Mathematical Model of a pH-gradient Creation at Isoelectrofocusing. Part IV. Theory
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Mathematical Model of a pH-gradient Creationat Isoelectrofocusing.Part IV: Theory
E. V. Shiryaeva, ∗ N. M. Zhukova, † and M. Yu. Zhukov ‡ Southern Federal UniversityRostov-on-Don, Russia (Dated: August 8, 2018)
Abstract
The mathematical model describing the non-stationary natural pH-gradient arising under theaction of an electric field in an aqueous solution of ampholytes (amino acids) is constructed. Themodel is a part of a more general model of the isoelectrofocusing (IEF) process. The presentedmodel takes into account: 1) general Ohm’s law (electric current flux includes the diffusive electriccurrent); 2) dissociation of water; 3) difference between isoelectric point (IEP) and isoionic point(PZC – point of zero charge). We also study the Kohlraush’s function evolution and discuss therole of the Poisson-Boltzmann equation.
PACS numbers: 82.45.-h, 87.15.Tt, 82.45.Tv, 87.50.ch ,82.80.Yc, 02.60.-xKeywords: isoelectrofocusing, mass transport ∗ Electronic address: [email protected] † Electronic address: zhuk˙[email protected] ‡ Electronic address: [email protected] . INTRODUCTION This paper continues the series of papers [9–11] about pH-gradient creation at isoelec-trofocusing (IEF). Here, the general mathematical model of IEF is obtained. The construc-tion of accurate electrophoresis mathematical models is described in the works [1–3, 6–8],in which, in particular, are described and classified the various methods of electrophoresis:zone electrophoresis, isotachophoresis, and isielectrofocusing. The models presented in theseworks are either very general or, on the contrary, describe very partial problems. Usually, atconstructing isoelectrofocusing model the simplifying assumptions are chosen. In particular,dissociation of water is not always taken into account, Ohm’s law does not include termsthat corresponds to diffusion current, etc . All such simplification can lead to the violationof the basic physical laws such as the law of conservation of mass or the law of conservationof electric charge. Despite the fact that the differences between isoelectric point (IEP) andisoionic (PZC) point are already described in [1, 2], this effect is usually omitted at theconstructing IEF. In [11] the results of the numerical investigation for IEF model are pre-sented. In this work it is shown that differences between IEP and PZC take the importantrole, especially for almost stationary regime. Mentioned effects are considered in this paper.Of course, the model is not complete. In particular, the influence of the ionic strength of asolution on the mobility of ions, the effect of Wien and others are not taken into account.The paper is organized as follows. In Sec. II we demonstrate the method of the generalmodel constructing. In Sec. II A we study the role of the difference between the isoelectricand isoionic points. In Sec. II B we obtain and study the Kohlraush’s function. Finally, inSec. II C we discuss the role of the Poisson-Boltzmann equation.
II. MATHEMATICAL MODEL OF IEF
To construct the mathematical model of IEF we use the theory of the local chemical equi-librium described in [1, 2, 8]. Generally, it is convenient to write the dissociation reactionsfor a solution consisting of n amphoteric substances as ( k = 1 , . . . , n ) a + k B k ⇋ a k + H + , a k A k ⇋ a − k + H + , (1)where a k is a zwitterion (‘neutral’ ion), A k and B k are dissociation constants for acid group a − k (negative ion) and based group a + k (positive ion) correspondingly.2he chemical kinetic equations have the following form da + k dt = r + k , da − k dt = r − k , da k dt = r k , (2)where r + k = − B + k a + k + B − k a k [H + ] , (3) r − k = − A + k a k + A − k a − k [H + ] ,r k = − r + k − r + k . Here, a + k , a − k , a k are the molar concentration; r + k , r − k , r k are the density of concentrationsources (we use the same symbol for denotation of the substance and its concentration),[H + ] is the analytical concentration of the hydrogen ion; A + k , B + k , A − k A − k are the velocitiesof the direct and reverse reactions.The ion concentrations are connect to analytical concentration of the substance a k withthe help of the relations: a + k = θ + k a k , a − k = θ − k a k , a k = a k + a + k + a − k , (4)where θ − k , θ + k are the dissociation degrees (further, we show that the dissociation degrees aredepend on [H + ] only, i. e. θ + k = θ + k ([H + ]), θ − k = θ − k ([H + ])).The mass transport under action of an electric field is described by the equations (di-mensionless variables): ∂ t a + k + div i + k = r + k , i + k = − εµ + k ∇ a + k + z + k µ + k θ + k ([H + ]) a + k E , k = 1 , . . . , n, (5) ∂ t a − k + div i − k = r − k , i − k = − εµ − k ∇ a − k + z − k µ − k θ − k ([H + ]) a − k E , k = 1 , . . . , n,∂ t a k + div i k = r k , i k = − εµ k ∇ a k , k = 1 , . . . , n, where i − k , i + k , i k are the flux densities, E is the intensity of electric field, z − k µ − k , z + k µ + k , εµ − k , εµ + k are the electrophoretic mobilities and diffusive coefficients of the ions, εµ k is thediffusive coefficients of the ‘neutral’ ion, z − k = − z + k = +1 are the ion charges (ion chargeper unit of the electron charge).We assume that µ − k = µ + k = µ k , εµ k = εµ − k = εµ + k = εµ k , k = 1 , . . . , n. (6)3n this case the summation equations (5) for each k give (see (3), (4)) ∂ t a k + div i k = 0 , k = 1 , . . . , n, (7) i k = − εµ k ∇ a k + µ k θ k a k E , k = 1 , . . . , n, (8)where a k = a k + a + k + a − k , (9) i k = i − k + i k + i + k ,θ k = θ k ([H + ]) = θ + k − θ − k . Here, θ k is the specific molar charge of the substance a k .Note, the equations (7) do not contain the density of source. Other words, these equationsare the conservative laws (not balance equations). The analytical concentrations a k areintegrals of the chemical kinetic equations (2).The system of equations (7), (8) is apparent, however we add two comments on therelated physical processes. First, the equations are written for the concentrations a k . Itmeans that these equations describe the distributions of some complex chemical substancesconsisting of ions and zwitterions (not ions and neutral substances separately). From thephysical viewpoint we deal only with such substances (not with their components) and we cannot (without employing of special methods) observe the components of the k -th substance.Second, the system (7), (8) is so called unclosed system, because the molar charge θ k isundefined (even if temporarily assume that the intensity of the electric field E is given).To close the system (7), (8) we use hypothesis of the local chemical equilibrium introducedin [1, 2] and developed in [8]. We assume that dissociation chemical reactions are very fast(which are completed almost instantly). It allows to believe that the conditions of thechemical equilibrium are valid: r − k = 0 , r + k = 0 , ( r k = 0) , (10)or a k [H + ] a + k = B + k B − k = B k , a − k [H + ] a k = A + k A − k = A k . (11)Solving this system we get the dependance of the dissociation degrees on hydrogen ionconcentration θ + k ([H + ]) = [H + ] [H + ] + B k [H + ] + A k B k , θ − k ([H + ]) = A k B k [H + ] + B k [H + ] + A k B k . (12)4e emphasize once more that this relations appear as the result of extremely fast chemi-cal reactions that are instant (mathematically) or much faster than any transfer processes(physically). In fact, we have two types of variables: fast variables ( a k , a − k , a + k ) and slowvariables ( a k ). The equations (4), (9)–(11) give connections between these variables.For further we need more additional equations. We must obtain the equation for deter-mining the electric field and the concentration of hydrogen ions.In the general case in addition to the reactions (1) one should take into account theautodissociation of water (as well as the autoionization of water or autoprotolysis)H O k + ⇋ k − H + + OH − . (13)At the local chemical equilibrium the concentrations of hydrogen ions (or hydronium ions)[H + ] and hydroxide ions [OH − ] are reflated as[OH − ] = k w [H + ] , (14)where k w is the autodissociation constant of water (the synonyms are: ionization constant,dissociation constant, self-ionization constant, and ion product of water; in dimensionalvariables k w = 10 − mol/l; it should be noticed here that water also represents an amphotericsubstance.).In chemistry the term autodissociation constant is used for K w = k w . However, theoret-ically it represents a confusion, since the standard dimension of the dissociation constant ismol/l. This is exactly the dimension of k w (not K w ).The electroneutrality equation has the following form n X k =1 ( a + k − a − k ) + [H + ] − [OH − ] = 0 . (15)Taking into account (4) we get n X k =1 (cid:8) θ + k ([H + ]) a + k − θ − k ([H + ]) a − k (cid:9) + [H + ] − [OH − ] = 0 . (16)This equation allows to determine the concentration [H + ].Obviously, the electric current flux densities of the ions have the following form j + k = z + k i + k = − εµ k ∇ a + k + µ + k a + k E , j − k = z + k i + k = + εµ k ∇ a − k − µ − k a − k E , (17)5 H = − εµ H ∇ [H + ] + µ H [H + ] E , j OH = + εµ OH ∇ [OH − ] − µ OH [OH − ] E , (18)where µ H , µ OH , εµ H , εµ OH are the electrophoretic mobilities and diffusive coefficients of thewater ions.Then, the electric current flux densities of the mixture is j = n X k =1 (cid:0) j + k + j − k (cid:1) + j H + j OH . (19)Taking into account (4) we get j = n X k =1 ( − εµ k ∇ ( θ k a k ) + µ k σ k a k E ) + (20)+ (cid:0) − εµ H ∇ [H + ] + µ H [H + ] E + εµ OH ∇ [OH − ] − µ OH [OH − ] E (cid:1) , where θ k = θ k ([H + ]) = θ + k − θ − k , σ k = σ k ([H + ]) = θ + k + θ − k . (21)Here, θ k is the specific molar charge of the substance a k , σ k is the specific molar conductivityof the substance a k .The constitutive relation (20 ) is so called generalized Ohm’s law which differs from theusual law by the presence of the diffusion terms (see also [11]).The current flux density j satisfy to the equation of the electric current continuitydiv j = 0 . (22)We also assume that the electric field is potential E = −∇ ϕ, (23)where ϕ is the electric potential.The equations (7), (8), (16), (22), (23), and constitutive relations (12), (14), (20), (21)are the complete system of equation that allows to determine the concentrations a k , [H + ],[OH − ], and potential ϕ .An important characteristic of the solution of amphoteric substances is pH; its value isdefined by the concentration of hydrogen ions H + expressed in mol / l with the use of therelation pH = − lg[H + ] .
6t is better to write this expression aspH = − lg (cid:18) [H + ] k w (cid:19) , (24)where k w is the autodissociation constant of water.In addition, instead of conventionally used function pH (that represents the measure ofthe acidity or alkalinity of a solution) we use acidity function ψ (that is linearly connectedwith pH), which is better adapted to our mathematical model[H + ] = k w e ψ , [OH − ] = k w e − ψ , (25)pH = − lg k w − ψ lg e. Usually, the value of pH varies from 0 to 14, which corresponds to the changes of ψ inthe interval from − .
118 to +16 . A k and B k use their negative decimaldegrees pA i , pB i that are given by relations:pA k = − lg (cid:18) A k k w (cid:19) , pB k = − lg (cid:18) B k k w (cid:19) . (26)We especially emphasize, that the replacement of the concentrations [H + ] and [OH − ] bythe acidity function ψ allows to write the system of equations in final form convenient forfurther mathematical investigation: ∂ t a k + div i k = 0 , k = 1 , . . . , n, (27) i k = − εµ k ∇ a k + µ k θ k ( ψ ) a k E , k = 1 , . . . , n, n X k =1 θ k ( ψ ) a k + 2 K w sinh ψ = 0 , (28)div j = 0 , E = −∇ ϕ, (29)where j = n X k =1 ( − εµ k ∇ ( θ k ( ψ ) a k ) + µ k σ k ( ψ ) a k E ) + (30)+2 k w µ ( − ε ∇ (sinh( ψ − ψ )) + cosh( ψ − ψ ) E ) ,θ k ( ψ ) = sinh( ψ − ψ k )cosh( ψ − ψ k ) + δ k = ϕ ′ k ( ψ ) ϕ k ( ψ ) , (31) σ k ( ψ ) = cosh( ψ − ψ k )cosh( ψ − ψ k ) + δ k = ϕ ′′ k ( ψ ) ϕ k ( ψ ) , k ( ψ ) = cosh( ψ − ψ k ) + δ k ,ψ k = 12 ln A k B k k w , δ k = 12 r B k A k , (32) µ = √ µ H µ OH , ψ = 12 ln µ OH µ H , where ψ i is the isoelectric point (electrophoretic mobility µ i θ i is equal to zero at ψ = ψ i , i.e. µ i θ i ( ψ i ) = 0), µ is the effective mobility of water ions, µ H , µ OH are the mobilitiesof hydrogen H + and hydroxide OH − ions, ψ is the value of ψ when water conductivity isminimal, δ i > ϕ k ( ψ ) is some auxiliary function.Specify connection parameters ψ k and δ k to the parameters used in chemistrypI k = 12 (pA k + pB k ) , pA k − pB k = 2 lg(2 δ k ) . (33)Here pI k is the electrophoretic point of amphoteric substance.The system (27)–(32) allows to determine concentrations a k , acidity function ψ , andelectrical potential ϕ (or electric field intensity E ) when parameters ε , µ k , µ , ψ k , ψ , δ k , k w are given.We should add comments on the roles of different equations in the system (27)–(32) aswell as different terms of these equations. The term 2 k w µ cosh( ψ − ψ ) in (30) describesthe contribution of water ions into the mixture conductivity, while the term 2 k w sinh ψ in(28) corresponds to the contribution of these ions into the mixture molar charge. As a rule,the contribution of water ions to the mixture conductivity and the charge of the mixture issmall enough, and these terms for the simplified models can be omitted (see, for example,[9–11]).The algebraic equation (28) represents the condition of the electroneutrality of mixture; itallows us to find ψ . In fact, this equation describes the instant control of medium properties(electrophoretic mobilities and molar conductivities) by the function ψ (that is linked to theconcentration of hydrogen ions or pH of mixture).We also assume that the maximal values of concentrations a k and the values µ , ψ , µ k , ψ k are all of the order O (1), while the parameters ε and k w are small. We should also mentionon some important properties of the physical processes. The absence of the concentrationflux ( i k = 0) does not mean that the k -th substance does not participate into an electriccurrent. 8or example, let us neglect the diffusion and take ε = 0 in (27) and (30). Then at theisoelectric point (when ψ = ψ k ) the charge θ k ( ψ k ) a k = 0 and the mobility µ k θ k ( ψ k ) = 0,so we get i k = 0. At the same time the density of the electric current at ψ = ψ k is j k = µ k σ k ( ψ k ) a k = 0. However this fact does not contain any contradiction since there aretwo equal (at ψ = ψ k ) but opposite fluxes i − k and i + k of the negative and positive ions thatboth are driven by the electric field. The flux of a k is i k = i + k + i − k = 0. The density ofthe electric current in this case is j k = z + k i + k + z − k i − k = 0. This fact plays a key role in thedescribing of the transport processes under action of an electric field. A. Difference between mobilities of the negative and positive ions
For more precise mathematical model we should take into account the difference betweenmobilities of the negative and positive ions, i. e. µ + k = µ − k (see (6)). In particular, if themobility of ions is different then the values of the function ψ , at which the molar charge andmolar mobility are equal to zero, are different. In fact, the molar charge is θ k ( ψ ) = θ + k ( ψ ) − θ − k ( ψ ) = sinh( ψ − ψ k )cosh( ψ − ψ k ) + δ k (34)and θ k ( ψ k ) = 0. The molar mobility is µ k Θ k ( ψ ) = µ + k θ + k ( ψ ) − µ − k θ − k ( ψ ) = µ k sinh( ψ − Ψ k )cosh( ψ − ψ k ) + δ k , (35)where µ k = q µ + k µ − k , ψ k − Ψ k = 12 ln µ + k µ − k . (36)We call Ψ k isoionic point. At ψ = Ψ k the quantity of the negative and positive ions ofsubstance is coincided.Difference between ψ k and Ψ k is well demonstrated by the example of water ions (see(32)): θ H O ( ψ ) = [H + ] − [OH − ] = 2 k w sinh( ψ − , (37)2 K w µ Θ H O ( ψ ) = µ H [H + ] − µ OH [OH − ] = 2 k w µ sinh( ψ − Ψ ) , Ψ = ψ . In the Tab. I the mobility of ions µ + k and µ − k are presented. Data in Tab. I are takenfrom software PeakMaster (see [13]) that includes a database based on Takeshi Hirokawa’stables with the data of many ions. 9 ABLE I: Parameters of amino acidspKb i pKa i pI i ψ i Ψ i ψ − Ψ i ∆pI i δ i µ − i µ + i Thr 2 .
14 9 .
200 5 . .
062 3 . − .
009 0 .
004 1694 .
22 3 .
09 3 . .
85 10 .
640 6 . .
739 1 .
633 0 . − .
046 12415 .
66 2 .
90 3 . .
25 9 .
857 6 . .
179 2 .
133 0 . − .
020 3180 .
31 3 .
22 3 . .
30 9 .
765 6 . .
228 2 .
183 0 . − .
019 2700 .
66 2 .
67 2 . .
26 9 .
728 5 . .
316 2 .
266 0 . − .
022 2710 .
00 2 .
64 2 . .
21 9 .
710 5 . .
395 2 .
353 0 . − .
018 2811 .
71 2 .
84 3 . .
13 9 .
262 5 . .
003 2 .
993 0 . − .
004 1840 .
64 2 .
69 2 . .
31 9 .
594 5 . .
413 2 .
409 0 . − .
002 2192 .
65 2 .
54 2 . .
13 9 .
344 5 . .
908 2 .
908 0 . − .
000 2022 .
88 2 .
93 2 . .
13 9 .
302 5 . .
957 2 . − .
003 0 .
001 1927 .
39 3 .
36 3 . .
10 9 .
224 5 . .
081 3 .
067 0 . − .
006 1823 .
77 2 .
88 2 . .
10 9 .
030 5 . .
304 3 .
298 0 . − .
003 1458 .
71 3 .
16 3 . β -Ala 3 .
42 10 .
241 6 . .
390 0 .
279 0 . − .
048 1286 .
68 3 .
08 3 . .
32 9 .
780 6 . .
188 2 .
160 0 . − .
012 2685 .
16 3 .
74 3 . B. The Kohlrausch’s function
In this section we obtain the analog of the Kohlrausch’s function for simplest IEF model( i. e. at K w = 0). The division of each equation (27) on the µ k and summarization over all k give R t − ε ∆ S = 0 , (38)where R = n X k =1 a k µ k , S = n X k =1 a k . (39)Here R ( x, t ) is the the analog of the Kohlrausch’s function.Especially interesting is the case, when µ k = µ , k = 0 , . . . , n . Then S = µR and takinginto account the boundary conditions and the initial conditions (see, for example, [9–11]) in10D case we have problem R t − εµR xx = 0 , R x (0 , t ) = 0 , R x ( L, t ) = 0 , R ( x,
0) = 1 µ n X k =1 M k . (40)Obviously, the solution of this problem is R ( x, t ) = 1 µ n X k =1 M k = const . (41)Notice that near the stationary state, when each concentration is almost localized in itsown region, for instance [ x l , x r ] we can write approximation ∂ t a k − εµ k ∂ xx a k = 0 , x ∈ [ x l , x r ] . (42)This allows to obtain the characteristic time of the steady state release t k ≈ ( x r − x l ) εµ k π = ( x r − x l ) λµ k π . (43)For example, at λ = 500, ( x r − x l ) = 0 . µ k = 1 we have t k ≈ C. Whether to ignore the Poisson-Boltzmann equation?
In the general case, instead of the electroneutrality equation q ≡ X θ k a k + 2 K w sinh ψ = 0 (44)and the electric current continuity equationdiv j = 0 (45)we should use the charge conservation law ∂ t q + div j = 0 (46)and the Poisson-Boltzmann equation ε div E = q, (47)where ε is the permittivity of water (for water the dimension value of the permittivity is ε ∗ ≈ · . · − F / m, F = C / (V · m)).11f the permittivity is small enough then (47) implies (44) and (46) implies (45). Wecompare the molar charge of water ([H + ] − [OH − ]) and the term ε div E in 1D case.Using the dimensional variables we can write (see, for example, [9–11])( ε ϕ xx ) ∗ = ϕ xx ε ∗ E ∗ L ∗ = ε ∗ R ∗ T ∗ F ∗ L ∗ ϕ xx λj (C / m ) , ([H + ] − [OH − ]) ∗ = 2 K ∗ w F ∗ sinh ψ (C / m )or ( ε ϕ xx ) ∗ ≈ . · − ϕ xx λj (C / m ) , ([H + ] − [OH − ]) ∗ ≈ .
019 sinh ψ (C / m ) . . . . . . − x − λ = 500 ϕ xx . . . . . x − − ϕ xx λ = 1000 FIG. 1: The distributions of the ϕ xx ( x ) at K w = 0. The current constant regime ( j = 1) at λ = 500(left) and λ = 1000 (right), t = 2 . L ∗ = 2 .
54 cm At λ = 500, ϕ xx ≈
200 (see Fig. 1, left), and ψ = 1 we have( ε ϕ xx ) ∗ ≈ . / m ) , ([H + ] − [OH − ]) ∗ ≈ .
023 (C / m )and at λ = 1000, ϕ xx ≈
200 (see Fig. 1, right), and ψ = 1 we have( ε ϕ xx ) ∗ ≈ . / m ) , ([H + ] − [OH − ]) ∗ ≈ .
023 (C / m ) . In particular, when λ = 1000 contribution of the term ( ε ϕ xx ) in charge of the mixture onlyin 4 times less than the contribution of the water ions.Thus, if we take into account the water ions, then the using of the Poisson-Boltzmannequation and the charge conservation law instead of the electroneutrality equation and theelectric current continuity equation can play a significant role in the describing of the IEF.12trictly speaking, the law of charge conservation (46) is always valid. Indeed, using (5),similar equation for ions H + and OH − , and (19) we obtain (46). If we assume that q = 0, it isclear that (46) is splitted into two equations: q = 0 and div j = 0. If we refuse to conditionsof electroneutrality mixture, then equation (46) should be used to define the function ψ .In this case, we rewrite (46) in the following form n X k =1 ∂q∂a k ∂ t a k + ∂q∂ψ ∂ t ψ + div j = 0 . Using (27), (28) we get ∂q∂ψ ∂ t ψ + div ( j − n X k =1 θ k ( ψ ) i k ) = 0 . Taking into account (30) and the relation θ ′ k ( ψ ) = σ k ( ψ ) − θ k ( ψ ) , finally, we have the evolution equation for the determination of function ψq ψ ∂ t ψ + div J = r, (48)where q ψ = n X k =1 a k θ ′ k ( ψ ) + 2 K w cosh ψ, (49) J = n X k =1 µ k a k θ ′ k ( ψ ) + 2 K w µ cosh( ψ − ψ ) ! ( E − ε ∇ ψ ) ,r = − n X k =1 i k · ∇ θ ′ k ( ψ ) = n X k =1 µ k θ ′ k ( ψ ) ( ε ∇ a k − θ k ( ψ ) E ) · ∇ ψ. Notice that, as expected, the value J formally coincides with the current flux density j for the stationary problem, and the multiplier in front of the term ( E − ε ∇ ψ ) is theconductivity σ stat for stationary case (see Sec. 5, and equations (28) in [11] ). Of course, thecontribution of water ions should be added in the conductivity σ stat . III. CONCLUSION
In detail, the described technique of constructing the mathematical models of elec-trophoresis is presented in [1, 2]. In this paper we emphasize the importance of the takinginto account the different physical and chemical effects. Using of the simple models can leadto inadequate description of experiments. 13 cknowledgments
This research is partially supported by Russian Foundation for Basic Research (grants10-05-00646 and 10-01-00452), Ministry of Education and Science of the Russian Feder-ation (programme ‘Development of the research potential of the high school’, contracts14.A18.21.0873, 8832 and grant 1.5139.2011). [1]
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