Analysis of Biochemical Equilibria Relevant to the Immune Response: Finding the Dissociation Constants

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1 Bull Math Biol 202) 74:7 206 DOI 0.007/s ORIGINAL ARTICLE Analysis of Biochemical Equilibria Relevant to the Immune Response: Fining the Dissociation Constants L.J. Cummings R. Perez-Castillejos E.T. Mack Receive: 5 May 20 / Accepte: 5 January 202 / Publishe online: 2 February 202 Society for Mathematical Biology 202 Abstract This paper analyzes the biochemical equilibria between bivalent receptors, homo-bifunctional ligans, monovalent inhibitors, an their complexes. Such reaction schemes arise in the immune response, where immunoglobulins bivalent receptors) bin to pathogens or allergens. The equilibria may be escribe by an infinite system of algebraic equations, which accounts for complexes of arbitrary size n n being the number of receptors present in the complex). The system can be reuce to just 3 algebraic equations for the concentrations of free unboun) receptor, free ligan an free inhibitor. Concentrations of all other complexes can be written explicitly in terms of these variables. We analyze how concentrations of key experimentallymeasurable) quantities vary with system parameters. Such measure quantities can furnish important information about issociation constants in the system, which are ifficult to obtain by other means. We provie analytical expressions an suggest specific experiments that coul be use to etermine the issociation constants. Keywors Antiboy Aggregation Bivalent ligan Immune system Introuction In the immune response, multivalent antiboies an antigens or pathogens) bin together an form noncovalent complexes, which may be large. The efficacy of the L.J. Cummings ) Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ 0702, USA lina.cummings@njit.eu R. Perez-Castillejos Department of Biomeical Engineering, New Jersey Institute of Technology, University Heights, Newark, NJ 0702, USA E.T. Mack BP Biofuels Global Technology Center, 4955 Directors Place, San Diego, CA 922, USA

2 72 L.J. Cummings et al. response of the immune system towar a pathogen is often relate to the number of antiboies that bin to a pathogen Murphy et al. 2008). This immune response can be particularly profoun in an allergic reaction Golstein 988), where the antigen is an allergen such as pollen; an the allergic response is a useful an much-stuie moel system for the immune response. In this paper, we follow a moel propose by Mack et al. 20), which itself generalizes one presente in a classic paper by Dembo an Golstein 978). We consier a system containing ientical symmetric bivalent receptors antiboies, or immunoglobulins), bivalent ligans antigens; pathogens or allergens), an monovalent ligans. Bivalent ligans can crosslink two or more bivalent receptors, forming aggregates or oligomers) comprising multiple ligans an receptors. The monovalent ligans act to inhibit the immune response, since while a site on a bivalent receptor is occupie by a monovalent ligan, no crosslinking reaction can occur. The original Dembo an Golstein moel Dembo an Golstein 978) for the ynamic equilibrium between complexes in this system mae several key assumptions to facilitate analytical progress, among them: ) bivalent ligan is present in the mixture in great excess, so that the amount of free unboun) ligan present may be assume to be the same as the al amount of ligan in the original mixture; an 2) the affinities of the monovalent an singly-boun bivalent ligans for the bining sites of bivalent receptors are ientical. Both of these conitions are relaxe in our moel. The Dembo an Golstein moel has also been generalize in several ifferent irections by these an other authors, to account for heterogeneous populations of bivalent receptors Golstein an Wofsy 980), for asymmetrical bivalent ligans Wofsy 980), anforcooperativityinthebining WofsyanGolstein987). More recently, Posner et al. 995a, 995b) stuie a time-epenent moel that preicts the concentrations of bivalent receptors an ligans in a mixture as functions of time, an Barisas 2003) propose a moel similar to that of Dembo an Golstein that escribes a population of bivalent receptors, an ligans with valences 2. Other approaches to moeling such complex multivalent systems inclue so-calle rulebase moeling ; see, e.g., Colvin et al. 2009), Faeer et al. 2005), an Hlavacek et al. 2006). Using the original moel of Dembo an Golstein, Golstein 988) reporte an exact result for the al concentration of bivalent ligan that maximizes the concentration of complexes containing two or more bivalent receptors. Such an explicit preiction is very useful, as an experiment may then be carrie out an compare with the moel preiction, enabling estimates to be mae of various issociation or bining constants. Mack et al. 20) carrie out a similar analysis for their extene moel to reuce the system to three algebraic equations, which are reaily solve numerically. Here, we consier the same extene system, but focus principally on specific asympic limits in which analytical progress can be mae. We analyze how concentrations of key experimentally-measurable) quantities vary as a function of the system parameters. Determining such functional relationships allows us to extract the values of system parameters such as the issociation constants, which are ifficult to measure irectly from experimentally-measurable ata Henrickson et al. 2002; Hlavacek et al. 999; Posner et al. 2002; Sklar et al. 2002). Such information is useful as it provies a means of quantifying the response of various antiboies to given

3 Analysis of Biochemical Equilibria Relevant to the Immune Response 73 pathogens. In contrast to previous approaches base on numerical calculations e.g., Mack et al. 20), we fin analytical expressions that can be use to etermine system parameters irectly from experimental ata see also Mack et al. 2008). We expect the simple analytical expressions presente here to be of great interest to experimentalists who are unfamiliar with numerical techniques. Specific experiments are suggeste that will allow the experimentalist to use the analytical formulae to etermine issociation constants for the system. In orer to facilitate the use of the paper by nonmathematicians, each section of mathematical analysis is followe by a separate subsection ientifie by the title Determining the issociation constants), suggesting how the results might be use to etermine certain of the issociation constants in the laboratory. Aitionally, two tables summarize the results of this paper: Table lists the efinitions an Table 2 the main analytical expressions etermine in this stuy. 2 Reaction Scheme an Mathematical Moel Following Mack et al. 20), we consier a moel system in which ientical symmetric bivalent receptors, ientical symmetric bivalent ligans, an ientical monovalent ligans inhibitor molecules) are present. The various possible complexes that may form in such a system, an the ynamic) equilibria between them, are etaile in Scheme of that paper, which appears in aapte form here as Fig.. Iniviual equilibria are epicte in Fig. as pairs of arrows pointing in opposite irections:. Each of the two bining sites of a bivalent receptor epicte in Fig. as ) can noncovalently bin to either one monovalent ligan epicte as ) or to one of the two bining moieties of a bivalent ligan epicte as ). The uppermost block of Fig. shows the complete set of seven possible receptor-ligan complexes that can form with one bivalent receptor: the unboun free) receptor ientifie in Fig. as Z ); the bivalent receptor boun to one of the bining sites of one bivalent ligan A ); the bivalent receptor boun to one of the bining sites of two bivalent ligans simultaneously B ); the bivalent receptor boun to the two bining sites of one bivalent ligan C ); the bivalent receptor boun to one monovalent ligan D ); the bivalent receptor boun to two monovalent ligans E ); an the bivalent receptor boun to one of the bining sites of one bivalent ligan an to one monovalent ligan simultaneously F ). The subscript inicates that these molecular complexes contain only one bivalent receptor. The central block in Fig. shows the complexes containing exactly two bivalent receptors: accoringly, the subscript for the complexes in this block is 2. In general, the seven complexes with n bivalent receptors are escribe as A n, B n, C n, D n, E n, an F n bottom block of Fig. ). The etaile escription of the equilibria of the system stuie here can be foun elsewhere Mack et al. 20). Briefly, the equilibria between all the molecular species in this system are escribe through the prouct of stoichiometric factors erive from the ratio of the number of bining moieties to the available number of bining sites) an four issociation constants Connors 987; Mack et al. 20). The monovalent issociation constant K mono relates the concentration of free unboun) monovalent ligan ientifie as I in Fig. ); the concentration of bivalent receptors either free or boun) with at least one available bining site A n, D n, Z n in Fig. );

4 74 L.J. Cummings et al. Fig. Summary of all possible ynamic equilibria between symmetric bivalent ligans ), symmetric bivalent receptors ), an monovalent ligans ), aapte from Mack et al. 20) an the concentration of the complex after the monovalent ligan is boun to the available receptor site F n, E n, D n, respectively, for the preceing list). The intermolecular issociation constant K inter relates the concentration of free unboun) bivalent ligan ientifie as L in Fig. ); the concentration of bivalent receptors either free or boun) with at least one available bining site A n, D n, Z n in Fig. ); an the concentration of the resulting complex when the bivalent ligan is boun to the available receptor site B n, F n, A n, respectively, for the preceing list). The

5 Analysis of Biochemical Equilibria Relevant to the Immune Response 75 intramolecular issociation constant K intra relates the concentration of complexes exhibiting a bivalent ligan with one free moiety, the other being boun to a complex presenting a bivalent receptor with an unoccupie bining site A n in Fig. ); an the concentration of circular or cyclic complexes with all bining moieties of bivalent ligans an all bining sites of bivalent receptors boun to each other C n in Fig. ). Finally, the cooperativity parameter α escribes the tenency of complexes with n ) bivalent receptors to form complexes with n receptors. Equations ) 6) escribe each of the equilibria in the bottom block of Fig. as a function of the issociation constants introuce above. In this manuscript, the presence of an asterisk ) enotes that a quantity is imensional we nonimensionalize below an rop the ); in all cases, the imensions are those of concentration, i.e., moles per volume. The overbrace in the equations enotes the cyclic complex, where a chain of alternating ligan-receptor pairs is joine up into a ring intramolecular boning) by a single bivalent ligan, which joins the two en receptors of the chain together. In our moel, the intramolecular issociation constant may epen on the size n of the cyclic complex forme: K,n intra. This size-epenence is another important ifference between our moel an that of Dembo an Golstein 978). We consier two cases to encompass intramolecular bining: i) The first case escribes the possibility that the bivalent ligan is too short to span the istance between the two bining sites of a bivalent receptor, which ultimately prevents the formation of mono-receptor cyclic complexes C ). Mathematically, this case is escribe with K intra, = while all K,n> intra are equal. ii) The secon case is that of bivalent ligans long enough to bin the two sites of one bivalent receptor simultaneously. Mathematically, this case has all K,n intra equal. The equilibrium between free bivalent receptors Z ) an complexes of two bivalent receptors with two available bining sites Z2 ) is escribe through the cooperativity parameter α as Z2 = α/k inter )Z A = 4α/K inter ) 2 )Z )2 L the secon equality on substitution for A = 4/K inter )Z L ). The general situation is summarize by the set of equations below, with the equilibrium between Zn an Z n given by 7): [ ] = 4 K inter [ ] = [ {}}{ ] = K inter 2nK intra,n [ ] 2 = K mono [ ] = 2K mono [ ] = 2 K inter [ ] [ ], A n = 4 K inter [ ] [ ], B n = [ ], C n = [ ] [ ], D n = 2 K inter 2nK intra,n K mono [ ] [ ], E n = 2K mono [ ] [ ], F n = 2 K inter Z n L, ) A n L, 2) A n, 3) Z n I, 4) D n I, 5) D n L, 6)

6 76 L.J. Cummings et al. [ ] 4α = K inter Zn = 4α ) 2 ) n [ ] n [ ] n, ) n L ) 7) n ) Z n. K inter ) 2 This system of equilibria is augmente by mass conservation conitions, which state that the al amount of receptor R ), ligan L ), an inhibitor I ) in the system must be conserve. Accounting for the amounts in complexes of all sizes, these conitions are R = n Zn + A n + B n + C n + D n + E n + F n ), 8) n= L = L + n )Z n + na n + n + )B n + nc n + n )D n n= + n )En + nf n ), 9) I + D n + 2En + F n ). 0) n= Normalizing all concentrations with K inter,) 7) become A n = 4Z n L, ) B n = A n L, 2) C n = where K mono = K mono /K inter, Kn intra imensionless moel), A n = A n /K inter 2nK intra n A n, 3) D n = 2 K mono Z ni, 4) E n = 2K mono D ni, 5) F n = 2D n L, 6) Z n = 4αLZ ) n Z, 7) = K,n intra/k inter we rop subscripts in the an similarly for all new unstarre quantities introuce. The mass conservation equations 8) 0) are unchange except that s are roppe as quantities become imensionless, an we can simplify by replacing 9) with 8) minus 9)). In imensionless form then, 8) 0) are equivalent to R = nz n + A n + B n + C n + D n + E n + F n ), 8) n= R L = L + I = I + Z n B n + D n + E n ), 9) n= D n + 2E n + F n ). 20) n=

7 Analysis of Biochemical Equilibria Relevant to the Immune Response 77 The complete mathematical moel of the equilibria shown in Fig. consists of ) 20). In general, we must solve this system of 0 equations for quantities A n, B n, C n, D n, E n, F n, Z n, Z, L, I, given the al amounts of receptor, ligan, an inhibitor, R, L, I ae to the system initially, an the values of the normalize issociation constants Kn intra, K mono. We simplify matters by noting that we may substitute for Z n from 7) into) an 4), an then for A n from ) into 2) an 3), an finally for D n from 4) into5) an 6). This sequence of operations gives expressions for all quantities A n F n, an Z n, in terms of Z, L an I. Moreover, on substitution of these expressions into the conservation equations 8) 20), we are able to compute the require infinite sums explicitly, assuming 4αLZ < this conition, which can be checke a posteriori, is mathematically necessary for convergence of the infinite sums; failure to satisfy this conition means that the sums o not converge, an woul point physically to some kin of bifurcation in the system, where all the mass ens up in extremely large complexes). We have Z n = Z 4αLZ ) k Z =, 4αLZ 0 A n = 4L Z n = 4LZ, 4αLZ B n = L A n = 4L2 Z, 4αLZ D n = 2I K mono I E n = 2K mono Z n = D n = 2IZ K mono 4αLZ ), I 2 Z K mono ) 2 4αLZ ), 4ILZ F n = 2L D n = K mono 4αLZ ) we o not require C n, though this can be evaluate). For the receptor conservation law 8), we also nee to evaluate nz n an similar quantities, for which we note, writing x = 4αLZ, that nz n = Z nx n = Z x Hence, ) x n Z nz n = 4αLZ ) 2, 0 4LZ na n = 4L nz n = 4αLZ ) 2, = Z x) ) = Z x x) 2.

8 78 L.J. Cummings et al. 4L 2 Z nb n = L na n = 4αLZ ) 2. For nc n,wehave nc n = /2) A n /Kn intra ), an as state, we wish to consier the possibility that the issociation constants Kn intra vary with n. For general n-epenence, we cannot compute the sum explicitly, but we consier the two special state cases of relevance: Case i), where K intra = an all other Kn intra are equal short bivalent ligan); an Case ii), where all the Kn intra, incluing K intra, are equal long bivalent ligan). In case i), the cyclic complex with just one receptor an one inhibitor,, is not forme for example, if the ligan is too short to span the two sites of a single bivalent receptor), but all other cyclic complexes are equally likely to form. We have ropping the subscript n from Kn intra henceforth) nc n = 2K intra A n = ) 4LZ 2K intra 4LZ 4αLZ an We also have = nc n = 2K intra 2 8αL 2 Z 2 K intra 4αLZ ) nd n = 2I K mono I ne n = 2K mono A n = nz n = nd n = case i) ), 2LZ K intra 4αLZ ) 2IZ K mono 4αLZ ) 2, case ii) ). I 2 Z K mono ) 2 4αLZ ) 2, 4ILZ nf n = 2L nd n = K mono 4αLZ ) 2. Substitution of these expressions into the conservation equations 8) 20) then reuces the whole system to three algebraic equations for Z, L an I, which after simplification take the form: [ Z R = 4αLZ ) 2 + I ) 2 ] + 2L + 8αL2 Z Kmono K intra 4αLZ ) in case i), 2) [ Z R = 4αLZ ) 2 + I ) 2 + 2L + 2L ] Kmono K intra 4αLZ ) in case ii), 22)

9 Analysis of Biochemical Equilibria Relevant to the Immune Response 79 Z R = L L + 4αLZ ) 2IZ I = I + K mono 4αLZ ) + I K ) + 2L mono ) + I + 2L Kmono where 23) an 24) apply in both cases i) an ii). + I 2L Kmono ), 23), 24) 3 Preictive Analysis We want to use our moel to extract information such as the values of issociation constants for the system. One approach is the following: A quantity that the experimentalist can etermine in any given experiment is the concentration of complexes of size n 2 by which we mean, complexes containing at least two receptor molecules) in the mixture. Consier a titration experiment in which ligan is slowly ae to known fixe) amounts of receptor an inhibitor. If we monitor the concentration of complexes as a function of ligan concentration methos for oing this are iscusse by, for example, Henrickson et al. 2002, Hlavacek et al. 999, Posner et al. 2002, Sklar et al. 2002, an references therein), then we shoul be able to compare this concentration profile, or at least key features of it, with preictions from our moel. Easy features to compare coul be the value of the ligan concentration at which the concentration of complexes is maximize an/or the value of this maximum concentration), or we might measure the initial graient of the titration concentration curve, an compare this with preictions from the moel. In our notation, the relative concentration of complexes of size n 2, which we call,is = R Z + A + B + C + D + E + F ) ), 25) R so that in case i) = Z + I ) 2 + 2L, 26) R Kmono while in case ii) = Z [ + I ) 2 + 2L + 2L R Kmono K intra ]. 27) Although in principle any require information a turning-point, or a graient of f at a given point) may be compute numerically for given values of the system parameters, etermining such information analytically, in terms of arbitrary system parameter values, using 2) 24), is very ifficult. In the following sections, we iscuss certain simplifie cases in which analytical progress may be mae, an the preictive powers of the reuce moels.

10 80 L.J. Cummings et al. 4 Case Without Monovalent Inhibitor The main simplification of relevance is when no inhibitor monovalent ligan) is present. In this case, 24) reuces to an ientity, an 2) or22)) an 23) simplify: [ ] Z R = 4αLZ ) 2 + 2L) 2 + 8αL2 Z K intra 4αLZ ) in case i), or 28) [ Z R = 4αLZ ) 2 + 2L) 2 + 2L ] K intra 4αLZ ) in case ii), an 29) R = L L + Z 4L 2 ) 4αLZ ). 30) Solving 30)forZ, L + R L Z = 4 α)l 2 3) + 4αLR L ) an substituting in 28) or29) gives a sixth-orer polynomial equation for the free ligan concentration L in case i), an a quartic in case ii). Neither have general analytical solution, but special cases may be solve exactly. 4. Cooperativity α =. Case ii), Long Bivalent Ligan Setting α = lowers by one) the egree of the polynomial satisfie by L, so that case ii) reuces to a cubic for L, with analytical solution. Case i) is governe by a quintic an is still intractable in general. So, we first consier case ii) with α = which, as we shall see, provies insight into the short bivalent ligan case i). In case ii) α = ), L satisfies 4 + 2K intra L ) L 3 2L 2 2R + 2L + K intra 4L R ) 2 2L + 4R ) )) + L 2L 2R K intra 4L R ) 2 2L + 4R ) )) + K intra L = 0 32) or, introucing convenient shorthan notation for the coefficients in 32) consiering R an K intra fixe but allowing L to vary) gl,l ) := b 0 L )L 3 + b L )L 2 + b 2 L )L + b 3 L ) = 0. 33) The cubic 33) has three solutions, given in terms of the following parameters: Q = 3b 2b 0 b 2 9b0 2, R= 9b 0b b 2 27b0 2b 3 2b 3 54b0 3, D = Q 3 + R 2, S = R + D ) /3 ) /3,, T = R D with ST = Q. The solutions are

11 Analysis of Biochemical Equilibria Relevant to the Immune Response 8 L L ) = S + T b, 34) 3b 0 L 2 L ) = 2 S + T) b + i 3 S T), 35) 3b 0 2 L 3 L ) = 2 S + T) b i 3 S T), 36) 3b 0 2 an here the iscriminant D 0 always, which leas to real an in general, istinct) roots S an T are complex conjugates). Hence, LL ) is given by the relevant root real, positive an, to guarantee convergence of the infinite sums, such that 0 < 4LZ < ), with Z then given by Z = L + R L 37) + 4LR L ) from 3). It is not ifficult to check that there is always exactly one relevant solution for L: although two solutions are positive an, therefore, possible caniates, one of these always leas to 4LZ > an so is unacceptable. The relevant solution is the smaller positive root though which of the three expressions 34), 35), or 36)thisis given by varies epening on the values of the parameters, as we shall see explicitly below). A special case of the system 32), 37) arises when L = R + /2. When this happens, the cubic equation 32) has a simple factorization: 2L) 2 L + K intra) + K intra R + /2) ) = 0, an the only relevant root is the repeate one, L = /2. This situation correspons to a maximum value with respect to L ) of the quantity efine in 27), as we shall now see. With L = R + /2 an L = /2, the solution 37) forz is unefine, an careful local analysis is require to etermine its value. This we o by examining the cubic equation 32) when L = R + /2 + ɛ for ɛ, an etermining the appropriate root L, correct to orer ɛ. Forɛ of either sign but small), we have L = 2 + ɛl + O ɛ 2), Z = L + Oɛ), 38) 2 + L ) where L = [ + + 8λK intra ) /2], λ= + 2K intra + 2R ). 39) 2λ Further asympic analysis enables us to etermine higher-orer corrections to Z an L. Substituting these asympic expressions for Z an L into our expression 27) setting I = 0) for, we fin that has no orer-ɛ contribution, an that its orer ɛ 2 correction term is strictly negative that is, consiere as a function of L, it has a local maximum at the value L = R + /2. The value of Mathematically, L increasing through the value R + /2 correspons to the crossing over of the two positive roots mentione above: the smaller root increases through the value /2 an then ceases to be relevant, while the larger root ecreases through /2 an becomes the relevant one. Moreover, at this value of L, examination of 37) givesz = /2 an unphysical result) except in the special case L = /2. Thus for L = R + /2 we always require L = /2.

12 82 L.J. Cummings et al. at this local maximum is foun by setting L = R + /2, I = 0, L = /2, an Z = L )/2 + L )) in 27), with L givenby39). This proceure gives after simplification),max = + 4Kintra )[2λ + 8λK intra ] 2K intra R [2λ λK intra ] with λ as given in 39) above. 40) 4.. Determining the Dissociation Constants If the value of the cooperativity parameter α is close to unity, then these results can be use to etermine the issociation constants K inter, K intra,asfollows.note that, since the moel is mae imensionless by scaling concentrations with the unknown K inter, we must here work with imensional quantities that the experimentalist woul know. Carry out a titration with no inhibitor, an a known concentration of receptor, R. At regular points throughout the experiment, measure the concentration C of complexes of size n 2 which, in the notation above, is given by C = R e.g. Henrickson et al. 2002; Hlavacek et al. 999; Posner et al. 2002; Sklar et al. 2002). Determine, as closely as possible perhaps using a curve-fit to extrapolate if only a few ata points are taken) the value of ae ligan, L, at which C is maximize, an the corresponing maximal value, Cmax = R f,max.we know, from the analysis above that the maximum is obtaine when L = R + /2 or, in imensional terms, when L = R + K inter. 2 Hence, from the recore maximizing value of L, we obtain K inter. Next, we know that at the maximum, 40) hols. With K inter now etermine, we know all quantities in 40) except for K intra R = R /K inter, an,max = Cmax /R ). Hence, 40) is a nonlinear equation to be solve for Kintra. After rearrangement, this reuces to a cubic, one root of which is the physically-relevant one it is possible that ambiguity will arise at this stage). Finally, then K intra = K inter K intra. 4.2 Cooperativity α =. Case i), Short Bivalent Ligan Remarkably, although case i) is governe by a ifferent, more complicate quintic, rather than quartic) equation for L in the case I = 0, α=, a similar simplification occurs at the same value L = R + /2 as in case ii). The system is governe by couple equations [ ] Z R = 4LZ ) 2 + 2L) 2 + 8L2 Z K intra 4LZ ), 4) Z = L L + R + 4LR L ), 42)

13 Analysis of Biochemical Equilibria Relevant to the Immune Response 83 set α = in28) an 3)). Substitution of Z from 42) into4) leas to a quintic for L, which factorizes in the special case L = R + /2, with a triple factor 2L ) emerging: 0 = 2L ) 3 4L 2 2LK intra + 2R ) + K intra + 2R ) ). 43) This equation has five real roots for L: L = /2 with multiplicity 3), an two others, neither of which are physically relevant they violate the convergence conition 0 < 4LZ <, leaing to Z = /2in42)). Again, with L = R +/2 an L = /2, Z is unefine in 42) an we nee a local analysis. Setting L = R + /2 + ɛ, L = /2 + ɛl + ɛ 2 L 2 +, ɛ in43), we can etermine L, L 2, successively; substitution in 42) then gives Z correct to orer ɛ just as in case ii), Z = L )/2 + L )) + Oɛ)). The correction L is foun to satisfy a cubic equation, 2K intra + 2R ) ) L K intra + 2R ) ) L K intra) L + 2K intra = 0. 44) With L = /2 + ɛl + Oɛ 2 ), Z = L )/2 + L )) + Oɛ), only one of the three roots L satisfies the convergence criterion 0 < 4LZ < ; this is the real root of 44) that lies in the interval 0, ). We substitute the physically-relevant expressions in = Z + 2L) 2 45) R which follows from 26) with I = 0 an α = ). Again explicit calculation reveals a local maximum in at ɛ = 0L = /2). The value of at this maximum is obtaine by setting L = R + /2, L = /2, Z = L )/2 + L )) in 45), giving,max = 2 L ) R + L ), 46) with L the unique root of 44) lying in 0, ). Ientifying the coefficients in 44) by c 0 = 2K intra + 2R ), c = 2 + K intra + 2R ) ), c 2 = + 2K intra, c 3 = 2K intra, the roots are foun in terms of parameters Q = 3c 2c 0 c 2 9c0 2, R= 9c 0c c 2 27c0 2c 3 2c 3 54c0 3, D = Q 3 + R 2, S = R + D ) /3 ) /3,, T = R D with ST = Q). The three solutions are L, = S + T c, 3c 0 L,2 = 2 S + T) c + i 3 S T), 3c 0 2 L,3 = 2 S + T) c i 3 S T), 3c 0 2

14 84 L.J. Cummings et al. an here D 0 always, so all roots are real. Which root lies in 0, ) varies accoring to the values of K intra an R. The results may be summarize as follows: Consiere as a function of al ligan concentration L, the maximum value of occurs for L = R +/2. The maximal value at this value of L is given by the following expressions from 46)): 0 <K intra < 2 + 2R ) :,max = 2 L,3) R + L,3 ), 47) K intra > 2 + 2R ) : f,max = 2 L,) R + L, ). 48) 4.2. Determining the Dissociation Constants As explaine for case ii) above, if cooperativity is close to unity, these results can be use to estimate the issociation constants K inter an K intra.thevalueofk inter is obtaine exactly as explaine at the en of Sect. 4., using the relation L = R + K inter 2 that hols when the concentration of complexes of size n 2, C, is maximize. The measure value of Cmax must equal R f,max, where,max = Cmax /R is givenby47) or48). Either of these relations gives the value of L at the maximum remember that R = R /K inter ). It is known that L satisfies 44); with L an R known, 44) is a linear equation for the remaining unknown, K intra,giving Hence, finally, K intra K intra = 4.3 A Posteriori Checking L L ) L )[L 2 + 2R ) ]. = K inter K intra. Without prior information, it is ifficult to know whether or not the cooperativity α. Some form of check may be provie by carrying out more than one experiment of the kin escribe in Sects. 4.., 4.2., with ifferent values of R. Each such experiment shoul yiel the same or approximately the same) values for the issociation constants K inter, K intra the first is easily checke by looking at the values of L R ) at the maximum of for all experiments; the values shoul be the same). Large eviations in the values obtaine for K inter an K intra woul suggest that α is not close to unity. However, if the values are close for all experiments, then the approximation is likely goo. 5 General Case Numerical Simulations Before consiering asympic limits that permit analytical progress enabling explicit preictions to be mae) we first present sample numerical computations that

15 Analysis of Biochemical Equilibria Relevant to the Immune Response 85 Fig. 2 Fraction of complexes with more than one bivalent receptor ) as a function of the al concentration of bivalent ligan L. Each curve correspons to a ifferent value of the al concentration of bivalent receptor R ; from the bottom curve upwar the values are: R = 0.0, 0.,, 0, 00 allow us to investigate the effect of iniviual system parameters for the general case in which 2) 24) hol. Such simulations corroborate the analytical results of the preceing section, an provie guiance as to which asympic limits might usefully be investigate. 5. Depenence on Normalize Total Receptor Concentration R Figure 2 shows the fraction of complexes with more than one bivalent receptor ) as a function of the al concentration of bivalent ligan L, at fixe values of the intramolecular issociation constant Kn intra an the cooperativity parameter α. This simulation correspons to the analysis of Sect. 4.2, since we simulate the simple case in which al monovalent ligan inhibitor) concentration I = 0; cooperativity α =, an K intra =, with all other Kn intra = K intra constant for n 2 short bivalent ligan). For this figure, we set K intra = 0., an use five ifferent values of normalize bivalent receptor concentration R as etaile in the caption. The figure bears out the analytical fining of a maximum in at L = R + / Depenence on Cooperativity Parameter α Figure 3 shows as a function of al bivalent ligan concentration L,atfixe values of intramolecular issociation constant K intra an al bivalent receptor concentration R, an al inhibitor concentration I = 0. We again consier case i) short bivalent ligan), with K intra =, an all other Kn intra = K intra = 0. n 2), an R = 0.. The plot shows curves for six ifferent values of the cooperativity α; from the bottom curve upwars the values are: α = 0.,, 0, 00, 000, We observe that the curves become left right asymmetric as α becomes large. This effect is not observe with the moel of Dembo an Golstein 978), an is attributable

16 86 L.J. Cummings et al. Fig. 3 Fraction of complexes with more than one bivalent receptor f ) as a function of the al concentration of bivalent ligan L. Each curve correspons to a ifferent value of cooperativity α; from the bottom curve upwar the values are: α = 0.,, 0, 00, 000, 0000 to our ealing explicitly with the concentration of free ligan L, which those authors assume fixe. 5.3 Depenence on Intramolecular Dissociation Constant K intra Figures 4, 5 show as a function of al bivalent ligan concentration L,atfixe values of α an R. Total receptor concentration R = 0.; al inhibitor concentration I = 0, an cooperativity α = in both plots. For Fig. 4, we consier case i), with intramolecular issociation constants K intra =, an all other Kn intra = K intra, constant for n 2; thus we are in the situation of Sect. 4.2 short bivalent ligan). The plot shows curves for seven ifferent values of K intra as etaile in the caption. For Fig. 5, we consier case ii), with all Kn intra equal for n ; thus we are in the situation of Sect. 4. long bivalent ligan). The plot shows curves for six ifferent values of K intra as etaile in the caption. Again, both Figs. 4 an 5 corroborate the analytical result of a maximum in at L = R + /2 with no inhibitor an α =. The epenence on K intra varies ramatically in the two cases i) an ii), with opposite trens being observe. In case i), as we ecrease the issociation constant K intra, the cyclic complexes are much more likely to form, an here, since the singleton cyclic complex C ) cannot form, cyclic complexes with size n 2 preominate in fact, the n = 2 cyclic complex will ominate all others). Hence, increases. In case ii), as we ecrease K intra it is easier for all cyclic complexes, incluing the singleton, to form. Since the singleton forms first, an is unlikely to issociate at small K intra, this will be the ominant complex, meaning that which measures complexes containing two or more receptors) goes to zero. This is iscusse in etail in Sects. 6. an 6.2 below.

17 Analysis of Biochemical Equilibria Relevant to the Immune Response 87 Fig. 4 Fraction of complexes with more than one bivalent receptor f ) as a function of the al concentration of bivalent ligan L. Here, the intramolecular issociation parameter for mono-receptor complexes K intra =, while the intramolecular issociation constant for complexes with more than one bivalent receptor are all equal, K intra n> = Kintra case i)). Each curve correspons to a ifferent value of K intra ; from the bottom curve upwar the values are: K intra = 00, 0,, 0., 0.0, 0.00, Fig. 5 Fraction of complexes with more than one bivalent receptor f ) as a function of the al concentration of bivalent ligan L. Here, all the intramolecular issociation constants K intra are equal case ii)), an each curve correspons to a ifferent value; from the bottom curve upwar the values are: K intra = 0.00, 0.0, 0.,, 0, Depenence on Normalize Total Inhibitor Concentration I Figure 6 shows f as a function of al bivalent ligan concentration L,atfixe values of K intra, K mono the imensionless form of the issociation constant relating to complexes between monovalent inhibitor an bivalent receptor, as escribe in Sect. 2), an R. Total inhibitor concentration I iffers for each curve in the plot.

18 88 L.J. Cummings et al. Fig. 6 Fraction of complexes with more than one bivalent receptor ) as a function of the al concentration of bivalent ligan L. Each curve correspons to a ifferent value of the al concentration of monovalent ligan inhibitor) I ; from bottom curve upwar the values are: I = 00, 0,, 0., 0.0, 0.00 We assume case i), short bivalent ligans, with K intra =, an all other Kn intra = 0. for n 2; the normalize inhibitor issociation constant K mono =, an R = 0.. The plot shows curves for five ifferent values of I as etaile in the caption. Strong inhibition suppresses the number of large complexes forme, as we woul expect, an also shifts the value of the maximum complex concentration to larger values of ae ligan. 6 Asympic Analysis The general case governe by 2) 24) is highly nonlinear, but consierable analytical progress may be mae usefully in certain asympic limits. We explore these below, noting the physical interpretation of the limits, an comparing results to the numerical computations of the previous section. 6. K intra = ɛ Asympically Small, Case i) Short Bivalent Ligan) If the parameter K intra is small cyclic complexes are favorable an form preferentially compare with linear complexes. In case i), ligans are short, an the singleton cyclic complex cannot form, so the ominant complex is the cyclic 2-form C 2 ). Figure 4 suggests that this asympic limit is nontrivial an might yiel a useful preictive result. 2 We consier K intra = ɛ in2), 23), an 24). Solutions epen on the 2 In case ii), consiere next, the ligan is long enough to form cyclic singletons, an this is the ominant complex in the equilibrium mixture; Fig. 5 suggests that in this limit 0 uniformly in K intra an that therefore this limit may be ifficult to use preictively.

19 Analysis of Biochemical Equilibria Relevant to the Immune Response 89 relative proportions of receptor an ligan present in the original mixture, since one of these will be entirely use up at leaing orer), so that either L orz. 6.. L <R In this case, all ligan is boun up in cyclic complexes, leaving only asympically small amounts of free ligan: L, Z = O), I = O), so that the asympic expansions procee as Z = Z 0 + o), L = o), I = I 0 + o). Equations 23) an 24) are simple at leaing orer, R L = Z 0 + I ) 2 0, I = I 0 + 2I 0Z 0 K mono K mono Eliminating Z 0, I 0 satisfies a quaratic equation, + I 0 K mono I0 2 + I 0 2R L ) I + K mono) I K mono = 0, 49) with unique positive solution I 0. Hence, Z 0 = R L + I 0 K mono ). 50) 2 Stuy of 2) suggests that ligan concentration L scales with ɛ /2 : L = ɛ /2 L 0 + Oɛ), an 0 is given by the leaing-orer terms in 26). Since L, it is clear that in this case 0 is given by 0 = L 0, ), R an we o not nee to calculate L 0 unless we wish to etermine higher-orer terms. This expression for 0 makes intuitive sense: with short ligans, but cyclic complexes favore, the ominant reaction will be the formation of the cyclic 2-complex, limite only by availability of ligan when L <R. So, if all available ligan ens up in cyclic 2-complexes then these will constitute all complexes of size n 2, an hence account for the entirety of 0. To fin the next orer correction, we expan variables in powers of ɛ /2 : Z = Z 0 + ɛ /2 Z +, L= ɛ /2 L 0 +, I = I 0 + ɛ /2 I +, in 2), 23), 24), an in the expression 26) for. After solving for Z, L 0, an I in terms of known) leaing orer quantities, substitution in 26) gives ɛ /2 L = L + R 2 2R K mono ) 2 L R ) α [ I 0 2R L ) 2K mono) I K mono) + K mono) 2 4αR L ) 2 4R L ) ) I K mono] + Oɛ), 5) ).

20 90 L.J. Cummings et al. Fig. 7 Comparison of numerical soli lines) an asympic solutions for f as a function of al concentration of bivalent ligan L. The four ifferent asympic regions are istinguishe as from left to right) L <R : ashe; L R : otte; L >R : ash-otte; L R : ashe. Parameter values are: al concentration of bivalent receptor R = 0.25; al concentration of monovalent inhibitor I = 0; intramolecular issociation constant K intra = 0 5 inepenent of the complex size, case ii)); monovalent issociation constant K mono = 0.25; an cooperativity parameter α = with I 0 as given by 49). For ɛ = K intra, this expression gives very goo agreement with the full numerical solution in the region L <R see the whole range of asympic approximations constructe in Fig. 7) L >R Here, ligan is in excess, so to leaing orer) all receptor is boun up in cyclic complexes, leaving only asympically small amounts: Z, L = O), I = O). Leaing-orer behavior is trivial, 23) an 24)giving L 0 = L R, I 0 = I 52) the only free ligan is the excess ligan; an since cyclic complexes are favore, inhibitor is left with nothing to bin to). The leaing-orer balance in 2) isr 8αL 2 Z 2/ɛ, suggesting that Z scales as ɛ /2 : Z = ɛ /2 Z 0 + Oɛ), an L = L 0 + ɛ /2 L + Oɛ), I = I 0 + ɛ /2 I + Oɛ). This leaing-orer balance, together with 52) gives R Z 0 = 8α L R ). Equations 23) an 24) at orer ɛ /2 give expressions for the corrections L, I, an substitution of the expansions into 26) gives the fraction of receptor boun up in complexes of size n 2 correct to orer ɛ /2 )as = ɛ/2 2L R ) + + 8αR L R ) I K mono ) 2 + Oɛ). 53)

21 Analysis of Biochemical Equilibria Relevant to the Immune Response L R The asympic expansions constructe in the regions L <R an L >R break own when L R. In this case, at leaing orer, all free receptor an ligan are use up in making the cyclic complex, so both Z an L are asympically small; an no inhibitor can bin to receptors. While many possible balances coul be explore, the crossover region between L <R an L >R is aequately escribe by stuying the asympic regime Equations 2) an 23) lea to L 0 = Z 0 which together yiel Z 0 = L = R + ɛ /4 l, Z = ɛ /4 Z 0 +, L = ɛ /4 L 0 +, I = I +. R 8α ) /2, L 0 = l + Z 0 + I ) 2 K mono, { [ ) /2 2 + I l + l 2 K ) R + I ) 2 ] /2 } 8α K mono, 54) where recall l = L R )/K intra ) /4 = O). The expression 26) for then gives = ɛ /4 Z 0 + I R K mono 2 + I )) K mono + O ɛ /2), 55) with Z 0 as given by 54) above L R When L is asympically large, the scaling for Z changes. With L = O/ɛ /2 ), leaing orer in 2) gives ) Z R 4αLZ ) 2 4L 2 + 8αL2 Z 4αLZ ). ɛ The exact balance of terms here epens on the size of LZ which, recall, must always be less than /4α)). Note first that 4αLZ ) cannot be asympically small, since this can give no balance of terms. Neither can we have both LZ an 4αLZ ) orer-one, since then 4L 2 Z / 4αLZ ) 2 with nothing to balance it. So, we must have LZ, an R 4L 2 Z + 8αL2 Z 2, ɛ this new balance is where we will first epart from the previous case, where R 8αL 2 Z 2/ɛ 4L2 Z ). The asympic expansions now procee as

22 92 L.J. Cummings et al. L = L 0 ɛ /2 + L + ɛ /2 L 2 + Oɛ), Z = ɛz 0 + O ɛ 3/2), I = I 0 + ɛ /2 I + Oɛ), an also L = L /ɛ /2, with L O). Leaing-orer analysis of 2) 24) yiels L 0 = L, I 0 = I, Z 0 = 4α + + 2αR ) /2 4α L 2, an substitution in 26) gives the leaing-orer expression for as = + L 2 + 2αR ) /2 ) αr L 2 + o). 56) Figure 7 shows all four asympic expressions for, plotte as functions of L ) against the full numerical solution, on the appropriate omains of L.Asthe figure shows, to the orers obtaine, the asympic expressions are quite accurate on their regions of applicability Determining the Dissociation Constants Since a primary goal of our analysis is to erive results that experimentalists might use to extract information about a given system, we now consier how the expressions erive above can be use if it is suspecte that K intra = K intra /K inter. The most easily-ientifie experimental regimes in Fig. 7 appear to be L <R, an L R ; accoringly, we focus on these regimes. We again consier a titration experiment in which ligan is ae to a receptor/inhibitor mixture, an the concentration, C = R, of complexes of size n 2, is measure at selecte stages. In the regime L R, the late stages of a titration, the expression 56) is vali, an thus in the imensional variables C = R + K intra L 2 αk inter ) 2 or, rearranging to make α/k intra α K intra = 2L 3 [ the subject, K inter ) 3 C R ) 2 + 2αK inter K intra R L R + C R ) L K inter ) /2 ], 57) ). 58) In a given experiment, the quantities C, R, an L can be measure at any stage, an we woul like to extract the unknowns K inter, K intra, an α that appear in 57), 58). Suppose we carry out an experiment with a fixe amount of receptor R, an make measurements at two ifferent concentrations of ae ligan, L an L 2.We measure the two corresponing values of complex concentration, C an C 2.The expression 58) applies to each measurement: if we take the ratio of the two expressions for the two measurements, we obtain a relation that contains only one unknown, K inter. Solving this relation for K inter an simplifying, we fin

23 Analysis of Biochemical Equilibria Relevant to the Immune Response 93 K inter = R [L 3 C 2 R )2 L 3 2 C R )2 ] C R )C2 R )[L 2 2 C R ) L 2 C 2 R )]. 59) With K inter etermine, we can now return to 58) an use either of the experimental measurements to etermine the ratio α/k intra. We cannot, however, etermine the two constants separately using these results. To make further progress we consier another experiment with less ligan than receptor, L <R, so that the expression 5) is vali this coul simply be a measurement taken from the earlier stages of the same experiment iscusse above). Consier first an experiment in which no inhibitor is present. Equation 5) then simplifies, an in terms of imensional quantities takes the form ) /2 L )/2 K intra 2α C = L 2 K inter R L ) [ 4α R ) 2 L 4K inter R L ) ) K inter 2 ]. 60) For a given measurement of C, at a ligan concentration L <R, since we alreay know K inter an K intra /α by the proceure outline above, this relation 60) is reaily solve for α. Finally, if we wish also to etermine the issociation constant K mono,wemust carry out a further experiment with nonzero concentration of monovalent ligan inhibitor), I 0. Rewriting 5) in imensional form, the only unknown is now K mono. The problem reuces to solving a cubic equation for K mono, the physicallyrelevant root of which must be selecte. 6.2 K intra = ɛ Asympically Small, Case ii) Long Bivalent Ligan) In this case, regarless of the relative sizes of L an R, the fraction of receptor boun up in complexes of size n 2,, is always asympically small. This makes intuitive sense because if there is less ligan than receptor in the initial mixture then all available ligan is immeiately taken up to form the cyclic -complex, leaving none free to form larger complexes. Available inhibitor can bin to remaining receptor that is not taken up in singleton cyclic complexes C ). On the other han, if there is more ligan than receptor in the initial mix then it will bin all available receptor into singleton cyclic complexes; there will be excess free ligan, an no inhibitor will be able to bin to receptor. No receptor is left over that can form any of the larger complexes. Since is always asympically small an reliable estimates of issociation constants woul therefore be ifficult to obtain), we o not pursue this case in etail; but we briefly emonstrate the above statements in the cases L <R, L >R, below L <R Here, stuy of 22) reveals ligan concentration L to scale with ɛ: L = ɛl 0 + Oɛ 2 ), an 0 is given by the leaing-orer terms in 27). In this case, we cannot simply neglect all terms in L at leaing orer, since 27) contains a term in L/ɛ; so we nee to fin L 0. This we o from 22), which gives

24 94 L.J. Cummings et al. [ R = Z 0 + 2L 0 + I 0 K mono 2 + I )] 0 K mono. 6) The fraction of receptor boun up in complexes of size n 2, 0, is then given to leaing orer) by 0 = Z [ 0 + 2L 0 + I 0 R K mono 2 + I )] 0 K mono = 0, using 6) in the final step. The ominant complex forme is the -cyclic complex; no complexes of size n 2 are forme to leaing orer) L >R In this case, ligan is present in excess an free-receptor concentration is asympically small. The leaing-orer balance in 22) isr 2LZ /ɛ, suggesting that Z scales as ɛ. It is then easily euce from 23) an 24) that Z = ɛz 0 + Oɛ 2 ), L = L R + Oɛ) an I = I + Oɛ);22) then gives Z 0 = R /2R L )). Hence, in 27), retaining only leaing orer terms, 0 = High Cooperativity α, Case i) Short Bivalent Ligan) Here, the formation of large complexes is favore. Figure 3 suggests that this is a nontrivial asympic limit worth exploring, which may yiel useful preictions. In this case, the conition 4αLZ < shows that one or both of L or Z must be asympically small. Again, the asympic scalings epen on the relative quantities of receptor an ligan in the system L <R With L <R receptor is in excess, an all free ligan is taken up to form larger complexes. In this regime, L is asympically small, while Z = O). We know that 0 < 4αLZ <, but we o not know whether this quantity is O) or asympically small. However, the ominant balance in 2) an 23) noting that αl 2 Z must be asympically small because αlz is at most O)) is Z R + I ) 2, 62) 4αLZ ) 2 K mono Z R L + I ) 2 4αLZ ) K mono, 63) showing that take the ratio of 63) an 62)) 4αLZ L. R So, with Z = O), L is of orer /α, an asympic expansions procee as Z = Z 0 + α Z +, L= α L 0 +, I = I 0 + α I +.

25 Analysis of Biochemical Equilibria Relevant to the Immune Response 95 Substitution into 2), 23), 24), an retaining only leaing-orer terms gives Z 0 = x2, L 0 = L ) R 4x 2, I 0 = K mono R L, 64) x where x is the unique positive root of 2 I K mono x2 + x K mono 2 ) K mono R L ) + R L ) = 0. The concentration of nontrivial complexes,, efine in 26), then has a simple leaing-orer expression in terms of L : = L ) 2 + o). 65) R L >R In this case, ligan is in excess, an all available free receptor is boun, leaving only asympically small amounts. Free ligan an inhibitor are present in O) concentrations, however. With R = O), 2) shows that 4αLZ must also be asympically small. Careful consieration of 2) an 23) reveals the correct scalings to be Z = α Z 0 + α 3/2 Z +, L= L 0 + α /2 L +, I = I ) α /2 I +, an substitution of these expansions into 2) an 23) leas to ) Z = 4αL R ) + O α 3/2, L= L R + O I = I + O α /2 ). Equation 26) then gives correct to O/α) as f = 4αR L R ) + 2L R ) + I K mono ) α /2, 67) ) ) This expression is clearly not vali as L R,orasL becomes asympically large. In these cases, a separate analysis is require below) L R When al ligan an receptor concentrations are approximately equal, the concentrations of free ligan an receptor are both asympically small, an the scalings iffer from the cases L <R an L >R consiere in Sects an above. Stuy of 23) shows that, as we transition between these cases, we require

26 96 L.J. Cummings et al. Z 4αLZ. The transition region may be aequately escribe by the istinguishe limit in which R L =Oα /4 ), L Oα /2 ), Z Oα /2 ), 4αLZ Oα /4 ), an I O). With expansions proceeing as Z = Z 0 α /2 + Z α 3/4 +, L= L 0 α /2 + L α 3/4 +, I = I 0 + I, 69) α/4 an 4αLZ = O α /4), 70) the conition 70) givesl 0 Z 0 = /4; 24) trivially gives I 0 = I at leaing orer, while 2) an 23) give, respectively, Z 0 R = 6L 0 Z + L Z 0 ) 2 + I ) 2 K mono, Z 0 α /4 R L = + I ) 2 4L 0 Z + L Z 0 ) K mono. Together these results give Z 0 = α/2 R L ) 2 R + I K mono ), L 2 0 =, I 0 = I. 7) 4Z 0 While these asympic solutions for Z, L, an I are clearly ifferent to those foun in regions L <R an L >R, we nonetheless fin on substitution in 26)) that = L ) 2 + o) vali when L R O α /4), 72) R an expression ientical to 65) from the region L <R L R The expressions erive in Sect for L >R cease to be vali when L Oα). Since ligan is greatly in excess assuming R = O)), L = Oα) also, an writing L = α L where L O)) we assume asympic expansions Z = Z 0 α 2 + Z α 3 +, L= αl 0 + L + L 2 α +, I = I 0 + I α + the scaling for Z comes from the requirement that 4αLZ O)). Leaing orer in 2) then gives R = while Oα) an O) in 23) give 4Z 0 L 2 0 4L 0 Z 0 ) 2, 73) L 0 = L, R = 4L2 0 Z 0 4L 0 Z 0 L