Fundamental Principles to Tutorials. Lecture 3: Introduction to Electrostatics in Salty Solution. Giuseppe Milano

Transcription

1 III Advanced School on Biomolecular Simulation: Fundamental Principles to Tutorials Multiscale Methods from Lecture 3: Introduction to Electrostatics in Salty Solution Giuseppe Milano

2 Reference Rob Phillips, Julie Theriot, Jane Kondev You can skip in the first reading red bar slides where some derivations are perfomed

3 Electrostatics in Salty Solutions Macromolecules of life inhabit a watery medium. Charge state of DNA and proteins is largely controlled by the ionic character of the surrounding solution Electrostatics of charges in water and how the energetics of such charges plays a role in dictating the binding and assembly reactions that are so central to biology

4 Water and Ions: long range interactions Water as the medium for life is its unusual ability to interact with charged molecules ranging from individual ions such as K+ and Na- to enormous charged macromolecules such as DNA. Charge interactions play a very special role in the function of biological molecules because they can act over long distances.

5 Electrostatics and Specificity biological function usually requires specificity. That is, a macromolecule will bind to some very specific partners but will not bind to others of similar shape and size. Dissecting protein RNA recognition sites Nucleic Acids Research 36(8): June 2008 In many cases, binding specificity arises because of complementary patterns of charge distribution on the surface of the two macromolecules involved in the interaction

6 Salt and Binding Many events in the lives of biological molecules involve specific recognition of one molecule by another. RNA polymerase binding to DNA specifically at promoter sites and receptors binding their specific ligands The importance of electrostatics in macromolecular binding is easily observed in the laboratory because nearly all such binding interactions are strongly dependent on the concentration of ions in the solution in which they are measured

7 Salt and Binding: DNA Protein Interactions Salt dependence of protein DNA binding. Equilibrium constant for Lac repressor binding to nonspecific DNA as a function of Na+ concentration. The range of in vivo salt concentrations is shaded. (Adapted from Y. Kao-Huang et al., Proc. Natl Acad. Sci. USA 74:4228, 1977.) There are several intuitive ways to begin to think about these results. Ions in salty solutions assemble in a screening cloud around the macromolecules that have the potential to bind. When this binding reaction occurs, there is a release of the ions in these screening clouds, which results in an entropy increase. With increasing ion concentration, the ions can actually interact with the receptor in a way that competes with the ligands.

8 Electrostatics: Water Screening electrostatic force is diminished between two point charges due to the intervening molecules, which have permanent dipoles. As will be shown later, in the presence of free charges, such as salts, this screening effect is even more extreme

9 Electrostatics: Electric Field Instead of considering the forces between charges, we can introduce the concept of an electric field E that the force acting on a test charge qtest placed in the field is qteste. Note that this change in perspective places the emphasis on the space surrounding charges (that is, on the fields) rather than on the charges themselves

10 Electrostatics: Electric Field that the force acting on a test charge qtest placed in the field is qteste. for the electric field at position r; er is a unit vector along the radial direction r.

11 Electrostatics: Electric Field II In the case of a charge distribution, where charges qi are placed at specified positions ri, the electric field at position r is obtained by summing contributions from each of the individual charges The fact that the total field is simply obtained by adding the fields thatc would have been produced by individual charges assuming all other charges were absent is the principle of superposition

12 Electrostatics: Charge Distribution The concept of a charge distribution is very useful for describing charged macromolecules such as proteins and DNA in solution. Instead of a discrete distribution like that described above, which is specified by charges qi and their positions ri, it is often useful to introduce a continuous charge distribution specified by a density ρ(r) Given the charge density, the charge Q within a small volume DV located at position r is ρ(r)dv. The only difference is an operational one whereby we substitute the sum over discrete charges in Equation 9.14 with an integral over volume elements dv

13 Charge Distributions: Discrete vs. Continuum Electric field of a charge distribution. (A) For a single charge, the field lines emanating from the charge are directed radially outward (positive charge) or inward (negative charge). (B) For the case of a discrete set of charges, the electric field at a given point is obtained by computing the vector sum of the electric fields due to the individual charges. (C) For a continuous charge distribution, the electric field at some point is obtained by adding up (by integration) the contributions to the field due to every little volume element in the chargedbody.

14 Electric Field: Flux For electrostatics, we now develop a local relation that connects the electric field at position r to the charge density at the same position. E(r) and ρ(r)? To uncover this relation, we consider the flux of the electric field. If we think of the lines of electric field distributed through space, we define the flux as the number of lines per unit area, where the area element is perpendicular to the field direction.

15 Electric Field: Flux For electrostatics, we now develop a local relation that connects the electric field at position r to the charge density at the same position. E(r) and ρ(r)? To uncover this relation, we consider the flux of the electric field. If we think of the lines of electric field distributed through space, we define the flux as the number of lines per unit area, where the area element is perpendicular to the field direction.

16 Electric Field: Flux Mathematically, this operation amounts to an integration in which we break the surface up into small area elements Ai with surface normals ni, and for eachelement we compute E(ri) =nia Area della Sfera

17 Electric Field: Flux The flux does not depend on the size of the sphere. This is in line with the intuitive definition of the flux as the number of field lines piercing the sphere. If we take into account the fact that field lines start and end only at charges, then the number of field lines through the sphere will not depend on how big it is. Moreover, this number will not change if we deform the sphere to an arbitrary shape. Superposition principle

18 Gauss Law This same result applies just as well to a spherical charge distribution that has a charge Q spread uniformly throughout its volume. Evaluate the flux through a surface with radius larger than that of the ball of charge itself. The total flux of the electric field through a sphere of radius r is 4πr^2 E(r)

19 Gauss Law: E(r) and ρ(r)? Charge distribution where the charge density only depends on the coordinate x while it is uniform in y and z nonuniform field in the x-direction The charge contained in the small volume The flux of the electric field through the outer surface of this volume element is

20 Gauss Law: E(r) and ρ(r)? Taylor Expansion Subsitute this in 9.17

21 Gauss Law: E(r) and ρ(r)? Repeat the same for y and z This is a result more general of Gauss law connection E(r) -> r(r)

22 Electric Potential V(r) E(r) is a vector field simpler description of the field is obtained in terms of the electric potential V(r), which is a scalar field. The electric potential at position r is defined as the work per unit charge done in bringing a test charge from a point at infinity to the position r. The process of transporting the charge is assumed to be very slow so that at any given instant the force applied on the charge is exactly cancelled by the force due to the electric field

23 Electric Potential V(r) From the symmetries of this charge distribution the electric field points along the x-direction. electric force on a unit test charge is Ex(x)ex ex unit vector x direction Differentiate both sides

24 Electric Potential V(r) using We obtain For the special case of a uniform field, the integral results in V(x) V(0) = Ex x. if we are interested in the difference in potential between two neighboring points, we have Equivalent to (9.23)

25 Electric Potential V(r): Poisson Eq Nelle 3 direzioni indipendenti Given the electric potential V the electric field E is a vector che pointing in the direction of the fastest derease of the potentiale. E module is the derivative of the potential along this direction. Remeber Gauss law We obtained Poisson Equation

26 Electric Potential V(r): Poisson Eq In particular, given a scalar field such as V(r), we can compute an associated vector (known as the gradient vector) of the form

27 Energy Cost of Collection of Charges Cells invest a great deal of their energy budget in moving charges around. To understand better the energetic implications of charge management, we introduce the energy associated with a charge distribution. What is the work done in bringing isolated charges from far away, where they do not interact, to form such a distribution? The electric energy is equal to this work. Positive value of the electrical energy means that another form of energy has to be expended in order to assemble the charge distribution. two point charges separated by a distance r.

28 Energy Cost of Collection of Charges In continuos form? As an example we compute the electrical energy of a ball of radius R with a charge Q uniformly distributed throughout its volume

29 Energy Cost: Sphere To do this calculation, we consider the work in assembling this ball by breaking it up into the work needed to put together spherical shells of thickness dr and charge dq to form the ball. If the charge of the assembled shells is q, the increase in electric energy when another shell is added is This equation says that when adding layers in the onion, adding the layer at radius r costs an energy dictated by the potential of the charge already there, V(r), times the amount of charge added in that shell, dq. The potential of the charge already there once the ball has been built up to a radius r is gotten by remembering that the electric field outside a charged ball is the same as for a point charge placed at its center.

30 Energy Cost: Sphere We need V(r) Using the definition of the potential as the work of the electric field, we find

31 Energy Cost: Sphere Summing up over all shells, which amounts to integrating Equation above over r, we find We used

32 DNA Condensation in Bacteriophage φ29 DNA inside a virus is contained in the volume provided by the viral capsid.

33 DNA Condensation in Bacteriophage φ29 DNA is a strong acid, it is highly charged in solution. Therefore, energy needs to be expended in order to bring all the charge carried by the DNA into close proximity within the capsid. For the bacteriophage φ29, this energy is provided in the form of mechanical work done by the portal motor (a protein machine that translocates DNA), which in turn is fueled by ATP hydrolysis. These measurements reveal that upon packaging of the DNA there is an internal force build-up. The work against this internal force can be estimated from the measurements by evaluating the area under the force length curve and is about This estimate uses the fact that length of the φ29 DNA is roughly 7μm and the maximum force is 60pN (for this rough estimate, we assume a simple linear rise of the force with the amount of DNA packaged).

34 DNA Condensation in Bacteriophage φ29 The total charge carried by the φ29 genome, which is roughly 20,000 base pairs long, is where e = C is the charge of an electron. Assuming that all this charge is spread uniformly throughout the viral capsid, which we approximate as a sphere of radius R 20 nm, This estimate of the charging energy is 2000 times greater than the measured work against the internal force!

35 DNA Condensation in Bacteriophage φ29 Apparent violation of energy conservation Forces between the charges on the DNA are screened by the presence of counterions, making them effectively much weaker than Equation 9.35 would lead us to believe. We need to understand the nature of screening and its biological implications

36 Charged Proteins To gain quantitative intuition about the charging of proteins in solution we consider a simple model of a globular protein as a ball of radius R composed of amino acids, which are represented as tightly packed smaller balls whose radius is r. All of the hydrophobic residues are sequestered within the protein and the surface is assumed to be composed strictly of polar residues, each of which is assumed to be able to surrender a single charge

37 Charged Proteins II All of the charge is concentrated on the shell of radius R We can show that if charges are only on the surface The number of residues is given by Let s write U in terms of Bjerrum length

38 Charged Proteins III Nanometer-sized objects can tolerate charge loss

39 Energy and Scale The nanometer is the key scale of biology

40 PAUSA

41 DNA Condensation in Bacteriophage φ29 Apparent violation of energy conservation Forces between the charges on the DNA are screened by the presence of counterions, making them effectively much weaker than Equation 9.35 would lead us to believe. We need to understand the nature of screening and its biological implications

42 Nature of Charge Screening: Bjerrum length The Energy to Liberate Ions from Molecules Can Be Comparable to the Thermal Energy The idea is illustrated in Figure, which shows two equal and opposite charges that have been separated by a distance (the socalled Bjerrum length) such that their electrostatic interaction energy is equal to the thermal energy kt.

43 Screening in Salty Solutions The distribution of ions around a given protein or DNA molecule depends upon the shape of that molecule and the concentration of these surrounding ions Develop a precise description of the distribution of such ions around the DNA molecule and to find how the presence of such molecules alters the electric potential set up by the molecule.

44 Screening in Salty Solutions II A charged macromolecule such as a protein or nucleic acid will acquire a screening cloud in a salty solution such that the total charge on the macromolecule is neutralized as shown in Figure on the left distance over which the cloud extends is determined by a competition Between? Energy and entropy of the ions. Total charge Q. In this case, positive counterions are drawn into the vicinity of the macromolecule, thus forming a screening cloud. These counterions adopt this configuration since by doing so they lower their electrostatic energy. At the same time, the local increase of the concentration of counterions in the screening cloud is characterized by an entropic penalty since they are effectively more confined.

45 Screening in Salty Solutions II A charged macromolecule such as a protein or nucleic acid will acquire a screening cloud in a salty solution such that the total charge on the macromolecule is neutralized as shown in Figure on the left distance over which the cloud extends is determined by a competition Between? Energy and entropy of the ions. Total charge Q. In this case, positive counterions are drawn into the vicinity of the macromolecule, thus forming a screening cloud. These counterions adopt this configuration since by doing so they lower their electrostatic energy. At the same time, the local increase of the concentration of counterions in the screening cloud is characterized by an entropic penalty since they are effectively more confined.

46 Screening in Salty Solutions III The Size of the Screening Cloud Is Determined by a Balance of Energy and Entropy of the Surrounding Ions To estimate the width of the screening cloud, we assume that the concentration of counterions is uniform in the cloud, and given by

47 Screening in Salty Solutions: Width of screening cloud the factor of Q/e is the number of charges and permits us to compute the concentration.

48 Screening in Salty Solutions: Width of screening cloud The average electrostatic potential within the screening cloud is therefore The average electrostatic potential within the screening cloud is therefore

49 Screening in Salty Solutions: Width of screening cloud In equilibrium, the chemical potential for the positive counterions is everywhere the same the chemical potential beyond the screening cloud, where the concentration of ions is c and the electrical potential is zero, is On the other hand, the chemical potential of an ion that finds itself in the screening cloud is Charge of Energy in the cloud

50 Screening in Salty Solutions: Width of screening cloud Equating the chemical potentials for the counterions inside and outside the screening cloud and substituting Equation 9.45 for c and Equation 9.47 for Vcloud leads to If we assume that the change in concentration of the positive ions in the screening cloud is small compared with c, we can expand the logarithm Remember that

51 Screening in Salty Solutions: Width of screening cloud For distances greater than the Debye length, the charged macromolecule and its associated screening cloud are effectively neutral. nature of the electrostatic interaction between two charged macromolecules. Interaction is shortranged. no interaction unless the two macromolecules are brought in sufficiently close proximity that their screening clouds overlap. At these short distances, the interaction is a combination of entropic and energetic effects associated with changing the screening cloud configuration

52 DNA Condensation in Bacteriophage φ29 DNA inside a virus is contained in the volume provided by the viral capsid. This estimate of the charging energy is 2000 times greater than the measured work against the internal force!

53 DNA Condensation: Screening The physical picture associated with screening can serve as the basis for making a new estimate of the energy expended in packing DNA into a viral capsid that goes beyond the estimate given earlier Screening cloud made up of positively charged counterions only Charge on the DNA is perfectly screened so that no electrical interaction between these charges remains Free-energy cost for packing DNA ìresult of decreasing the entropy of the charges that make up the cloud of counterions as they are transferred to the confines of the capsid

54 DNA Condensation: Screening The size of the screening cloud is given by the Debye length, which under physiological conditions with a characteristic salt concentration of 100mM results in a screening length λd 1 nm. When the DNA is packed into the viral capsid, the counterions enter the capsid as well to neutralize the DNA. The energy cost associated with this process corresponds to the pv work for squeezing the counterions into a smaller volume than they occupied when the DNA was outside the capsid

55 DNA Condensation: Screening The volume of the screening cloud is obtained by thinking of the DNA as a long cylindrical rod of radius RDNA surrounded by a cloud of thickness λd. This results in where we have taken L = 20, nm = 6800nm for the length of the DNA, RDNA = 1nm is the DNA radius, and λd =1nm is the Debye length. The volume available to the counterions in the capsid is the difference between the volume of the capsid and that of the DNA, f29 W charge 65,000 kbt, which is roughly a factor of 3 larger than the work measured in single-molecule experiments

56 Screening: Poisson Boltzmann Eq Given the crudeness of our approach, this is a satisfactory agreement, especially when we compare it with the estimate we obtained by completely ignoring the screening of the DNA charge by counterions. One factor that is missing in our estimate is the lowering of the electrostatic energy for the counterions in the capsid since they will now be on average closer to the DNA than they were in the screening cloud around the DNA free in solution. The more formal theory of charge rearrangements in solution is based upon the Poisson Boltzmann equation. This approach merges two of the most important results of theoretical physics, namely, the Poisson equation, which relates the electric potential to the charge density, and the Boltzmann distribution (andiamole a rivedere)

57 Screening: Poisson Boltzmann Eq The Distribution of Screening Ions Can Be Found by Minimizing the Free Energy the charged object in solution is a membrane of negative charge with charge per unit area given by σ distribution of ions around a charged object such as a macromolecule in solution containing positive and negative ions. To simplify matters, we consider ions that have a charge +z e or ze

58 Screening: Poisson Boltzmann Eq Close to the surface there will be an increase in the density of positive ions, which are attracted, and a decrease in the density of negative ions, which are repelled. Far from the macromolecule, electrical neutrality of thesolution demands that the number densities for the two ion species are the same, c+ = c = c More generally, in the case of many different ionic species, the neutrality condition can be written as where zi is t z is the valency of the ith species. The number densities for the ions in solution are then given by the Boltzmann formula

59 Screening: Poisson Boltzmann Eq

60 Screening: Poisson Boltzmann Eq The electric potential itself is determined by the distribution of ion charges The total charge density at position x is obtained by summing over the contributions from both the negative and positive charges and is given by This charge density is related to the electric potential at the same position by the Poisson equation and is given by Substituting Equation 9.54 into Equation 9.55 and the resulting expression for ρ(x) into the Poisson equation, we arrive at Poisson Boltzmann equation

61 Screening: Poisson Boltzmann Eq The Screening Charge Decays Exponentially Around Macromolecules in Solution

62 Poisson Boltzmann Eq: solution If we differenciate this function we obtain the same function This is a property of the exponential function. Indeed, we can check that

63 Poisson Boltzmann Eq: solution The constants A and B are fixed by boundary conditions. The first condition states that the potential far from the charged membrane is zero, which implies that B = 0. To determine the remaining integration constant, we can make use of the fact that the electric field at the surface of the membrane is (Gauss law) Substituting this solution into the equation relating the potential to the charge density away from the charged membrane, Equation 9.56

64 Poisson Boltzmann Eq: solution Key idea of the Debye length: beyond this length the potential is essentially zero and the charge distribution is uniform. For a charged protein in a salt solution with charge density c = 200mM, typical for potassium ions in the cell interior, the Debye screening length is roughly 0.7 nm. Beyond this distance the charge on the protein will not be felt by other charges. This signals the importance of the three-dimensional shapes of proteins in matters electrical, since for two proteins to experience electric interactions, their charged residues have to be able to come in close proximity of each other. This will be allowed geometrically if their shapes are complementary to each other.

65 Virus as charged spheres A second class of important problems concerns the charge state around proteins and macromolecular assemblies such as ribosomes and viral capsids. In this case, a useful idealization of these structures is a charged sphere. We use this result to estimate the electrostatic energy associated with the assembly of a viral capsid. In particular, we address the experimentally relevant question of the salt dependence of the equilibrium constant for this process.

66 Virus as charged spheres Two eq are very similar rv(r) is an exponentially decaying function, and hence A can be determined by

67 Virus as charged spheres Substituting this formula for the constant A into Equation 9.70, we arrive at an expression for the potential at the surface of a charged sphere, The electrostatic energy of the spherical shell is now simply

68 Nature of Viral Assembly

69 Nature of Viral Assembly The protein building blocks making up the capsid are held together by contact forces, such as the hydrophobic force, that arise between the surfaces of the protein units that are in contact. These protein building blocks are charged in solution. Important contribution to the energy budget of capsid assembly: electrostatic cost for bringing like charged capsomers in close proximity to form the capsid. Figure which shows the human hepatitis B virus, which is built up of 120 dimer capsomers

70 Viral Assembly Model A simple model that accounts for the electrostatic and contact (typically hydrophobic in origin) forces can be written as

71 Viral Assembly Model This model can be tested against experiments that measure theequilibrium constant Kcapsid of capsid assembly at different salt concentrations The data in Figure show a linear increase in ln Kcapsid as a function of temperature. This is to be expected given the small range of temperatures over which assembly of the virus is observed. Furthermore, we see that the slope of the curves is independent of salt concentration, indicating that the temperature dependence of Gcapsid comes primarily from the contact interactions.

72 Viral Assembly Model This model can be tested against experiments that measure theequilibrium constant Kcapsid of capsid assembly at different salt concentrations

73 Conclusions Charged macromolecules are surrounded by a nonuniform distribution of screening ions Presence of ions has many consequences for cellular life, evidenced by the host of phdependent processes in cells and the strong salt dependence of many important binding reactions Key theoretical tool that is used to think about these problems involves a merger of two important results from electrostatics and statistical mechanics, namely, the Poisson equation, which relates charge density to electric potential, and the Boltzmann distribution which assigns probability to states The resulting model involves a nonlinear partial differential equation for the electric potential. However, for small potentials, this equation can be approximated by a linear equation resulting in the Debye screening length. For simple planar and spherical geometries, these equations can be solved all the way to the end, and they reveal an exponential decay of the screening charge and potential