The maximum likelihood degree of a chemical reaction at the equilibrium
aa r X i v : . [ s t a t . O T ] A ug THE MAXIMUM LIKELIHOOD DEGREE OF A CHEMICALREACTION AT THE EQUILIBRIUM
SIMONE CAMOSSO
Abstract.
The complexity of a maximum likelihood estimation is measuredby its maximum likelihood degree ( ML degree). In this paper we study themaximum likelihood problem associated to chemical networks composed byone single chemical reaction under the equilibrium assumption. Contents
1. Introduction 12. Preliminaries 22.1. The Maximum likelihood estimation problem (MLE) 22.2. Chemical reactions 32.3. Chemical equilibrium 33. Results 44. Conclusions 9References 91.
Introduction
The maximum likelihood estimation (MLE) is a method of estimating the param-eters of a statistical model given observations. MLE problems appear frequently inexperimental sciences. Examples of this diffusion are [1] and [5]. In these works theauthors consider substances in small concentrations and a discrete random model forchemical reactions (“chemical networks”). Using the maximum likelihood methodand numerical tecniques they found an estimation of the rate constants associatedto the chemical model. Inspired by these works we offer a new point of view on thetopic.This article is based on certain analogies between the chemical language andthe algebraic statistical formalism. The aim here is modest compared to the worksmentioned earlier and, in what follows, we will reduce our analysis to consider toymodels. To begin, let us list the main differences between our assumptions andthose adopted by these authors. First of all our model is not discrete as in [1] andconcentrations are not only “small”. Second, we want to use algebraic statisticmethods instead numerical. Third, we assume the initial concentrations know andthe work is done in order to determine a theoretical index (the maximum likelihooddegree, denoted by
M L degree) associated to the chemical kinetics. There is a lastassumption that concerns the situation of “equilibrium” where the chemical reaction ∗ e-mail : [email protected] s in a privilegiated condition from the kinetics point of view. The key idea is tointerpret the concentration of some chemical substance as a “frequency” and wewill see how it is possible to associate to a chemical process a MLE problem. Thisformal trick permits to consider a large number of examples. Results are obtainedusing the methods from the algebraic statistics (as references the reader can consult[11], [13], [14], [7], [12], [15] and [6]) with the auxiliary support of a math software asMaple. Conclusions and considerations are discussed in the last part of the paper.In the preliminaries section necessary math and chemical notations are introducedand explained. 2. Preliminaries
The Maximum likelihood estimation problem (MLE).
In algebraicstatistic a statistical model is a subset of ∆ n = { p = ( p , . . . , p n ) ∈ R n +1 : p , . . . , p n > , p + . . . + p n = 1 } called the probability simplex. The real num-bers p , . . . , p n are frequencies and given a statistical model we shall consider theZariski closure in P n denoted by V as the complex solutions of a system of homo-geneous polynomial equations. The maximum likelihood problem consist to find( p , . . . , p n ) in the model V > = V ∩ ∆ n which “best explains” the parameter u = ( u , . . . , u n ) ∈ N n +1 . This can be obtained maximazing the function: L u = p u · · · p u n n ( p + . . . + p n ) u + ··· + u n , (2.1)with the constraint that p ∈ V > . Let λ = P ni =0 u i be the dimension of the sample,the problem can be solved using the method of Lagrange multipliers. Furthermorewe present another formulation of the same problem in terms more “algebraic”.Let H = { ( p , . . . , p n ) ∈ P n : p · · · p n ( p + . . . + p n ) = 0 } be the arrangiament of n + 2 hyperplanes then we are interested for critical points of L u in P n \ H . We alsorestrict our attention for regular points of the model V reg = V \ V sing . We have allelements to define the maximum likelihood degree ( M L ) associated to a statisticalmodel.
Definition 2.2.
The maximum likelihood degree
M L of V is the number of com-plex critical points of L u on V reg \ H , for some u .In particular for the case of a curve in P the following theorem tell us how tocalculate the M L degree.
Theorem 2.3.
Let V be a smooth curve of degree d in P and a = X ∩ H ) thenumber of points in the distinguished arrangement, then the M L degree of X is d − d + a . Observation 2.4.
We observe that if the curve V is sufficiently generic then a = 4 d and the M L degree is d · ( d + 1) as predict by B´ezout theorem for the equations: f ( p , p , p ) = 0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u p u p u p ∂f∂p ∂f∂p ∂f∂p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . Here f is the homogeneous polynomial generating the curve V .The previous theorem is a particular case of a general result for very affinevarieties (see [10]). heorem 2.5 (Huh) . If the very affine variety X \ H is smooth of dimension d ,then the M L degree is equal to ( − d χ ( X \ H ) . Chemical reactions.
A chemical reaction is represented by reactants placedto the left of an arrow and products placed to the right. We have that both reactantsand products are denoted by capital letters
A, B, C, . . . . The arrow in a chemicalreaction can be of three types ← , → and ↔ , denoting respectively the directionof evolution of the chemical process. The last case is usually used to denote asystem that is in equilibrium. Denoting by A , A , . . . , A n some reactant and by B , B , . . . , B m some product, we represent the chemical reaction with the followingequation: α A + α A + · · · + α n A n → β B + β B + · · · + β m B m , (2.6)where α , . . . , α n , β , . . . , β m are called stoichiometric coefficients. We define theorder of a chemical reaction the sum of α , . . . , α n . It is possible to define on theset of chemical substances an evaluation map [ · ] that gives the molar concentrationof a particular substance represented by the dot · . Another useful definition inchemical kinetics is the reaction velocity: v = 1 α i d [ A i ] dt , (2.7)where [ A i ] is the molar concentration of product or reactant A i and α i the stoi-chiometric coefficient in the reaction. For example the reaction: I + Br → IBr, has a reaction velocity that is the same whether we look at I ( α = − Br ( β = − IBr ( γ = +2): − d [ I ] dt = − d [ Br ] dt = 12 d [ IBr ] dt . Often, the reaction velocity can be written in terms of a rate law, a power lawin the reactant concentrations (or product concentrations), with a concentration–independent coefficient called the (direct) rate constant K d : v d = K d [ A ] α · · · [ A n ] α n , (2.8)or v i = K i [ B ] β · · · [ B m ] β m , (2.9)where the v d and v i stands for “direct” and “inverse” velocity. For details see [4].2.3. Chemical equilibrium.
There are situations in which both reactants andproducts are present but have no further tendency to undergo net change, these kindof reactions are called equilibrium reactions. We assume that we are in presence ofan equilibrium represented by the following equation: α A + α A + · · · + α n A n ↔ β B + β B + · · · + β m B m . (2.10)Thus we can write the two velocity associated to the kinetic system: v d = K d [ A ] α · · · [ A n ] α n , (2.11) nd v i = K i [ B ] β · · · [ B m ] β m . (2.12)By the equilibrium assumption we have the equality between v i and v d that canbe written as: K i [ B ] β · · · [ B m ] β m = K d [ A ] α · · · [ A n ] α n , and isolating the constants terms we find that: K i K d = [ A ] α · · · [ A n ] α n [ B ] β · · · [ B m ] β m . We denote the term K i K d by K e and call it the equilibrium constant associated tothe reaction (2 . Results
Proposition 3.1.
Let [ A ] and [ B ] be the concentrations of certain substances inthe following chemical equilibrium reaction of first order: A ↔ B, (3.2) then the M L degree is equal to for K e = − and for K e = − .Proof. Let x and y be quantities associated respectively to [ A ] and [ B ]. This is aline in P . We must study the M L degree of: X = V ( − y + K e x ) . Let ϕ : C ∗ → ( C ∗ ) be the map that p ( p , K e p ) with the constraint p (1+ K e ) = 1, then the likelihood–log function is L u ,u = u log p + u log K e p .Studing the critical points of the likelihood–log under the constraint we find that: p = u + u λ (1 + K e ) . The conclusion follows. (cid:3)
Proposition 3.3.
Let [ A ] , [ B ] and [ C ] be the concentrations of certain substancesin the following chemical equilibrium reaction of second order: A + B ↔ C, (3.4) then the M L degree is for K e = 4 , for K e = 0 and in the other cases.Proof. For the equation (3 .
4) the equilibrium constant is given by: K e = [ C ] [ A ][ B ] . Under the assumption that [ A ]+[ B ]+[ C ] = c with c >
0, calling x = [ A ] c , y = [ B ] c and z = [ C ] c we have: x + y + z = 1 . he variety of interest is: X = V ( K e xy − z ) , and the case of K e = 4 is the Hardy–Weinberg law (details are in [8], [3] and [9]).The case K e = 0 gives as points in the intersection: X ∩ H = { (1 : 0 : 0) , (0 : 1 : 0) } , so the M L degree is 0. What remain to examine is the case of K e = 0 ,
4. In thiscase we have the following equations system: ( K e xy = z − zu y + K e yu z + u K e − u K e + zu x − K e xu z = 0 . This leads to two solutions of the following form:( ε i : 1 : K e ε i ) , where ε i for i = 1 , K e z u − u K e +( − K e u − u + u + u K e ) z = 0. This proves that the M L degree is 2. (cid:3)
Proposition 3.5.
Let [ A ] and [ B ] be the concentrations of certain substances inthe following chemical equilibrium reaction: nA ↔ nB, (3.6) for n = 2 , then the M L degree is .Proof. Let x and y be quantities associated respectively to [ A ] and [ B ]. Let K e x n − y n = 0 be the equation defining X . In order to determine the ML degree we considerthe map ϕ : C ∗ → ( C ∗ ) that p ( p , n √ K e p ) with the constraint p (1+ n √ K e ) =1, then the likelihood–log function is L u ,u = u log p + u log n √ K e p . Studingthe critical points of the likelihood–log under the constraint we find that: p = u + u λ (1 + n √ K e ) . (cid:3) Proposition 3.7.
Let [ A ] , [ B ] , [ C ] and [ D ] be the concentrations of certain sub-stances in the following chemical equilibrium reaction: A + B ↔ C + D, (3.8) then the M L degree is .Proof. By the total conservation of the quantities [ A ] + [ B ] + [ C ] + [ D ] = c we set x = [ A ] c , y = [ B ] c , z = [ C ] c and t = [ D ] c . Our models is the well know independencemodel of [7] § X = V ( K e xy − zt ) ⊂ P . The variety X is isomorphicto P × P with coordinates (( K e x : y ) , ( z : t )). We have that: X \ H = P × P \ { K e xyzt ( K e x + y )( z + t ) = 0 } = ( P \ { K e xy ( K e x + y ) = 0 } ) × ( P \ { zt ( z + t ) = 0 } )= ( P \ { } ) × ( P \ { } )and by theorem 2 . M L degree is χ ( X \ H ) = ( − · ( −
1) = 1. (cid:3) bservation 3.9. We observe that the equilibrium constant K e is given by theArrenius formula: K e = e ∆ GRT , where T is the temperature, R the gas constant and ∆ G the Gibbs free energy. Forthis reason it makes sense only consider the case of strictly positive K e . Proposition 3.10.
Let [ A ] , [ B ] and [ C ] be the concentrations of certain substancesin the following chemical equilibrium reaction: A + B ↔ C, (3.11) then the M L degree is .Proof. By the total conservation of the quantities [ A ] + [ B ] + [ C ] = c , we set x = [ A ] c , y = [ B ] c and z = [ C ] c . The variety of interest is: X = V ( z − K e xy ) . For convenience we fix K e = 1. We have the following transformation ϕ :( C ∗ ) → ( C ∗ ) given by ( p , p ) (cid:0) p , p , p p (cid:1) with the constraint p + p + p p =1. We study the critical points of the log–likelihood function under the constraint: L u ,u ,u = 3 u log p + 3 u log p + u log p + u log p , (3.12)where u , u , u are a set of parameters. The critical equations are:3 u p + u p = λ (3 p + p ) , (3.13)and 3 u p + u p = λ (3 p + p ) . (3.14)We can denote the polynomial (3 .
13) by f and the polynomial (3 .
14) by g . TheSylvester matrix is:Syl( f, g, p ) = λ λp − u − u λp λp − u − u λp λp − u − u
00 0 λp λp − u − u , and the resultant Res( f, g, p ) = det (Syl( f, g, p )) is a polynomial of nine degreein p , for the fundamental theorem of algebra we have 9 solutions. (cid:3) Proposition 3.15.
Let [ A ] and [ B ] be the concentrations of certain substances inthe following chemical equilibrium reaction: A ↔ B, (3.16) then the M L degree is . roof. By the total conservation of the quantities [ A ] + [ B ] = c we set x = [ A ] c and y = [ B ] c . The variety of interest is: X = V ( y − K e x ) . For convenience we fix K e = 1. We have the following transformation ϕ :( C ∗ ) → ( C ∗ ) given by ( p ) (cid:0) p , p (cid:1) with the constraint p + p = 1. We studythe critical points of the log–likelihood function under the constraint: L u ,u = 3 u log p + 2 u log p , (3.17)where u , u are a set of parameters. The critical equation is:3 u p + 2 u p = λ (3 p + 2 p ) . (3.18)This is a polynomial in p of third degree and for the fundamental theorem ofalgebra there are 3 solutions. (cid:3) Proposition 3.19.
Let [ A ] , [ B ] and [ C ] be the concentrations of certain substancesin the following chemical equilibrium reaction: nA + mB ↔ pC, (3.20) then for n = m = p = 2 the M L degree is , n = m = 2 and p = 1 the M L degreeis , n = p = 1 and m = 2 the M L degree is , n = m = p = 3 the M L degree is .Proof. For convenience we fix K e = 1. We have the following transformation ϕ :( C ∗ ) → ( C ∗ ) given by ( t , t ) ( t p , t p , t n t m ) with the constraint t p + t p + t n t m =1. We study the critical points of the log–likelihood function under the constraint: L u ,u ,u = ( pu + nu ) log t + ( pu + mu ) log t , (3.21)where u , u , u are a set of parameters. From the critical equations we find thetwo polynomial: f = λpt p + nλt n t m − nu − pu , (3.22)and g = λpt p + mλt n t m − mu − pu . (3.23)We start considering the case n = m = p = 2 with f ( t , t ) = 2 λt + 2 λt t + a , g ( t , t ) = 2 λt + 2 λt t + b , a = − nu − pu and b = − mu − pu . The Sylvestermatrix is:Syl( f, g, t ) = λ + 2 λt a
00 2 λ + 2 λt a λt λt + b
00 2 λt λt + b , withRes( f, g, p ) = 16 λ t + 16 λ t b + 4 λ b + 32 λ t + 32 λ t b + 8 λ t b − λ t a − λ t ab + 16 λ t + 16 λ t b + 4 λ t b − λ t a − λ t ab + 4 λ t a , a polynomial of eight degree in p and for the fundamental theorem of algebra wehave 8 solutions.For the case n = m = 2 and p = 1 with f ( t , t ) = λt + 2 λt t + a , g ( t , t ) = λt + 2 λt t + b , a = − nu − pu and b = − mu − pu , the Sylvester matrix is:Syl( f, g, t ) = λt λ a
00 2 λt λ a λt λt + b
00 2 λt λt + b , withRes( f, g, p ) = 4 λ t +8 λ t b +4 λ t b − λ t a − λ t ab +2 λ t +2 λ t b +4 λ t a , a polynomial of six degree in p and for the fundamental theorem of algebra wehave 6 solutions but with only 4 different from zero.In the case n = p = 1 and m = 2 with f ( t , t ) = λt + λt t + a , g ( t , t ) = λt + 2 λt t + b , a = − nu − pu and b = − mu − pu , the Sylvester matrix is:Syl( f, g, t ) = (cid:18) λ + λt a λt λt + b (cid:19) , with Res( f, g, p ) = λ t + λb + λ t + λt b − aλt , a polynomial with only 2 solutions different from zero.In the last case n = m = p = 3 with f ( t , t ) = 3 λt + 3 λt t + a , g ( t , t ) =3 λt + 3 λt t + b , a = − nu − pu and b = − mu − pu , the Sylvester matrix is:Syl( f, g, t ) = λ + 3 λt a λ + 3 λt a
00 0 3 λ + 3 λt a λt λt + b λt λt + b
00 0 3 λt λt + b , with Res( f, g, p ) = (cid:0) λ t + 3 λb + 9 λ t + 3 λt b − aλt (cid:1) , a polynomial with 9 solutions. (cid:3) Proposition 3.24.
Let [ A ] , [ A ] , . . . , [ A n ] and [ B ] , [ B ] , . . . , [ B n ] be the concen-trations of certain substances in the following chemical equilibrium reaction: A + A + · · · + A n ↔ B + B + · · · + B n , (3.25) then the M L degree is .Proof. We observe that the number or reactants is equal to the number of productsthat is n . We consider the following transformation ϕ : ( C ∗ ) → ( C ∗ ) n given by t (cid:16) t , . . . , t , n p K e t , . . . , n p K e t (cid:17) . roceeding in a similar way as other results we find that the M L must be 1. (cid:3)
Example 3.26.
As example we can consider the synthesis of ammonia at thepressure of 800 atm and at T = 500 ◦ C . At the equilibrium: N + 3 H ↔ N H . The transformation map ϕ is given ( t , t ) (cid:0) t , t , √ K e t t (cid:1) . As in the proofof previous results we consider the likelihood–log function: L u ,u ,u = (2 u + u ) log t + (2 u + 3 u ) log t + u log p K e , with the constraint t + t + √ K e t t = 1. The procedure leads to the determinantof the Sylvester matrix to be a polynomial of degree 8 with the numeric coefficientdifferent from 0. In this example the M L degree is equal to 8.4.
Conclusions
In the previous results the interpretation of chemical concentrations as “fre-quencies” leads to different examples of
M L degree problems. In each examplea solution has been proposed. The propositions provide a partial classification ofcertain chemical reactions by its
M L degree and we can observe qualitatively thegrowth of the
M L degree to varying complexity of chemical reactions. The fact thatno higher order reaction has been considered is due principally by the motivationthat reactions with high molecularity are “rare” because the probability of effectivecollision between particles decreases. Another interesting study regards chemicalreactions with half order or with no a “perfect” equilibrium, in adjoint we don’tknow how to treat the case when a reaction is composed by more steps in order toarrive to the final products.In other words how to treat the case of chemical networks? In [5] they intro-duced the multinomial model. The method used here works well only under theequilibrium assumption and it is not possible to use it for general chemical networks.In conclusion what emerges on this study is that the
M L problems are generallyconnected to the problem of solving polynomial equations in order to find projectivepoints. It is interesting that chemical reactions of high order seems rare in the sameway as to find solutions of higher degree equations is not quite obvious (we refer tothe Galois famous result on the solvability by radicals). In fact the
M L degree isthe degree of the extension K / Q ( u ) obtained adjoining all solutions of the likelihoodequations to Q ( u ). In this notation Q ( u ) is the field of rational functions and u isthe indeterminate vector of parameters u = ( u , . . . , u n ) as observed by [12] § References [1] A.Andreychenko, L.Mikeev, D.Spieler, V.Wolf, “Approximate maximum likelihood estima-tion for stochastic chemical kinetics”, EURASIP J. Bioinformatics and Syst Biology, (2012),2012(1): 9.[2] P.Atkins, J.de Paula, “Atkins’Physical Chemistry”, W.H. Freeman and Company New York(2006), Eighth edition, pp 200–202.[3] S.Camosso, “Considerations on the genetic equilibrium law” , IOSR Journal of Mathematics(IOSR-JM), Volume 13, Issue 1, Ver. I (Jan.-Feb. 2017), pp 01–03.[4] A.Cooksy, “Physical Chemistry: Thermodynamics, Statistical Mechanics, & Kinetics”, Pear-son (2014).[5] G.Craciun, C.Pantea, G.Rempala (2009), “Algebraic methods for inferring biochemical net-works: a maximum likelihood approach”, Comput. Biol. Chem. 33(5), pp. 361367.
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