Modeling of Ion and Water Transport in the Biological Nanopore Clya

Supporting Information: Modeling of Ion and Water Transport in the Biological Nanopore ClyA

Kherim Willems,†,‡ Dino Rui´c,¶,‡ Florian Lucas,§ Ujjal Barman,‡ Johan

Hofkens,† Giovanni Maglia,∗,§ and Pol Van Dorpe∗,†,‡

†KU Leuven, Department of Chemistry, Celestijnenlaan 200F, B-3001 Leuven, Belgium ‡imec, Kapeldreef 75, B-3001 Leuven, Belgium ¶KU Leuven, Department of Physics and Astronomy, Celestijnenlaan 200D, B-3001 Leuven, Belgium §University of Groningen, Groningen Biomolecular Sciences & Biotechnology Institute, 9747 AG, Groningen, The Netherlands

E-mail: [email protected]; [email protected]

S-1 Contents

1 Extended materials and methods S-3 1.1 ClyA-AS mutations w.r.t. the wild-type ...... S-3 1.2 Fitting of the electrolyte properties ...... S-3 1.3 Surface integration to compute pore averaged values ...... S-5 1.4 Weak forms of the ePNP-NS equations ...... S-7 1.5 Estimating the bulk conductance of ClyA ...... S-10

2 Extended results S-12 2.1 Ionic current rectiﬁcation ...... S-12 2.2 Peak values of the radial potential proﬁles inside ClyA ...... S-12 2.3 Pressure distribution inside ClyA ...... S-13

References S-14

S-2 1 Extended materials and methods

1.1 ClyA-AS mutations w.r.t. the wild-type

Table S1. Mutations of the ClyA-AS variant compared to the S. tyhpii wild-type.

Position WT AS

8 Lys Gln 15 Asn Ser 38 Gln Lys 57 Ala Glu 67 Thr Val 87 Cys Ala 90 Ala Val 95 Ala Ser 99 Leu Gln 103 Glu Gly 118 Lys Arg 119 Leu Ile 124 Ile Val 125 Thr Lys 136 Val Thr 166 Phe Tyr 172 Lys Arg 185 Val Ile 212 Lys Asn 214 Lys Arg 217 Ser Thr 224 Thr Ser 227 Asn Ala 244 Thr Ala 276 Glu Gly 285 Cys Ser 290 Lys Gln

1.2 Fitting of the electrolyte properties

The parameters of the fitting functions used to interpolate the experimental ion diffusion coefficients, mobilities and transport numbers, and the electrolyte viscosity, density and relative permittivity are given in Tab. S2 and the resulting curves are plotted in Fig. S1.

S-3 Note that since most of these functions are merely empirical ﬁts with no physical meaning, and that they are solely used to interpolate and represent the experimental data. Finally, for the concentration dependence of the relative permittivity we made use of the model proposed by Gavish et al.4

3α εr(¯c) = εr,0 − (εr,0 − εr,ms) L c¯ (S1) εr,0 − εr,ms

a b c 9 -3 Di(c) ×10 ρ(c) ×10 2 -1 -3 (m ·s ) (kg·m ) D(d)/D0 Ion diffusion 1 2.03 coefficient Density 1.18 Na+, Cl– Cl– Relative diffusion 1.33 0.997 coefficient 1.11 Na+ 0.836 4 η(c) ×10 0 8 (Pa·s) μi(c) ×10 2 -1 -1 1.66 (m ·V ·s ) η0/η(d) Dynamic 1 7.90 Ion mobility viscosity H O – 2 5.19 Cl 0.890 Inverse 3.34 0.384 relative + Na 1.81 ε(c)/ε0 viscosity

78.15 Relative 0 5 permittivity 0 2 cbulk (M) dwall (nm)

43.40

0 5

cbulk (M) Figure S1. Concentration and positional dependent electrolyte properties. (a) Dependency of the ion self-diffusion coefficients (top, Na+: and Cl– : ) and the ion electrophoretic mobilities (bottom, Na+: and Cl– : ) on the bulk NaCl concentration. Lines represent empirical fits to experimental literature data (markers). The number next to the left and right axes correspond to the values at infinite dilution (0 M) and saturation (≈5 M), respectively. (b) Dependency of electrolyte density (top, ), viscosity (middle, ) and relative permittivity (bottom, ) on the bulk NaCl concentration. Lines represent empirical fits to experimental literature data (markers). (c) Dependency of the relative ion diffusion coefficient and the mobility (top, ) and the relative viscosity (bottom, ) on the distance from the nanopore wall. The relative diffusion coefficient (and its mobility) declines sharply when an ion approaches within ≈1.0 nm of the protein wall. For water molecules, this phenomenon is modelled as a sharp increase of the fluid viscosity within ≈1.0 nm distance from the wall. The values for the diffusion coefficient were taken directly from ref. 1, who used an empirical fit on molecular dynamics data from ref. 2. The inverse relative viscosity data was taken directly from the molecular dynamics study in ref. 3, and fitted with a logistic function after offsetting for the protein hydrodynamic radius.

S-4 Table S2. Overview of the NaCl ﬁtting parameters used for interpolation.

Fitting parameters

2 Property P0 P1 P2 P3 P4 R References

−1 −1 −1 −2 +(c) 1.334 2.02 0.14 10 3.05 0.41 10 2.19 0.38 10 3.13 1.08 10 >0.99 5 D ± × − ± × ± × − ± × −1 −2 −2 −2 −(c) 2.032 1.49 0.30 10 4.94 9.04 10 3.40 8.26 10 1.43 2.30 10 >0.99 5 D ± × − ± × ± × ± × −1 −1 −1 −3 µ+(c) 5.192 7.91 0.06 10 3.53 0.17 10 1.46 0.15 10 9.23 3.89 10 >0.99 6–9 ± × − ± × ± × ± × −1 −1 −1 −2 µ−(c) 7.909 6.29 0.06 10 4.29 0.17 10 2.12 0.14 10 1.07 0.37 10 >0.99 6–9 ± × − ± × ± × − ± × −2 −3 −2 −3 t+(c) 0.3963 9.38 1.64 10 2.86 3.24 10 1.88 6.52 10 4.51 2.75 10 0.98 7,9–12 ± × ± × − ± × ± × η(c) 0.8904 7.56 0.27 10−3 7.77 0.04 10−2 1.19 0.01 10−2 5.95 0.35 10−4 >0.99 13 ± × ± × ± × ± × %(c) 0.997 4.06 0.01 10−2 6.39 0.16 10−4 >0.99 13 ± × − ± × ε(c) 78.15 3.08 0.00 101 1.15 0.00 101 4 ± × ± × +(d) 6.2 0.01 1,2,14 D −(d) 6.2 0.01 1,2,14 D µ+(d) 6.2 0.01 1,2,14

µ−(d) 6.2 0.01 1,2,14 η(d) 3.36 0.23 1.47 0.23 10−1 0.97 3 ± ± ×

1.3 Surface integration to compute pore averaged values

The average pore values for quantity of interest X was computed by

ZZ βαX dr dz Vα hXiα = ZZ , (S2) βα dr dz Vα

where   p, d ≥ 0 nm, average over the entire pore   α = b, d > 0.5 nm, average over the pore ‘bulk’ (S3)    s, d ≤ 0.5 nm, average over the pore ‘surface’

S-5 and

  1, if − 1.85 ≤ z ≤ 12.25 and r ≤ rp(z) βp = (S4)  0, otherwise   1, if − 1.85 ≤ z ≤ 12.25 and r ≤ rp(z) and d > 0.5 βb = (S5)  0, otherwise   1, if − 1.85 ≤ z ≤ 12.25 and r ≤ rp(z) and d ≤ 0.5 βs = (S6)  0, otherwise

with d the distance from the nanopore wall and rp(z) is the radius of the pore at height z.

S-6 1.4 Weak forms of the ePNP-NS equations

To solve partial diﬀerential equations with the ﬁnite element method, we must be derive their weak form. This is achieved through multiplication of the equation with an arbitrary test function and integration over their relevant domains and boundaries (Fig. S2). The

full computational domain of our model (Ω) is subdivided into domains for the pore (Ωp), the lipid bilayer (Ωm) and the electrolyte reservoir (Ωw). The relevant boundaries of these domains are also indicated, i.e. the reservoir’s exterior edges at the cis Γw,c) and trans Γw,t)

sides, the outer edge of the lipid bilayer (Γm) and the interface of the ﬂuid with the nanopore and the bilayer (Γp+m). The following paragraphs detail the equations that were used in this paper, together with their corresponding weak forms.

Figure S2. Computational domains and boundaries. The full computational domain (Ω) of the model is subdivided into various subdomains (Ωx) and their limiting boundaries (Γx).

S-7 Poisson equation. The global potential distribution is described by the Poisson equation (PE)15

f ∇ · D = − ρpore + ρion with D = ε0εr∇ϕ , (S7)

with D the electrical displacement ﬁeld, ϕ the electric potential, ε0 the vacuum permittivity

12 1 f (8.85419 × 10− F m− ), and εr the local relative permittivity of the medium. ρpore and ρion are the ﬁxed (due to the pore) and mobile (due to the ions) charge distributions, respectively. Multiplication of Eq. (S7) with the potential test function ψ and integration over the

entire model Ω = Ωw + Ωp + Ωm gives

Z Z f [∇ · D] ψ dΩ = − ρpore + ρion ψ dΩ , (S8) Ω Ω which, after applying the Gauss divergence theorem, yields the ﬁnal weak formulation

Z Z Z Z f [∇ψ · D] dΩ − [ψD · n] dΓPE = ψρpore dΩ + [ψρion] dΩ , (S9) Ω ΓPE Ω Ω

with boundaries ΓPE = Γw,c + Γw,t + Γm and n their normal vector. The boundary integrals at Γw,c and Γw,t are evaluated using the Dirichlet boundary conditions (BCs) ϕ = 0 and

ϕ = Vb, respectively. A zero charge BC, n · D = 0, is used for the integral at Γm.

Size-modified Nernst-Planck equation. The total ionic flux J i of ion i at steady-state is expressed by the size-modified Nernst-Planck equation (smNPE)15

3 3 X 3 ai /a0 aj ∇cj ∂ci j = −∇·J i = −∇·(Di∇ci + ziµici∇ϕ + βici − uci) with βi = , (S10) ∂t X 3 1 − NAaj cj j

with ion diﬀusion coeﬃcient Di, concentration ci, charge number zi, mobility µi, electrostatic

potential ϕ, steric saturation factor βi and ﬂuid velocity u. NA is Avogadro’s constant

S-8 23 1 (6.022 × 10 mol− ) and ai and a0 are the limiting cubic diameters for ions and water,

respectively. Using the ion concentration test function di, the weak form of Eq. (S10) becomes

Z Z ∂ci di dΩw = [−∇ · J i] di dΩw Ωw ∂t Ωw Z Z Z ∂ci di dΩw = [∇di · J i] dΩw − [diJ i · n] dΓNP Ωw ∂t Ωw ΓNP Z = [∇di · (Di∇ci + ziµici∇ϕ + βici − uci)] dΩw Ωw Z − [di (Di∇ci + ziµici∇ϕ + βici − uci) · n] dΓNP . (S11) ΓNP

The integrals on the boundaries ΓNP = Γw,c + Γw,t + Γp+m are evaluated using the Dirichlet boundary condition ci = cs for Γw,c and Γw,t, and the no ﬂux boundary condition n · J i = 0

for Γp+m

Variable density and viscosity Navier-Stokes equation. The steady-state, laminar fluid flow of an incompressible fluid with a variable density and viscosity is given by the system of equations16

u · ∇% = 0 (S12)

h Ti (u · ∇)(%u) + ∇ · σij = F with σij = pI − η ∇u + (∇u) (S13)

∇ · (%u) − u · ∇% = 0 , (S14)

with ﬂuid velocity u, density %, hydrodynamic stress tensor σij, viscosity η, pressure p and body force F . The pressure test function q is used to derive the weak forms of Eq. (S12)

Z Z [u · ∇%] q dΩw = [qu · ∇%] dΩw = 0 , (S15) Ωw Ωw

S-9 and Eq. (S14)

Z Z Z [∇ · (%u) − u · ∇%] q dΩw = [∇ · (%u)] q dΩw − [qu · ∇%] dΩw (S16) Ωw Ωw Ωw Z Z Z = [∇q · (%u)] dΩw − [q (%u) · n] dΓNS − [qu · ∇%] dΩw , Ωw ΓNS Ωw

while for Eq. (S13) we use the velocity test function v = [vr, vφ, vz]

Z Z [(u · ∇)(%u) + ∇ · σij] · v dΩw = F · v dΩw Ωw Ωw (S17) Z Z Z Z [(u · ∇)(%u) · v] dΩw − [σij · ∇v] dΩw + [v · σij · n] dΓNS = F · v dΩw , Ωw Ωw ΓNS Ωw

with boundaries ΓNS = Γw,c + Γw,t + Γp+m. The no-slip Dirichlet BC u = 0 is applied to

Γp+m, and the no normal stress σijn = 0 is used for Γw,c and Γw,t.

1.5 Estimating the bulk conductance of ClyA

The ionic current ﬂowing through a nanopore can be computed using Ohm’s law

I(cs,Vb) = Gbulk(cs)Vb (S18)

with Gbulk the nanopore conductance. A naive estimate of Gbulk can be obtained by modelling it as a fluidic resistor, represented by a fluid-filled cylinder.17 Because ClyA has an asymmetric shape, it is best approximated by a two resistors in series: a tall, wide cylinder (cis lumen) on top of a narrow short cylinder (trans constriction).18 Hence,

σ lcis ltrans Gbulk(cs) = (cs) 2 + 2 , (S19) 4π dcis dtrans

S-10 with σ the (concentration dependent) bulk electrolyte conductivity. The cis and trans cham-

bers have have heights and diameters of lcis = 10 nm and ltrans = 4 nm, and dcis = 10 nm and

18 dtrans = 4 nm, respectively.

S-11 2 Extended results

2.1 Ionic current rectiﬁcation

The ionic current rectiﬁcation (ICR, α, Fig. S3) represents the relative conductivity of the nanopore at identical, but opposing bias voltages

G(+Vb) α(Vb) = , with Vb ≥ 0 . G(−Vb)

Hence, for α > 0, the current is higher at positive bias voltages, and vice versa for α < 0. The ICR phenomenon results from an asymmetry in the ionic conductance pathways (e.g., moving from cis to trans is not equivalent as moving from trans to cis). In turn, this stems from the intrinsic asymmetries of the nanopore itself (i.e., geometry and charge distribution) or its response to the electric ﬁeld (i.e., gating or electrostriction). The heatmap of the simulated (ePNP-NS) α (Fig. S3a) reveals that α > 0 between 0 mV to 200 mV and 0.005 M to 5 M. It increases monotonically with increasing bias voltage and exhibits a maximum as a function of salt concentration at ≈0.15 M. The concentration dependency of α at 50 mV, 100 mV and 150 mV (Fig. S3b) shows that it increases monotonically with cs until it reaches a peak value, after which it decreases to approach unity

(α = 1) at saturating salt concentrations (cs ≈ 5 M). Even though the values produced by the ePNP-NS simulations do not fully match quantitatively to the experimental results, at

least for cs < 0.5 M, they do 1) reproduce the observed trends qualitatively and 2) exhibit much smaller errors relative to the PNP-NS results (data not shown).

2.2 Peak values of the radial potential proﬁles inside ClyA

The peak values of the radial electrostatic potential hϕirad at the cis entry, middle of the lumen and the trans constriction for 0.005, 0.05, 0.15, 0.5 and 5 M NaCl are summarized in Tab. S3.

S-12 a b

Rectiﬁcation α = G(+V )/G( V ) 50 mV 100 mV 150 mV b − b expt. sim. (ePNP-NS) 0.15 M 1.75 1 1.25 1.5 1.75 1.65 5 1.01 1.05 1 1.1 1.46

1.42 α 1.25 1.5 (M) 1.28

s 0.1 1.75 c 1.20 1.14 0.01 1

0 200 0.01 0.1 1 5 Vb (mV) cs (M)

Figure S3. Measured and simulated ionic rectiﬁcationof single ClyA nanopores. (a) Contourplot of the ionic current rectiﬁcation α = G(+Vb)/G(−Vb), computed from the ionic conductances in the ePNP- NS simulation, as a function of Vb and cs. (b) Comparision between the simulated (ePNP-NS) and measured (expt.) α as a function of cs for 50 mV, 100 mV and 150 mV.

Table S3. Peak radial potential.

hϕirad (mV) cis lumen trans cs (M) z ≈ 10 nm z ≈ 5 nm z ≈ 0 nm 0.005 −80 −108 −144 0.05 −34 −50 −86 0.15 −19 −29 −57 0.5 −9.3 −14 −30 5 −1.9 −1.7 −4.2

2.3 Pressure distribution inside ClyA

The electro-osmotic pressure distribution inside ClyA, as a consequence of the strong local enhancement of the ion concentration, is given in Fig. S4.

S-13 a b Hydrodynamic pressure p (atm)

1 0 1 2 5 10 20 30 − 15 cs = 0.15 M, Vb = 0 mV

0.05 M 0.15 M 0.5 M 5 M lumen

4.5 z (nm)

0 constriction

-5

-7.5 0 7.5 -5 0 25 r (nm) p (atm) h irad

Figure S4. Pressure distribution inside ClyA. (a) Contourmap of the pressure at cs = 0.15 M and + Vb = 0 mV, showing that the Na concentration ‘hotspots’ near the pore wall result in build-up of electro- osmotic pressure (5 atm to 30 atm) inside the confined fluid. Pressure drops of such magnitude over the course of a few nanometers could potentially exert a significant force on a captured protein. 19 (b) The axial pressure profile and averaged along the the entire radius of the pore at Vb = 0 mV.

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