Derivation Chemical Kinetics and . . . of Towards . . . - PowerPoint PPT Presentation

Molecular Binding in . . . Hill’s Equation: . . . Chemical Kinetics: . . . Chemical Kinetics and . . . Derivation Chemical Kinetics and . . . of Towards . . . Mathematical Analysis . . . Hill’s Equations Generalized . . . Conclusions from Home Page Scale Invariance Title Page ◭◭ ◮◮ Andres Ortiz ◭ ◮ Department of Mathematical Sciences Page 1 of 11 University of Texas at El Paso El Paso, Texas 79968, USA Go Back aortiz19@miners.utep.edu Full Screen supervised by Vladik Kreinovich vladik@utep.edu Close Quit

Molecular Binding in . . . Hill’s Equation: . . . 1. Molecular Binding in Biochemistry Chemical Kinetics: . . . • Many biochemical reactions involve binding of a smaller Chemical Kinetics and . . . molecule L (called ligand ) to a large macromolecule P : Chemical Kinetics and . . . Towards . . . L + P ↔ LP. Mathematical Analysis . . . • Examples: Generalized . . . Conclusions – oxygen binds to haemoglobin: this is one of the Home Page most important biochemical reactions; Title Page – acid content in the stomach regulated by histamine ◭◭ ◮◮ binding to histamine receptor (special protein); – human serum albumin, protein in human blood ◭ ◮ plasma, carries nutrients as ligands. Page 2 of 11 • It is desirable to predict the proportion of the bound Go Back [ LP ] def macromolecules θ = [ P ] + [ LP ]. Full Screen Close Quit

Molecular Binding in . . . Hill’s Equation: . . . 2. Hill’s Equation: Description and Challenge Chemical Kinetics: . . . • Reminder: it is desirable to predict the proportion θ of Chemical Kinetics and . . . the bound macromolecules: Chemical Kinetics and . . . [ LP ] Towards . . . def θ = [ P ] + [ LP ] . Mathematical Analysis . . . Generalized . . . • In many cases, this proportion is described by a for- Conclusions mula (called Hill’s equation ) Home Page [ L ] n Title Page θ = K d + [ L ] n . ◭◭ ◮◮ • In this formula, K d and n are empirical parameters. ◭ ◮ • Since its invention in 1910, Hill’s equation remains a Page 3 of 11 semi-empirical formula. Go Back • It is desirable to provide a theoretical explanation for Full Screen this formula. Close Quit

Molecular Binding in . . . Hill’s Equation: . . . 3. Chemical Kinetics: Reminder Chemical Kinetics: . . . • The quantitative results of chemical reactions are usu- Chemical Kinetics and . . . ally described by equations of chemical kinetics (CK). Chemical Kinetics and . . . Towards . . . • In CK, the reaction rate is proportional to the product Mathematical Analysis . . . of the concentrations [ c ] of all reactants c . Generalized . . . • Examples: Conclusions Home Page – for the reaction A + B → C , the reaction rate is proportional to the product [ A ] · [ B ]: Title Page d [ A ] = − k · [ A ] · [ B ]; d [ B ] = − k · [ A ] · [ B ]; d [ C ] ◭◭ ◮◮ = k · [ A ] · [ B ] . dt dt dt ◭ ◮ – for the reaction 2 A + B → C , the reaction rate is Page 4 of 11 proportional to [ A ] · [ A ] · [ B ]. Go Back • These formulas explain specific cases of Hill’s equation, corresponding to the case when n is an integer. Full Screen Close Quit

Molecular Binding in . . . Hill’s Equation: . . . 4. Chemical Kinetics and the n = 1 Case of Hill’s Chemical Kinetics: . . . Equation Chemical Kinetics and . . . k a k d • We have two reactions: L + P → LP and LP → L + P . Chemical Kinetics and . . . Towards . . . • Thus, equilibrium is when Mathematical Analysis . . . d [ L ] Generalized . . . dt = − k a · [ L ] · [ P ] + k d · [ LP ] = 0 . Conclusions Home Page • So, k a · [ L ] · [ P ] = k d · [ LP ] and [ LP ] = k a · [ L ] · [ P ]. k d Title Page � 1 + k a � ◭◭ ◮◮ • Here, [ P ] + [ LP ] = · [ L ] · [ P ], hence k d ◭ ◮ k a Page 5 of 11 · [ L ] [ LP ] k d θ = [ P ] + [ LP ] = . Go Back 1 + k a · [ L ] Full Screen k d Close Quit

Molecular Binding in . . . Hill’s Equation: . . . 5. Chemical Kinetics and the n = 1 Case of Hill’s Chemical Kinetics: . . . Equation (cont-d) Chemical Kinetics and . . . • Reminder: chemical kinetics implies that Chemical Kinetics and . . . Towards . . . k a · [ L ] Mathematical Analysis . . . k d θ = . 1 + k a Generalized . . . · [ L ] k d Conclusions Home Page • Multiplying both numerator and denominator by = k d [ L ] Title Page def K d , we get θ = K d + [ L ]. k a ◭◭ ◮◮ • This is Hill’s equation for n = 1. ◭ ◮ • Reactions like L + 2 P → LP + P can explain n = 2 Page 6 of 11 and other cases when n is integer. Go Back • In practice, we often observe non-integer values n . Full Screen • Such values are difficult to explain by chemical kinetics. Close Quit

Molecular Binding in . . . Hill’s Equation: . . . 6. Towards Generalization of Chemical Kinetics Chemical Kinetics: . . . • In the traditional chemical kinetics, the rate r of the Chemical Kinetics and . . . reaction A + B → C is r = const · [ A ] · [ B ]. Chemical Kinetics and . . . Towards . . . • This formula only explains the n = 1 , 2 , . . . cases. Mathematical Analysis . . . • To explain the general case of Hill’s equation, we need Generalized . . . to consider a more general formula r = f ([ A ] , [ B ]) . Conclusions • Idea: the numerical value of a quantity depends on the Home Page choice of a measuring unit; e.g., 2 m = 200 cm. Title Page • If we replace a unit for [ A ] by a λ A times smaller one, ◭◭ ◮◮ we get a new numerical value [ A ] ′ = λ A · [ A ]. ◭ ◮ • Similarly, for B , we get [ B ] ′ = λ B · [ B ]. Page 7 of 11 • It makes sense to require that the dependence is the Go Back same in the new unit if we appropriately re-scale r . Full Screen • So, for every λ A > 0 and λ B > 0, there exists a µ s.t. if r = f ([ A ] , [ B ]), then µ ( λ A , λ B ) · r = f ( λ A · [ A ] , λ B · [ B ]). Close Quit

Molecular Binding in . . . Hill’s Equation: . . . 7. Mathematical Analysis of Scale Invariance Chemical Kinetics: . . . • Reminder: the dependence f ([ A ] , [ B ]) is such that for Chemical Kinetics and . . . every λ A and λ B , there exists a µ for which: Chemical Kinetics and . . . – if r = f ([ A ] , [ B ]) then r ′ = f ([ A ] ′ , [ B ] ′ ) Towards . . . – where [ A ] ′ = λ A · [ A ], [ B ] ′ = λ B · [ B ], and r ′ = µ · r . Mathematical Analysis . . . Generalized . . . • In math. terms: f ( λ A · x, λ B · y ) = µ ( λ A , λ B ) · f ( x, y ). Conclusions • Natural assumption: f ([ A ] , [ B ]) is differentiable. Home Page • For λ B = 1, diff. w.r.t λ A and taking λ A = 1, we get Title Page x · f ′ ( x, y ) = α · f ( x, y ), i.e., x · d f def dx = α · f , w/ α = µ ′ (1). ◭◭ ◮◮ ◭ ◮ • Separating the variables, we get d f = α · dx f x . Page 8 of 11 • Integrating, we get ln( f ( x, y )) = α · ln( x ) + c 1 ( y ), so Go Back f ( x, y ) = exp(ln( f ( x, y ))) = c 2 ( y ) · x α , w/ c 2 = exp( c 1 ). Full Screen • Similarly, f ( x, y ) = c 3 ( x ) · y β , so c 2 ( y ) = const · y β and Close f ( x, y ) = const · x α · y β . Quit

Molecular Binding in . . . Hill’s Equation: . . . 8. Generalized Chem. Kin. Explains Hill’s Eq. Chemical Kinetics: . . . • Reminder: for A + B → C , the rate is k a · [ A ] α · [ B ] β . Chemical Kinetics and . . . Chemical Kinetics and . . . • Similarly: for C → A + B , the scale-invariant reaction rate is f ([ C ]) = k d · [ C ] γ . Thus, equilibrium is when Towards . . . Mathematical Analysis . . . d [ L ] dt = − k a · [ L ] α · [ P ] β + k d · [ LP ] γ = 0 . Generalized . . . Conclusions • So, k a · [ L ] α · [ P ] β = [ LP ] γ and [ LP ] = C · [ L ] n · [ P ] β/γ , Home Page k d � 1 /γ Title Page � k a with C = and n = α/γ . k d ◭◭ ◮◮ • When β = γ , we have [ P ] + [ LP ] = (1 + C · [ L ] n ) · [ P ], ◭ ◮ hence Page 9 of 11 C · [ L ] n [ LP ] θ = [ P ] + [ LP ] = 1 + C · [ L ] n . Go Back • Dividing both numerator and denominator by C , we Full Screen [ L ] n def get Hill’s equation θ = K d + [ L ] n , with K d = 1 /C . Close Quit

Molecular Binding in . . . Hill’s Equation: . . . 9. Conclusions Chemical Kinetics: . . . • In biochemistry, the proportion of the bounded macro- Chemical Kinetics and . . . [ L ] n Chemical Kinetics and . . . molecules is often described by Hill’s eq. θ = K d + [ L ] n . Towards . . . • When n is an integer, this eq. can be explained by Mathematical Analysis . . . chem. kin., where the rate of A + B → C is k · [ A ] · [ B ]. Generalized . . . Conclusions • However, in practice, n is often not an integer, and so Home Page the chemical kinetics explanation is not applicable. Title Page • We assume that the reaction rate f ([ A ] , [ B ]) is scale- invariant but can be more general than the product. ◭◭ ◮◮ • As a result, we get a family of formulas that include ◭ ◮ Hill’s equation as a particular case. Page 10 of 11 • Thus, we get a theoretical explanation for Hill’s equa- Go Back tion. Full Screen • We also get a more general formula that may be useful Close to explain possible deviation from Hill’s equation. Quit