Supporting Information. Binding of solvent molecules to a protein surface in binary mixtures follows a competitive Langmuir model

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1 Supporting Information Binding of solvent molecules to a protein surface in binary mixtures follows a competitive Langmuir model Tobias Kulschewski 1, Jürgen Pleiss 1* 1 Institute of Technical Biochemistry, University of Stuttgart, Allmandring 31, Stuttgart, Germany * Corresponding author: Prof. Dr. Jürgen Pleiss Institute of Technical Biochemistry University of Stuttgart Allmandring 31 D Stuttgart Germany Phone: (+49) Juergen.Pleiss@itb.uni-stuttgart.de Short title: Molecular model of protein hydration Keywords: model; protein hydration; thermodynamic activity; methanol; toluene;

2 1. Derivation of equation 3 (competitive Langmuir without water-water interaction) The binding of a molecule M 1 to a free binding site P on the protein surface forming a complex M 1 P is described by the binding equilibrium and a binding constant K 1 M 1 + P M 1 P (eq. S1a) M 1, P, and M 1 P is quantified by concentration or thermodynamic activity, and related by M 1 P = K 1 M 1 P (eq. S1b) Thus: P = M 1 P / K 1 M 1 (eq. S1c) Similarly, molecule M 2 binds to a binding site P on the protein surface forming a complex M 2 P M 2 + P M 2 P (eq. S2a) The binding equilibrium is characterized by a binding constant K 2 M 2 P=K 2 M 2 P (eq. S2b) Inserting (eq. S1c) into (eq. S2b) results in M 2 P = K 2 M 2 M 1 P / K 1 M 1 (eq. S2c) The fraction Θ 1 ' of all binding sites binding molecule 1 is calculated by the ratio Θ 1 ' = M 1 P / (P+ M 1 P + M 2 P) (eq. S3a) Inserting (eq. S1c) and (eq. S2c) into (eq. S3a) results in Θ 1 ' = M 1 P / (M 1 P / K 1 M 1 + M 1 P + K 2 M 2 M 1 P / K 1 M 1 ) = K 1 M 1 / (1+K 1 M 1 + K 2 M 2 ) (eq. S3b) Expressing M 1 and M 2 as thermodynamic activities a 1 and a 2, respectively, results in the classical multi-langmuir isotherm model (Markham 1931): ' Θ 1 (a 1, a 2 ) = K 1 a 1 / (1+K 1 a 1 + K 2 a 2 ) (eq. S3c)

3 In pure solvent consisting of molecule 1, Θ 1 ' (a 1 =1, a 2 =0) = K 1 / (1+K 1 ) The ratio Θ 1 of the number of molecules 1 bound to the protein surface in the binary mixture to the number of molecules 1 bound to the protein surface in pure solvent 1is calculated by ' ' Θ 1 (a 1, a 2 ) = Θ 1 (a 1, a 2 ) / Θ 1 (a 1 =1, a 2 =0) = (1+K 1 ) a 1 / (1 + K 1 a 1 + K 2 a 2 ) (eq. 3a) Similarly, the ratio Θ 2 of the number of molecules 2 bound to the protein surface in the binary mixture to the number of molecules 2 bound to the protein surface in pure solvent 2 is calculated by ' ' Θ 2 (a 1, a 2 ) = Θ 2 (a 1, a 2 ) / Θ 1 (a 1 =0, a 2 =1) = (1+K 2 ) a 2 / (1+ K 1 a 1 + K 2 a 2 ) (eq. 3b)

4 2. Derivation of equation 4 (competitive Langmuir with water-water interaction) The competitive Langmuir model of two molecules binding to binding sites on the protein surface (eqs. S1, S2) was extended by an additional equilibrium of molecule 1 (M 1 ) binding to a molecule 1 that is already bound to a binding site (M 1 P) on the protein surface forming complex M 2 1 P M 1 + M 1 P M 2 1 P (eq. S4a) The binding equilibrium is characterized by a binding constant K 11 M 1 2 P = K 11 M 1 P M 1 (eq. S4b) ' The fraction Θ 1 of all binding sites binding molecule 1 is calculated by the ratio ' Θ 1 = (M 1 P + M 2 1 P) / (P+ M 1 P + M 2 P + M 2 1 P) (eq. S5a) Inserting (eq. S1c), (eq. S2c), and (eq. S4b) into (eq. S5a) results in Θ 1 ' = (M 1 P + K 11 M 1 P M 1 ) / (M 1 P / K 1 M 1 + M 1 P + K 2 M 2 M 1 P / K 1 M 1 + K 11 M 1 P M 1 ) = (1 + K 11 M 1 ) K 1 M 1 / (1 + K 1 M 1 + K 2 M 2 + K 1 K 11 M 1 2 ) (eq. S5b) Expressing M 1 and M 2 as thermodynamic activities a 1 and a 2, respectively, results in Θ 1 ' (a 1, a 2 ) = (1 + K 11 a 1 ) K 1 a 1 / (1 + K 1 a 1 + K 2 a 2 + K 1 K 11 a 1 2 ) In pure solvent consisting of molecule 1, Θ 1 ' (a 1 =1, a 2 =0) = (K 1 +K 1 K 11 ) /(1+K 1 +K 1 K 11 ) The ratio Θ 1 of the number of molecules 1 bound to the protein surface in the binary mixture to the number of molecules 1 bound to the protein surface in pure solvent 1 is calculated by ' ' Θ 1 (a 1, a 2 ) = Θ 1 (a 1, a 2 ) / Θ 1 (a 1 =1, a 2 =0) = (1+K 1 +K 1 K 11 ) (1+K 11 a 1 ) a 1 / (1+ K 11 ) (1+K 1 a 1 +K 2 a 2 +K 1 K 11 a 2 1 ) (eq. 4a) ' The fraction Θ 2 of all binding sites binding molecule 2 is calculated by the ratio ' Θ 2 = M 2 P / (P+ M 1 P + M 2 P + M 2 1 P) (eq. S6a) Inserting (eq. S1b), P = M 2 P/K 2 M 2 from (eq. S2b) and (eq. S4b) into (eq. S6a) results in Θ 2 ' = M 2 P / (M 2 P/K 2 M 2 + K 1 M 1 M 2 P/K 2 M 2 + M 2 P + K 11 K 1 M 2 P/K 2 M 2 M 1 2 ) = K 2 M 2 / (1+K 1 M 1 +K 2 M 2 +K 1 K 11 M 1 2 ) (eq. S6b)

5 Expressing M 1 and M 2 as thermodynamic activities a 1 and a 2, respectively, results in Θ 2 ' (a 1, a 2 ) = K 2 a 2 / (1 + K 1 a 1 + K 2 a 2 + K 1 K 11 a 1 2 ) In pure solvent consisting of molecule 2, Θ 2 ' (a 1 =0, a 2 =1) = K 2 /(1+K 2 ) The ratio Θ 2 of the number of molecules 2 bound to the protein surface in the binary mixture to the number of molecules 2 bound to the protein surface in pure solvent 2 is calculated by ' ' Θ 2 (a 1, a 2 ) = Θ 2 (a 1, a 2 ) / Θ 2 (a 1 =1, a 2 =0) = (1+K 2 ) a 2 / (1+K 1 a 1 +K 2 a 2 +K 1 K 11 a 2 1 ) (eq. 4b)

6 3. Simulation of binding by the one-parameter Margules activity model For an ideal binary mixture (A=0), the thermodynamic activity is equal to the mole fraction a 1 = χ 1 (Fig. S2a). For identical binding affinities (K 1 =K 2 ), θ 1 is equal to a 1 (Fig. S3a). If molecule 1 binds better than molecule 2 (K 1 >K 2 ), the binding curve θ 1 (a 1 ) is concave (Fig. S3b), if molecule 2 binds better than 1 (K 1 <K 2 ), θ 1 (a 1 ) is convex (Fig. S3c). For a non-ideal, miscible mixture (A=2.45), a 1 (χ 1 ) is sigmoidal with a γ 1 (a 1 0) > 1 and γ 1 (a 1 1) = 1 (Fig. S2b). The binding curve θ 1 (a 1 ) is slightly convex for K 1 =K 2 (Fig. S4a), concave for K 1 >K 2 (Fig. S4b), and considerably convex for K 1 <K 2 (Fig. S4c). For a partially immiscible mixtures (A > 2.45), the miscibility gap is indicated by a region with a 1 (χ 1 ) 1 (Fig. S2c). In the miscible region, the binding curve θ 1 (a 1 ) is similar to a nonideal, miscible mixture. However, as a 1 (χ 1 ) approaches 1 from both sides of the miscibility gap, θ 1 (a 1 ) deviates considerably from the completely miscible scenario, resulting in a sensitive dependency of θ 1 (a 1 ) for K 1 =K 2 (Fig. S5a), K 1 >K 2 (Fig. S5b), and K 1 <K 2 (Fig. S5c), as a 1 approaches 1.

7 4. Supplementary table Table S1 Dimensions of the molecular systems (C.antarctica lipase B in binary mixtures) Mixture Number of atoms Number of molecules Box length [nm] water/methanol methanol /toluene water/ toluene toluene /water

8 5. Supplementary figures (a) (b) Fig. S1 Density ρ=dn/dr of (a) water and (b) methanol bound to CALB in binary water/methanol mixtures at different mole fractions of (a) water and (b) methanol.

9 a) b)

10 c) Fig. S2 Thermodynamic activity of one component in a binary mixture as a function of its mole fraction as calculated by the Margules model (eq. 6) with a) A=0, b) A=2, and c) A=2.6

11 a) b)

12 c) Fig. S3 Binding curve of component 1 to the protein surface as a function of its thermodynamic activity as predicted by the Langmuir model for ideal binary mixtures (A=0). a) Both components have the same binding affinity (K 1 =1;K 2 =1) b) Component 1 has a higher binding affinity than component 2 (K 1 =10;K 2 =1) c) Component 1 has lower binding affinity than component 2 (K 1 =1;K 2 =10)

13 a) b)

14 c) Fig. S4 Binding curve of component 1 to the protein surface as a function of its thermodynamic activity as predicted by the Langmuir model for non-ideal, miscible binary mixtures (A=2). a) Both components have the same binding affinity (K 1 =1;K 2 =1) b) Component 1 has a higher binding affinity than component 2 (K 1 =10;K 2 =1) c) Component 1 has lower binding affinity than component 2 (K 1 =1;K 2 =10)

15 a) b)

16 c) Fig. S5 Binding curve of component 1 to the protein surface as a function of its thermodynamic activity as predicted by the Langmuir model for non-ideal mixtures with a miscibility gap (A=2.6). a) Both components have the same binding affinity (K 1 =1;K 2 =1) b) Component 1 has a higher binding affinity than component 2 (K 1 =10;K 2 =1) c) Component 1 has lower binding affinity than component 2 (K 1 =1;K 2 =10)

17 Fig. S6 Binding curves predicted by the Langmuir model for non-ideal mixtures (A=1). In mixture 1 (dashed line), both component have the same binding affinity (K 1 =K 2 =1). In mixture 2 (dotted line), the component 2 has a higher binding affinity (K 1 = 2, K 2 = 5).

18 Fig. S7 Binding curves predicted by the Langmuir model for ideal (A=0, long dashed line) and non-ideal binary mixtures (A=1, A=2, A=3) with component 2 having a higher binding affinity than component 1 (K 1 =1, K 2 =2)

19 Fig. S8 Binding curves predicted by the Langmuir model for a slightly non-ideal mixture (A=0.59, similar to water/methanol) and a non-ideal mixture (A=2.08, similar to methanol/toluene) with component has a higher binding affinity towards the protein (K 1 =5, K 2 =2)

20 Fig. S9 Binding curves predicted by the Langmuir model for a non-ideal mixture (A=2) for different values of K 1 and K 2 : K 1 =1.0, K 2 =2.97 long dashed line, K 1 =2.0, K 2 =5.0 short dashed line, K 1 =4.0, K 2 =9.18 heavy dotted line, K 1 =8.0, K 2 =17.6 short dotted line

21 Fig. S10 Linear dependency of K 1 and K 2 for the values in Fig. S9

22 a) b)

23 c) Fig. S11 Binding to the CALB surface: simultaneous fit of the 6 simulated binding curves by a competitive Langmuir model with water-water interaction by varying K Wat in the range of 0.25 and 16 and optimizing K Met, K Tol, and K WatWat. (a) K MeOH (circles) and K TOL (square) depend linearly on K Wat ; (b) K WatWat ; (c) root mean square deviation

24 a) b)

25 c) Fig. S12 Binding to the high affinity water binding sites of CALB: simultaneous fit of the 6 simulated binding curves by a competitive Langmuir model with water-water interaction by varying K Wat in the range of 0.25 and 64 and optimizing K Met, K Tol, and K WatWat. (a) K MeOH (circles) and K TOL (square) depend linearly on K Wat ; (b) K WatWat ; (c) root mean square deviation

26 Fig. S13a Relative binding Θ Wat of water to CALB in binary water/methanol mixtures as a function of thermodynamic activity of water (a Wat ). MD data (dots) and fit by competitive Langmuir model without water-water interaction according to eq. 3 (K Met =6.17; K Wat =2.31) (line). Fig. S13b Relative binding Θ Met of methanol to CALB in binary water/methanol mixtures as a function of thermodynamic activity of methanol (a Met ). MD data (dots) and fit by competitive Langmuir model without water-water interaction according to eq. 3 (K Met =6.17; K Wat =2.31) (line).

27 Fig. S13c Relative binding Θ Met of methanol to CALB in binary methanol/toluene mixtures as a function of thermodynamic activity of methanol (a Met ). MD data (dots) and fit by competitive Langmuir model without water-water interaction according to eq. 3 (K Met =6.17; K Tol =2.20) (line). Fig. S13d Relative binding Θ Tol of toluene to CALB in binary methanol/toluene mixtures as a function of thermodynamic activity of toluene (a Tol ). MD data (dots) and fit by competitive Langmuir model without water-water interaction according to eq. 3 (K Met =6.17; K Tol =2.20) (line).

28 Fig. S13e Relative binding Θ Wat of water to CALB in binary water/toluene mixtures as a function of thermodynamic activity of water (a Wat ). MD data (dots) and fit by competitive Langmuir model without water-water interaction according to eq. 3 (K Wat =2.31; K Tol =2.20) (line). Fig. S13f Relative binding Θ Tol of toluene to CALB in binary water/toluene mixtures as a function of thermodynamic activity of toluene (a Tol ). MD data (dots) and fit by competitive Langmuir model without water-water interaction according to eq. 3 (K Wat =2.31; K Tol =2.20) (line).

29 a) b) Fig. S14 Binding to the surface of CALB: simultaneous fit of the 6 simulated binding curves by a competitive Langmuir model without water-water interaction by varying K Wat in the range of 0.25 and 16 and optimizing K Met and K Tol. (a) K MeOH (circles) and K TOL (square) depend linearly on K Wat ; (b) root mean square deviation

30 Fig. S15a Relative binding Θ Wat of water to high affinity water binding sites of CALB in binary water/methanol mixtures as a function of thermodynamic activity of water (a Wat ). MD data (dots) and fit by competitive Langmuir model without water-water interaction according to eq. 3 (K Met =17.09; K Wat =10.75) (line). Fig. S15b Relative binding Θ Met of methanol to high affinity water binding sites of CALB in binary water/methanol mixtures as a function of thermodynamic activity of methanol (a Met ). MD data (dots) and fit by competitive Langmuir model without water-water interaction according to eq. 3 (K Met =17.09; K Wat =10.75) (line).

31 Fig. S15c Relative binding Θ Met of methanol to high affinity water binding sites of CALB in binary methanol/toluene mixtures as a function of thermodynamic activity of methanol (a Met ). MD data (dots) and fit by competitive Langmuir model without water-water interaction according to eq. 3 (K Met =17.09; K Tol =2.20) (line). Fig. S15d Relative binding Θ Tol of toluene to high affinity water binding sites of CALB in binary methanol/toluene mixtures as a function of thermodynamic activity of toluene (a Tol ). MD data (dots) and fit by competitive Langmuir model without water-water interaction according to eq. 3 (K Met =17.09; K Tol =2.20) (line).

32 Fig. S15e Relative binding Θ Wat of water to high affinity water binding sites of CALB in binary water/toluene mixtures as a function of thermodynamic activity of water (a Wat ). MD data (dots) and fit by competitive Langmuir model without water-water interaction according to eq. 3 (K Tol =2.20; K Wat =10.75) (line). Fig. S15f Relative binding Θ Tol of toluene to high affinity water binding sites of CALB in binary water/toluene mixtures as a function of thermodynamic activity of toluene (a Tol ). MD data (dots) and fit by competitive Langmuir model without water-water interaction according to eq. 3 (K Tol =2.20; K Wat =10.75) (line).

33 a) b) Fig. S16 Binding to the high affinity water binding sites of CALB: simultaneous fit of the 6 simulated binding curves by a competitive Langmuir model without water-water interaction by varying K Wat in the range of 0.25 and 64 and optimizing K Met and K Tol. (a) K MeOH (circles) and K TOL (square) depend linearly on K Wat ; (b) root mean square deviation

34 Fig. S17 Relative binding Θ Wat of water to CALB in various binary mixtures as a function of thermodynamic activity of water a Wat. Binary water/methanol (black dots) and water/toluene mixtures (open black dots) from this study are compared with results on binary mixtures of water/hexane (red squares), water/mtbe (blue triangles), water/methanol (green diamonds) and water/tert-butyl-alcohol (yellow circles) from Wedberg et. al, 2012.

35 Fig. S18 RMSD for different molecular systems consisting of CALB, water, methanol and toluene. Except for one simulation, all systems were stable and partial unfolding was not observed.

36 Fig. S19: Location of the high affinity water binding sites on the surface of CALB (red = high water affinity, blue = low water affinity), using the tool MegaMol (Krone et al, manuscript under revision)

37 Fig. S20a Relative binding Θ Wat of water to CALB in binary water/methanol mixtures as a function of thermodynamic activity of water (a Wat ) for a cutoff of 0.4 nm. MD data (dots) and previously fitted competitive Langmuir model (line) with a cutoff of 0.3 nm. Fig. 20b Relative binding Θ Met of methanol to CALB in binary water/methanol mixtures as a function of thermodynamic activity of methanol (a Met ) for a cut-off of 0.4 nm. MD data (dots) and previously fitted competitive Langmuir model (line) with a cutoff of 0.3 nm.

38 Fig. 20c Relative binding Θ Met of methanol to CALB in binary methanol/toluene mixtures as a function of thermodynamic activity of methanol (a Met ) for a cut-off of 0.4 nm. MD data (dots) and previously fitted competitive Langmuir model (line) with a cutoff of 0.3 nm. Fig. 20d Relative binding Θ Tol of toluene to CALB in binary methanol/toluene mixtures as a function of thermodynamic activity of toluene (a Tol ) for a cut-off of 0.4 nm. MD data (dots) and previously fitted competitive Langmuir model (line) with a cutoff of 0.3 nm.

39 Fig. S20e Relative binding Θ Wat of water to CALB in binary water/toluene mixtures as a function of thermodynamic activity of water (a Wat ) for a cut-off of 0.4 nm. MD data (dots) and previously fitted competitive Langmuir model (line) with a cutoff of 0.3 nm. Fig. 20f Relative binding Θ Tol of toluene to CALB in binary water/toluene mixtures as a function of thermodynamic activity of toluene (a Tol ). MD data (dots) and previously fitted competitive Langmuir model (line) with a cutoff of 0.3 nm.

40 Fig. S21a Relative binding Θ Wat of water to the high-affinity water binding sites of CALB in binary water/methanol mixtures as a function of thermodynamic activity of water (a Wat ) for a cut-off of 0.4 nm. MD data (dots) and previously fitted competitive Langmuir model (line) with a cutoff of 0.3 nm. Fig. 21b Relative binding Θ Met of methanol to the high-affinity water binding sites of CALB in binary water/methanol mixtures as a function of thermodynamic activity of methanol (a Met ) for a cut-off of 0.4 nm. MD data (dots) and previously fitted competitive Langmuir model (line) with a cutoff of 0.3 nm.

41 Fig. 21c Relative binding Θ Met of methanol to the high-affinity water binding sites of CALB in binary methanol/toluene mixtures as a function of thermodynamic activity of methanol (a Met ) for a cut-off of 0.4 nm. MD data (dots) and previously fitted competitive Langmuir model (line) with a cutoff of 0.3 nm. Fig. 21d Relative binding Θ Tol of toluene to the high-affinity water binding sites of CALB in binary methanol/toluene mixtures as a function of thermodynamic activity of toluene (a Tol ) for a cut-off of 0.4 nm. MD data (dots) and previously fitted competitive Langmuir model (line) with a cutoff of 0.3 nm.

42 Fig. S21e Relative binding Θ Wat of water to the high-affinity water binding sites of CALB in binary water/toluene mixtures as a function of thermodynamic activity of water (a Wat ) for a cut-off of 0.4 nm. MD data (dots) and previously fitted competitive Langmuir model (line) with a cutoff of 0.3 nm. Fig. 21f Relative binding Θ Tol of toluene to the high-affinity water binding sites of CALB in binary water/toluene mixtures as a function of thermodynamic activity of toluene (a Tol ) for a cut-off of 0.4 nm. MD data (dots) and previously fitted competitive Langmuir model (line) with a cutoff of 0.3 nm.