51
PHENOMENA
CASE 1: a, = 0 AND a&,
> /%/I%
Case I is for partial substrate inhibition. (Without take /?,, = 1.) y = (a~X+a~X2)/(1+~~X+~~X2>.
loss of generality
we (3)
We find turning points (if any) where
x=xo=/9z(ala;‘-j?ij?;‘)
where
(4)
G = B2(al a; ’ -0.5B1 /?; 1)2 + (1 - 0.258: /3; ‘). Consideration
of the inequality
0 < G < 1 leads to the inequality alaZ’
< j&B;’
and hence no positive turning points; whereas G > 1 implies a1 a;’ and leads to just one positive turning point. Let al a2- ’ = /I1 fl; ‘8,
8 > 1 for maximum
turning point
> j?J?;’
(5)
and B2 = j?: 4, Then substitution
4 > 0 for initially
sigmoid curves.
(6)
of equation (4) into (3) gives (for 0 > 1):
1
(e-1)2
= p.a2.j3;’
(say), p > 1.
Y(X = xo) = 2 See Fig. 2(a). Rearrangement
’ + (2++e-i)-2p(++e2-e)+
1
(7)
gives the following relation between 0,+ and p : 4 _ (e-p)2 4P(P-1)
(8)
and so ~de--P) xo = 2/?&-l)
p2 = -A--
&p - 1)’
Thus from an experimental plot of y versus X, we can measure x0, p and a2/jJ2 which enables us to calculate /I2 and hence a2 and to find 4 as a function of 8. From a double reciprocal plot we can measure the slope (So) of the
52
R.
E.
CHILDS
AND
W.
G.
BARDSLEY
straight line asymptote as X-I + co [Fig. 2(b)] so giving PO
8=
/.&
=
B:xxP-l)
x0 - 2P~ow32);
MO
@a2
and then a1 = Bla2(e/S2). Zn$exions
Points of inflexion occur where /MA a2-alB2)x3+382a2xz+382al x+0& a1 --a21 = 0 which on substitution for 0 and #Jdefined above may be written y3-3ay-a-
By means of the trigonometric
Bl l+- 24 = 0 (9) p2 ( e-1 ) method we can solve equation (9) to give:
x = ~ A4 1+ 2(e2+4-e)‘.cos0 n ~2w) { J& nI where
8, = *cos--~
e+2+1
I 2n7-c.
2&(e2++-e)+
>
3 ’
n = 1,2,3.
Note that we require e+2+i 2J&e2
< 1
+ 4 - e)*
for all three roots to be real, i.e. 4, > 4. By this method al, a2, PI, /I2 can be determined and points of inflexion verified. CASE II:
a0 = 0; a1 = 0
In an effort to explain sigmoid v/A curves in enzyme kinetics, rate equations without linear terms have been derived (Sweeny & Fisher, 1968). Enzyme mechanisms where a substrate adds twice in a sequence uninterrupted by product release give rate equations of this type. This case gives a double reciprocal plot which is parabolic. The curve y(x) is aZway.ssigmoid and has no turning point in the first quadrant. This case is easily distinguished from that of a sigmoid curve where al is finite since in this latter case, the double reciprocal plot will reach a linear asymptote which actually has a negative y-l intercept. In fact, this seems easier to spot experimentally than a sigmoid inflexion.
ALLOSTBRIC
AND
RELATED
53
PHENOMENA
CAsEm:a~=0;a,=O Here we are dealing with a case of “dead-end substrate inhibition” and find y(x) can neuer be sigmoid but that there is always a turning point. Experimentally, it is often difficult to distinguish between dead-end substrate inhibition where y + 0 as x + cc and the case of a full 2 : 2 function (or 3 : 3 etc.) where we get partial substrate inhibition, particularly if the horizontal asymptote for large x is very small. A useful discriminatory plot for distinguishing between these two cases is that of y -’ versus x; since if CI~ = 0 [Fig. 2(c)], then there can be no intlexion in the positive quadrant for this plot whereas for the partial substrate
---_
(12, P;’
X
(b)
(cl I:2 / --
Y-'
y
/I 2:2 -A
/
X
ho. 2. (a), (b) 2 : 2 case showing experimentally obtuinable parameters of text. (c) Plot of y-l/x for 1 : 2 and 2 : 2 functions showing dit%rence x+03.
used in analysis in behaviour as
54
R.
E.
CHILDS
AND
W.
G.
BARDSLEY
inhibition case (Fig. 2), we find that there must be exactly one inflexion as shown by the following analysis : a2
a2
#
0,
PI a2
= 2{af+3a,
a2x+3a3x2+a2(/4 (a,x+a2x2)3
For this second case (partial substrate inhibition) of the second derivative can be written: x3--x2-bx-c
= 0;
a2-al
/12)x3)
.
the cubic in the numerator
a, b, c > 0,
and since there must be at least one inflexion in the positive quadrant, then it factorizes as follows : (x-rl)(x2+px+c/r,)
rl > 0.
= 0;
The quadratic factor here can be shown to have no positive root. CASE
IV:
I&,
=
0;
p2
=
0
This situation occurs in mixed dead-end and partial inhibition now inhibitor concentration. Consideration of the function
where x is
a,x+a2x2
= ~(x+ala;‘-l);‘-(ala;‘-8;‘)/(l+B1x)) Y = 1+&x 1 and its derivatives shows that there can be no sigmoid inflexion. CASE
v:
a0 = 0; PI = 0
This gives a sigmoid inflexion always together with a maximum. CASE
VI:
a,, = aI = PI = 0
This gives a sigmoid inflexion but no maximum. 10. Some Allosteric Models In all of the models discussed in the folIowing section, the coefficients are such that there are no turning points but in most cases a necessary constraint is found before sigmoid curves are possible. (A)
HILL
EQUATION
Kx” Y =1+Kx”
ALLOSTBRIC
AND
RELATED
PHENOMENA
55
This has an inflexion where x = “J(n- 1)/K@+ 1) and has no turning point but steadily approaches y = 1. The experimental value of n is thought to bc a measure of the allosteric linkage (Wyman, 1%7). For the case n = 2, for example, the Hill equation is really an approximation to equation (3) in the case where there is no turning point (i.e. ai #?z < cQi) and in this case a2 = B2 as it is modelling a saturation function. A similar analysis to the following will apply in the case where there is a V,,, instead. Experimental workers plot lo&/l -JJ) versus log x expecting to obtain a straight line of slope n if the data actually obeys the Hill equation. If they are actually plotting values which satisfy qx+a2x2
Y = 1+pix+o12x2
where u2 = B2
i.e. a1+cr2x log(-J!:hX+hl 1-Y > (1 +a -4x
>
then it is reasonable to suppose that the “straight” line chosen is the tangent to the curve at the point of inllexion and thus n is the gradient of this tangent. The inflexion to equation (11) occurs at x = [ai/a2& -cl,)]* and the slope of the tangent there is: -44 -al> a2+a1 (Bi-d+2(~1 a2>+(Bi -I)*’ We see that if a1 = & = 0 then the slope is 2 and the function is that of the Hill equation. Similar considerations could be applied for cases where n > 2. a2
l+
(B)
ADAIR
MODBL
FOR
HABMOGLOBIN
The saturation curve given by Adair [1925, equation (4)] is: Y=
0*25KI x +0’5K2 X2+0*75K3 l+KIx+K2x2+K3x3+K4x4
X3
+ K,
X4
from which we see by equation (2) that a sigmoid inflexion occurs if 2K2 > Kf. (C)
PAULING
MODEL
Pauling (1935) gives two saturation equations [equation
PaperI. K’p + (2a + 1)K’2p2 + 3a2K’3p3 -~ct~K’~p~
(0 Y = 1 + 4K’p
+ (4a + 2)K’“p2 + 4c?K”p3 + r~~K’~p~
(3) and (4) in his
56
R.
E.
CHILD.7
AND
W.
G.
BARDSLEY
which is sigmoid if a > 4 and (ii) y =
K’p + 3aK”p’
+ 3a3K13p3 +a6K’4p4
1 + 4K’p +6aK2p2 + 4a3K’3p3 + a6K’4p4 which is sigmoid if a > 4. (D)
MONOD
MODEL
In Monod et al. (1965) y = Lca(l +ca)“-‘+a(1 F L(1 +ca)“+(l
+ay-l +a)
which is sigmoid if: (n - l)(Lc’ + l)(L + 1) - n(Lc + 1)2 > 0. A number of interesting cases here are: c = 1: in which case YF is never sigmoid. L = 1; in which case yF is sigmoid if n(c- l)2 -2(c2 + 1) > 0. n = 4; in which case Yp is sigmoid if 3(Lc2 + l)(L+ 1) > 4&c+ 1)2. (E)
KOSHLAND
MODELS
In the tetrahedral model [Koshland et al., 1966, equation (13)], sigmoid curves arise if KBB > 4 K&. In the square model [equation (18)] if KBB > 3 KiB. In the linear model [equation (24)] if K
BB
> KABW:B
+ 3?4~3 + 1)
2+KAB
In the concerted model [equation
*
(30)], there can be sigmoid curves if:
KfC < 3.
(F)
In Volkenstein
VOLKENSTEIN
AND
GOLDSTEIN
MODEL
& Goldstein (1966) v = 2E K
1
k2K2S+k4S2 K1 K2+2K2 S+S2 > (
_ k2+k-!. kI
K
’
=
b+k-3
’
k,
giving a2-a1B1
= Ki
k4 K2
- -2kz G
ALLOSTBRIC
AND
i.e. k3k4(k2 +k-,) > 2k, kz(k4+k-,) a maximum is possible if k2 > 2k,.
RELATED
57
PHENOMENA
for sigvnoid inflexion. Note also that
11. An Inductive Proof for Sigmoid Carves An inductive proof that sigmoid curves cannot result from II independent 1 : 1 Michael&Menten isoenzymes or one enzyme with n independent active active sites each of 1 : 1 mechanism is as follows. Assume that a typical site or isoenzyme has a rate equation of the form: +AK;‘) -1 "r = V&%4(1 then the sum of n such terms will be of the form
with (~1~-.x1 /3J8 > 0 for sigmoid intlexions. that such curves are non-sigmoid. Assume
We now prove by induction
(a2-w%)s = - ,tl J-F;‘. Then, from:
we see that (a2
-alBl)n+l
=
(a2-alBlX-K+iG+21
?I+1 = - z1 v,q2. _ Nowforn
= 1: (a2
andforn
-a1 /II) = - VI K;’
= 2:
- wV2 = -(V,Iq2+V2K3. Thus the proof is complete. In the further case of 2 : 2 functions (nonsigmoid) it can similarly be shown that resulting sums of curves must be nonsigmoid. Similar analysis by induction can be used to prove that the addition of n 1 : 1 functions can never lead to an n : n function having a maximum. (a2
REFERENCES G. S. (1925). J. b&d Chcm. 63,529. G. R., SHILL, J. P. & NEET, K. E. (1972). J. bid. AINSWORTH, S. (1968). J. tbeor. B&d. 19, 1. ANENS, E. J. (1964). Molecular Phanmcdgy, vol. 1. New ADAIR, AIMLE,
Chem. 247,7088. York:
Academic
Press.
58
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E.
CHILDS
AND
W.
G.
BARDSLEY
ATKINSON, D. E. (1966). Rev. Biochem. 35, 85. BOTTS, D. J. (1958). Trans. Faraday Sot. 54, 593. CHA, S. (1968). J. biol. Chem. 243, 820. CLELAND, W. W. (1963). Biochim. biophys. Acta 67, 104, 173, 188. FERDINAND, W. (1966). B&hem. J. 98,278. HARPER, E. T. (1971). J. theor. Biol. 32,405. HARPER, E. T. (1973). J. theor. Biol. 39, 91. HILL, A. V. (1910). J. Physiol., Lord. 40, 190. KING, E. L. & ALTMAN, C. (1956). J.phys. Chem. 60, 1375. KOSHLAND, D. E., NEMETHY, G. & FILMER, D. (1966). Biochemistry, N. Y. 5, 365. LIJNDQUIST, F. & WOLTHERS, H. (1958). Acta pharmac. tox. 14,265. MONOD, J., WYMAN, J. & CHANGEUX, J. (1965). J. molec. Biol. 12, 88. NEAL, J. L. (1972). J. theor. Biol. 35, 113. PARKER, R. B. & WAND, D. R. (1971). J. Pharmac. 177, 1. PATON, W. D. (1970). In Molecular Properties of Drug Receptors (R. Porter, M. J. O’Conner & A. Churchill, eds). London. PAIJLING, L. (1935). Proc. natn. Acad. Sci. U.S.A. 21, 186. RABIN, B. R. (1967). Biochem. J. 102,22C. SESHAGRI. N. (1972). J. theor. Biol. 34. 469. STEIN, w. D.‘(196?). The Movement’of Molecules Across Cell Membranes. New York: Academic Press. SWEENY, J. R. & FISHER, J. R. (1968). Biochemistry, N. Y. 7, 561. TEIPEL, J. & KOSHLAND, D. E. (1969). Biochemistry, N. Y. 8, 4656. VOLKENSTEIN, M. V. & GOLDSTEIN, B. N. (1966). Biochim. biophys. Acta 115,471. WAGNER, J. G. (1973). J. Pharmacokinetics Biopharmaceutics 1, 103. WYMAN, J. (1967). J. Am. them. Sot. 89,2202.
