(Electrostatics in Biology)

Transcription

1 Practice-oriented, student-friendly modernization of the biomedical education for strengthening the international competitiveness of the rural Hungarian universities TÁMOP C-13/1/KONV (Electrostatics in Biology)!X Alajos Bérczi Institute of Biophysics, BRC HAS, Szeged, Hungary 14th September, 2016! X

2 ... Most biophysics courses and textbooks (at least in Hungary) deal with electric potential differences (e.g. diffusion potential, Donnan potential, action potential, Goldman-Hodgkin-Katz potential, etc.) but neglect the electrostatics of point charges. This lecture is intended to make up for this lack by giving a selected introduction into the parts of electrostatics of point charges that will then be employed in explaining some experimental observations.... Introduction - Why electrostatics? - Science-historical background (Maxwell equations) Electrical charges in aqueous media - The Poisson-Boltzmann equation - The Debye-Hückel approximation - Electric dipoles in aqueous media - Counterion condensation Surface charges and surface potentials at interface - The Gouy-Chapman theory - Surface charge layers Employing electrostatics in biology - Determination of surface charge density and surface potential of membrane vesicles (electrophoresis, fluorescence method) - Monovalent salt dependence of enzyme reactions - Like-charge attraction in polyelectrolytes - Variety of redox potential of the very same redox center - Zn 2+ -transporting ATPase Outlook! X

3 Introduction Why electrostatics? Martin Rees (1999): Just six numbers -/- Csak hat szám =0.007; 2p+2n He -- but m=-0.7% actual / critical (?) ~0.04 (1) l the cosmological constant (?) very small Q ratio of two energies (gravitational/ rest-mass ) ~10 5 D spatial dimension 3 N the ratio of F el to F gr ~ X

4 Introduction Gravitational field: Electrostatic field: F = f m m r F = f q q r f = N m kg f = 1 = N m 4πε C always attractive attractive or repulsive For two protons or two electrons at l = 1 nm (=10 Å) distance: 2 protons F = N F = N F F = electrons F = N F = N F F =

5 Introduction Fundamental interactions in nature Gravitational Electromagnetic Weak Strong??? dark??? Relative strength ???

6 Why electrostatics in biology? Biological objects: mass (m), electrical charge(s) (q, Q), typical distances and sizes are in 0.5 nm to 1 μm range (R, r), colloid system, dielectric media ( m, p, a ), Na + Example: Na + and Cl, r =2 nm (20 Å), in water ( r =80), gravitation (Newton s law): F g f g m m r F g = N r Cl - electrostatic (Coulomb s law): F e r q q r F e = N F e >>> F g (F e = F g ) Why electrostatics? motion by diffusion: m s 1 propagation of the electromagnetic effect: m s 1

7 Why charge-charge interactions firstly? Let T=298 K (25 C), r =80, q = C, r =1 nm (5 nm), l =0.3 nm (0.5 nm), =90, thermal energy (for each degree of freedom): potential energy in charge-charge interaction: potential energy in dipole-charge interaction: potential energy in dipole-dipole interaction: E T 2 kt E E E qq d q d d r r r q q 1 2 r p1q 2 r 2 p p r where p ql + 1 3cos E T = J E q-q = J (E q-q = J) E d-q = J (E d-q = J) E d-d = J (E d-d = J) Charge induced dipole and dipole induced dipole interactions are weaker (depend on polarizability a of molecules) and their distance dependence is r 4 and r 6, respectively.

8 Electromagnetism: Maxwell equations Formulation in terms of total charge and current; The Maxwell s microscopic equations in vacuum Gauss' law: Name Differential form Integral form James Clark Maxwell Gauss' law for magnetism: Maxwell-Faraday equation (Faraday's law of induction): Ampère's circuital law (with Maxwell's correction): The four Maxwell s microscopic equations describe the electric and magnetic fields arising from varying distributions of electric charges and currents, and how those fields change in the vacuum in time. In the presence of matter, however, the following equations will modify the Maxwell s microscopic equations:

9 The Poisson-Boltzmann equation If (1) there are no time-dependent changes of the field parameters, (2) there is no magnetic field (B = 0) and (3) no electric current (J = 0), then the following 3 equations will describe the electrostatic field: (a) No curl (b) No divergence The combination of the above 4 equations will result the most general form of the Poisson equation that can be used as strating point in the discussions of almost all biophysical problems: If, furthermore, (4) there is no polarization (P = 0), (5) the aqueous medium is homogeneous and isotropic ( = 0 r = const.), (6) the distribution of ions are according to the Boltzmann statistics, and (7) under the influence of electrostatic interactions only, then the Poisson equation gets its most frequently used form the Piosson-Boltzmann equation: It should be noted that general solution of this equation does not exists.

10 3D-Coordinate systems

11 The Debye-Hückel approximation The Debye-Hückel (DH) limiting law for electrolytes. For small potentials, i.e. where ey<<kt, the exponential factor in the Boltzmann equation can be linearized (i.e. application of Taylor series expansion for all exponential terms in the Boltzmann equation), e x ~1+x+ and the Poisson-Boltzmann equation can be solved. If (i) only Coulomb interaction exists between ions (the other types of interaction are neglected), (ii) spherical symmetry can be applied for all ions, (iii) which are considered as point charges and ions around a point charge is approximated by ionic clouds with continuous charge distribution, and (iv) the medium is considered as a homogenous and isotropic continuum, we obtain the following equation for the potential distribution around a charge Ze where The Debye length is a useful measure of the spatial extent of the diffuse charge layer. The Debye lengths are in the nanoscale regime and significantly reduced by increasing ionic strength. and is the reciprocal Debye length, (1/ D ). A Debye sphere is a volume whose radius is the Debye length ( D ), outside of which charges are electrically screened. When the size of the central ion Ze is also taken into account, the potential distribution at r > R is:

12 Ion association temporary ion pair formation Ion association (temporary ion pair formation): it occurs when - the electrostatic energy of attraction between unlike charges is larger than the thermic energy; or with other words, - cations (z (+) ) and anions (z (-) ) in an aqueous medium are mutually present in each other s electrostatic field and the centre of an anion is closer to the centre of a cation than a critical distance (l crit ): l B + where is the Bjerrum length. 2 2l B 4l B l B Since l B 7 Å in an aqueous phase and at room temperature, ion pairs play important role in the properties of polyelectrolytes.

13 Dipoles and set of point charges in a well-defined space The electrostatic field of two point charges with opposite sign is the superposition of the two electrostatic fields of the point charges. Far from the dipole (i.e. when L<<r) the potential distribution will be Since r i q i R d i If +q i then The last equation would describe the superposition of the electrostatic fields of an electric point charge Q and of a permanent electric dipole p. In such case the p is called the dipole moment of a space charge.

14 Counterion condensation theory Polyelectrolytes (or polyions) are macromolecules in which a substantial portion of the constitutional units have ionizable and/or ionic groups (proteins, DNA, RNA). Polyelectrolytes which bear both cationic and anionic groups are called polyampholytes (proteins). Ion association theory predicts that such molecules exist always together with associated counterions. According to the original counterion condensation theory (G.S. Manning, 1969), if (i) the contour length of a linear polyelectrolyte is L, and (ii) this polymer chain bears N charged groups of valence z N, (the linear charge density of a polyelectrolyte molecule is z N en/l), and (iii) the valence of counterions is z ci, then counterions should condense around the polyelectrolyte molecules as long as the reduced charge spacing, l fulfils the following inequality: Remarks: (1) While the condensation of counterions is an assumption in the Manning s theory it automatically shows up in the solution of the Poisson- Boltzmann equation in cylindrical coordinates employed for rod-like polyelectrolytes. (2) While the original counterion condensation theory deals only with the polyelectrolytes (without any additional solutes in the medium), the second counterion condensation theory (Manning et al , Manning 2011) study the problem in the presence of a bias electrolyte solution. Charge renormalization: Counterion concentration profile vs. radial distance around an infinitely long rod with radius of 5 Å and linear charge density equal to the B-DNA in 0.01 M electrolyte of 1:1 type. The Poisson-Boltzmann model (solid line) and the counterion condensation model (broken line) is compared. where is the fraction of condensed z ci -valent counterions per polymer charged site. Thus, for instance, on B-DNA (L/N)1.7 Å and since l B 7 Å, z ci 0.8 in case of z ci =1 (Na + ), and z ci 0.9 in case of z ci =2 (Mg 2+ ).

15 The Gouy-Chapman theory The Gouy-Chapman theory for charged surfaces preceded the Debye-Hückel theory for electrolytes by about 10 years. The basic assumptions are identical in the two theories. Let us assume that (i) there is an infinite surface with uniform surface charge density of at the YZ plane and (ii) a homogeneous and isotropic aquous medium in the X>0 half space with (iii) point charges and the distribution of charges follows the Boltzmann statistics. In such case the Poisson-Boltzmann equation can be integrated once and we obtain the following one-dimensional form: Under the following boundary conditions the relationship between the surface charge density (), surface potential ( 0 ) and the ion concentrations (c i ) in the medium will be given as: Red curve refers to ionic medium of 20 mm 1:1 electrolyte. Blue curve refers to a medium of 20 mm (M + ), 30 mm (A ) and 5 mm (D 2+ ). is about 10 times that of the thylakoid membranes.

16 The Gouy-Chapman theory If there is only one biner z:z electrolyte present in the aqueous medium (e.g. z = 1 in case of NaCl and z = 2 in case of MgSO 4 ), and the bulk concentration of the electrolyte is c( ), the one dimensional potential distribution, ψ(x), is given by the following equations: where 2 mm MgATP -2 - anion r 0 =5 Å T=300 K 0.15 M NaCl r>r (r ion ) (1/)=0.256 nm 0.1 As m -2 Red arrows and curve 1 Gouy-Chapman theory Blue arrows and curve 2 Debye-Hückel theory Distribution of the potential in the electric double layer at 4 different surface charge densities in a 100 mm 1:1 electrolyte solution (left panel) and in 4 different 1:1 electrolyte solutions at fixed surface charge density (right panel)

17 Surface charge layers The Gouy-Chapman theory has been improved by taking into account the spatial distribution of surface charges at the interface and the penetration of electrolytes into the membrane phase. The interface region between aqueous and membrane phases is a region of continuous changes from point of view of both size and composition. Charged groups (e.g. phosphate), dipoles (e.g. =C=O and water), and induced dipoles determine both the relative permittivity ( r ) and the Debye length ( 1 ) in this region. Analytical form for the potential function thus can hardly be derived, however, numerical approximations do exist for some special cases.

18 Surface charge layers z:z electrolite With the following boundary conditions of and ( ) = 0 the problem can be solved completely. After introducing the dimensionless potential (y) and the reciprocal Debye length ( and ) in both compartments the potential distribution will be given as Potential distribution at membrane surface with space charge layer of thickness d = 10, 5, 0 Å (curves 1, 2, and 3, respectively). z = 1, c() = 0.1 M, e d d = = Cm 2 and m = a (solid lines) or m = 0.5 a (broken lines).

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20 Surface charge density and surface potential determination (1) According to the Helmholtz-Smoluchowski equation, a particle s electrophoretic velocity (v el ) is proportianal to the applied electric field strength (E), the zeta potential (the potential at the hydrodynamic plane of shear SE on the picture - z), and inversely proportianal to the dynamic viscosity of the medium () Charge distribution around an electrophoretic particle (liposome) in electrolyte. The equation has been derived with the following assumptions: the radius of the vesicle is much higher than the Debye length (generally higher than 100 nm; e.g. colloid particles), the charges are at the particle/solution interface, the surface is smooth, and there is no specific adsorption influencing the hydrodynamic plane of shear (this is hardly fulfilled). A more general formulation is the Henry s equation which takes into account the ionic strength of the medium (via the Debye length 1/) and can be used for charged particles with radius R Uncertainity of surface potential with identical zeta potential; the real surface charge density cannot be inferred from zeta potential. where f H (R) is the Henry s function (which approaches 1 for small R and 3/2 for large R). Problem unsolved here: the relation between and 0?

21 Surface charge density and surface potential determination (2) The simplest spectroscopic method for determining the surface charge density of membrane vesicles uses the relative fluorescence intensity values of a water soluble dye which are obtained when the quenched fluorescence in the electric double layer is released by replacing the dye molecules with monovalent or divalent cations. We have seen that With a proper choice of fluorescent dye (e.g. 9-aminoacridine: 9AA), and ionic solutions [e.g. KCl and (HM)Br 2 or (DM)Br 2 ], and assuming that fluorescence quenching of fluorescent molecules in the electric double layer is independent of the valence of cations used for the titration, both s and y 0 can be determined as follows... PM vesicles (bar 1 nm) Flourescence titration

22 Surface charge density and surface potential determination (3) Let us introduce K = 2000FkTε ε e and y = exp F RT then the equation for surface charge density given before will be as follows: σ = K c y 1 where c i and z i is the concentration and the valency of the i-th ion in the solution. Titration of the surface potential with 1:1 and 2:1 electrolytes in the presence of some buffer solution we may write for the surface charge density and surface potential σ = K c + c y + y 2 σ = K c y + y 2 + c y + 2y 3 for the 1:1 electrolyte, and for the 2:1 electrolyte. The surface potential ( o ) and the surface charge density () can be calculated as ψ = 2kT e arc sinh 2 H c and σ = K c + c H c where H c = c c 3 c c 2 and c m and c d are the monovalent and divalent cation concentrations, respectively. Bérczi A and Møller IM (1993) Eur. Biophys. J. 22:

23 Monovalent salt dependence of enzyme reactions According to the Michaelis-Menten kinetics We have seen that The surface potential depends both on the surface charge density and on the concentration of all ions in the medium. Thus any change in the concentration of any ion will affect the apparent value of the Michaelis-Menten constants. where and z S is the valence of the substrate. (1) Shift of the ph optimum by monovalent salts. Let 0, 01 0, 1:1 electrolyte, z S 0, and let change the monovalent salt concentration c m2 =10c m1. As consequence, but 02 0, and thus the surface ph value increases (the surface concentration of H + decreases) and the optimum ph value will shift to a lower ph value. (2) Activation, inhibition by monovalent salts. Let 0, 01 0, 1:1 electrolyte, z S 0, and let change the monovalent salt concentration c m2 =10c m1. As consequence, but 02 0, and thus [S] 0 increases and K M, app decreases.

24 Like charge attraction of polyelectrolytes (1) At the beginning it was known that multivalent ions generate attractions between like-charged polyelectrolytes in a wide range of systems, while monovalent ions do not. It was also known that different ion valences are required to condense different polyelectrolytes. How multivalent does an ion have to be before it can condense a given polyelectrolyte? Using the charge-tunable M13 filamentous virus system and a family of artificial homologous divalent ions of different effective sizes, a multivalent ion-polyelectrolyte phase diagram was constructed, and an experimentally motivated general criterion for like-charged attraction based on the ion size and the Debye length was established. M13 virus with ~2700 major coat proteins Dimethonium cations (n=1,2,3,4) M13 virus 5 mm Tris-HCl, 1 mm NaN 3 Small Angle X-ray Scattering (SAXS) measurements When >d (the Gouy-Chapman length around the polyelectrolyte charges is larger than the size of divalent cations), local charge inversion is possible. This may result in condensation ( association ) of polyelectrolytes.

25 Like charge attraction of polyelectrolytes (2) The basic elements of the second counterion condensation theory (Ray and Manning, ) are as follows. For two parallel polyions (separated by distance r and in z:z electrolyte), there are three components of the free energy: - the ionic interaction within each polyion, W p (r); summation over all Debye-Hückel pairwise interactions among the renormalized charges within each polyion, - the ionic interaction between the two polyions, W pp (r) K 0 (r); summation over all Debye-Hückel pairwise interactions among the renormalized charges, one member of the pair on one polyion, the other on the other polyion, - the work of transfer of a subpopulation of counterions from the bulk solution to the condensed layer on the two polyions, W transfer (r); According to the second counterion condensation theory, W(r) comes in three separated pieces: - the near distances: a < r < (e) 1 where a is the cutoff distance of the closest approach, 1 is the Debye length, - the intermediate distance: (e) 1 < r < 1 where 1 is the Debye length, - the far region: r > 1 where 1 is the Debye length. In the three different regions different approximations are taken for the expansion of the zeroth-order modified Bessel function of the second kind, K 0 (r).

26 Born model for solvation... and the redox potential of proteins The simples model for charcterizing a charged particle in a medium is based on the Born model. According to this model the electrostatic self-energy of an ion with charge Q and effective radius R (Born radius of ion) in free space (vacuum) is given as and in a medium as Thus the Born free energy of hydration or solvation (i.e. the energy needed for moving the charge from the free space into the medium) can be defined as the difference of W m and W v Similarly to this, the energy needed for moving the charge Q from the aqueous medium into a hydrophobic medium can be defined as the difference of W h and W a Problems: - infinite media, - the dipole character of the medium is neglected,

27 Zn 2+ -transporting ATPase (ZntA); a putative mechanism (Nature 2014, doi: /nature13618) Putative zinc transport mechanism of ZntA from Shigella sonnei. Phosphorylation events in the intracellular domains drive large conformational changes that permit alternating access to transport sites in the membrane about 50 Å from the ATP-targeted catalytic aspartate. According to the model, (1) a high-affinity state (E1), which is open to the intracellular space, binds to Zn 2+ and enters an occluded state. (2) This state then undergoes phosphorylation. (3) Completion of this event (E1P) triggers the release of the Zn 2+, establishing an outward-facing, low-affinity state (E2P). (4) Release of the inorganic phosphate (Pi) yields the fully dephosphorylated conformation (E2), which is followed by... (5)... restoration of the inward-facing conformation (E1), which initiates a new reaction cycle.

28 Zn 2+ -transporting ATPase (ZntA); a putative mechanism (Nature 2014, doi: /nature13618) Putative zinc transport mechanism of ZntA from Shigella sonnei. A transport cycle based on schematic models of the E1 and E1P states and the E2P and E2-P i structures. (1) In the presence of intracellular zinc, Zn 2+ enters the ATPase through the electronegative funnel (red). (2) Upon Zn 2+ binding to the intramembranous ion-binding site (grey circle),... (3)... F210 and M187 occlude the ion entry funnel, preventing backflow of Zn 2+. (4) Substantial domain rearrangements in transition to the E2P state open the extracellular pathway, lowering the affinity for Zn 2+ and... (5)... mediating Zn 2+ release,... (6)... possibly stimulated by K693. (7) Dephosphorylation triggers closure of the transmembrane domain, in which K693 (as a built-in counter ion) forms a salt bridge with D714. Upon dephosphorylation, the side chains move to their initial positions before an E2 to E1 transition is stimulated by the presence of intracellular Zn 2+.

29 Outlook Deficiencies in and/or problems with the standard Poisson-Boltzmann approaches (the linearized and the non-linearized forms) are well known and originate from (or go back to) the two basic assumptions of the theory. The Poisson- Boltzmann theory, for instance, underestimates the actual potential at highly charged surfaces by 15 25% for monovalent and divalent electrolytes. Neglecting of (a) ion correlation, (b) dielectric saturation and (c) finite size of ions and solvent molecules largely contribute to the inaccuracy of the theory. The way to overcome the problem is possible only after 1) abandoning the assumption (constraint) for the dielectric permittivity of the medium; as soon as the length scale of the order of a few Ångstroms becomes important, approximation of aqueous solution as a uniform dielectric medium is no longer sufficient and the Poisson-Boltzmann equation has the following more general, nonhomogeneous form: ε ψ r + P = ρ r ε r ψ r = ρ r F z ε 1000ε c exp z eψ r kt where f (r) is the charge density function of fixed charges (on molecules), r (r) refers to the electrostatic inhomogeneity of the medium (space-dependent relative dielectric constant), otherwise as in earlier, 2) taking into account the finite size of ions and the charge distribution inside the macromolecules (see the term f (r) in the above equation), and 3) introducing non-electrostatic component in the model. All of these three modifications have been introduced by nowadays and modeling of any biological processes on the basis of electrostatic interactions between the participants has been the most promising and fruitful in theoretical biophysics and provides the background for molecular modelling. The past 100+ years have proven that electrostatic interaction is the only interaction among the 4 fundamental interactions in nature that governs the biological processes and provide perfect starting point for quantitative discussion of any biological problems. We must, however, also be aware of that as soon as we want to go as deep as atomic level, the only proper theoretical background is quantum physics (quantum electro-dynamics).

30 Thank you for your attention! This work is supported by the European Union, co-financed by the European Social Fund, within the framework of " Practiceoriented, student-friendly modernization of the biomedical education for strengthening the international competitiveness of the rural Hungarian universities " TÁMOP C-13/1/KONV project.