68 MACROMOLECULES. Z + ln Z (2.233) C = T ds dt. Nk B (ln G) 2 [ 1 12 N 1. Thus, the specific heat per bond diverges in the thermodynamic limit.

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1 68 MACROMOLECULES Hence S = T ln Z + ln Z (2.232) ( T = x ln G ) Z + ln Z (2.233) x x = θ ln G x + ln(xn 1) ln(x 1). (2.234) S θ ln G (2.235) for Nɛ << 1. The entropy is proportional to the order parameter. For the specific heat we find C = T ds dt (2.236) in the neighbourhood of T c C k B (ln G) 2 d θ dɛ Nk B (ln G) 2 [ 1 12 N N 3 ɛ ] (2.237). (2.238) Thus, the specific heat per bond diverges in the thermodynamic limit. 2.5 Polyelectrolyte Polyelectrolytes are one of the least understood states of condensed matter, in contrast to neutral polymer solutions. Recall that a chemical compound composed of ions, in a solid, liquid or dissolved state is called a electrolyte. Such a system exhibits electrolytic conductivity and interionic interaction. The ions typically have charges of magnitude, equal to their valency z multiplied by the electronic charge e. A polymer having sufficient ionic substituents along the chain to be water-soluble is called a polyelectrolyte. Thus, polyelectrolytes are polymers with ionizable groups that can dissociate in solution, leaving ions of one sign bound to the chain

2 2.5 POLYELECTROLYTE 69 Figure 2.27: Electrolyte coil and anions in solution. They are soluble in water, in contrast to most neutral hydrocarbon polymers, which are only soluble in organic solvents. This a drastic effect on the polymer conformation. Whereas for the uncharged polymer we found a random coil we now will get more extended conformations as shown schematically in figure Also the the solution of the electrolyte and solvent molecules will be more viscous. A biological ion such as DNA, which is negatively charged due to phosphate groups will be surrounded by the positive anions K +, Na +, Ca +2 and Mg +2. What we are interested in are natural polyelectrolytes and chemically modified polyelectrolytes. Examples of natural polyelectrolytes are Nucleic acids (anionic) poly(l-lysine) (cationic) poly((l-glutamic acid) (anionic) The starting point of our development of a theory is the Debye-Hückel potential [43] (see figure 2.28). It form is given by

3 70 MACROMOLECULES The Deby-H ckel function compared to 1/r U_DH(r) ,5 1 1,5 2 2,5 3 r Deby-H ckel function 1/r Figure 2.28: The Debye-Hückel potential U DH (r) = k B T l B r exp( κr). (2.239) It is a short-range potential, where κ, the screening length, controls how fast the potential drops. κ is connected to the Bjerrum length by κ 2 = 4πl B I, (2.240) with I being the ionic strength of the solution (I = ionic species Z2 i c i, Z i being the valence and c i the concentration of species i). κ can vary considerably over at least two orders of magnitude The Poisson-Boltzmann equation We consider a collection of fixed charges surrounded by a gas of ions. The description of the equilibrium distribution of the electrolytic ions starts off with an approximate description. This is because for the full problem we would have to study the thermal motion of a strongly interacting many-body system with

4 2.5 POLYELECTROLYTE 71 long ranged Coulomb force. Hence the usual approach is to take the Poisson- Boltzmann equation as a starting point. Let c i (x) denote the number density of the ions. The charge is z i e. The density is given by the Boltzmann equation c i (x) = C exp( βφ(x)z i e) (2.241) where the internal energy is given by the product of the charge and the potential φ(x). The normalization constant c is to be determined from the condition that the concentration of each species c i eventually is that of the bulk concentration c 0 i (the bulk concentration of the i-th ionic component in the bulk electrolyte). Next the electrostatic potential is given by the Poisson equation d 2 φ(x) dx 2 = ρ(x)/ɛ. (2.242) The density can be related to the concentrations of the ion species by ρ(x) = i z i ec i (x) (2.243) Substituting equation (2.241) into (2.243) we obtain ρ(x) = i z i ec 0 i exp( βφ(x)z i e). (2.244) Finally, substituting the density into the Poisson equation, we arrive at the Boltzmann- Poisson equation. d 2 φ(x) dx 2 = 1 z i ec 0 i exp( βez i φ(x)) (2.245) ɛ i This equation can readily solved if we linearize by expanding the exponential term d 2 φ(x) dx 2 The solution to this equation is = 1 z i ec 0 i ɛ (1 βez iφ(x)). (2.246) i with d 2 φ(x) = l 2 dx 2 Dφ(x), (2.247)

5 72 MACROMOLECULES e2 β l D = zi 2 ɛ c0 i (2.248) being the Debye length l D. The Debye length is a measure for the region of a potential around a point charge, which causes screening of interactions. i Annealed polyelectrolytes The free energy of an annealed, fully stretched polyelectrolyte chain in a salty solution can be written as βf = + N/2 N/2 N/2 N/2 ds [f(s)(log f(s) 1) µf(s) (2.249) } ds f(s)f(s 1 b/λ ) D s s b e s s. (2.250) Here f(s) is the local charge distribution along the chain and λ D is the Debye screening length. The first term of eq is the entropy of a unidimensional ideal gas. The second term fixes the charge on the chain by the chemical potential µ and the third term represents the electrostatic interaction between the charges on the chain. Justification for eq can be obtained only if the average charge density < f > /b fulfills certain conditions. First we note, that < f > has to be small in order to make non-linear effects unimportant < f > l B b < f > u < 1 (2.251) Here, u = l B /b is a dimensionless parameter. Second < f > has to be large to make sure that a sufficiently large number of charges interact simultaneously < f > λ D b > 1, (2.252) i.e., the Manning parameter for the condensation of counterions on a partially charged rigid rod need to large enough. Minimizing eq the equilibrium charge density distribution on an annealed, fully stretched polyeletrolyte chain, up to first order in < f > l B /b is given by

6 2.5 POLYELECTROLYTE 73 Here f(s)/ < f > = 1+ < f > l B b + E 1 [( N 2 s ) b λ D E 1 (x) = x { E 1 [( N 2 + s ) b dt e t t ] 2 λ } D Nb λ D ] (2.253). (2.254) (2.255) Hence, we find a charge accumulation at the ends of the rod within a region of the order of the screening length. To generalize this to weakly charged flexible chains we use the most probable chain conformation r 0 (s) and the entropy of the freely jointed chain model βf [r 0 (s)] = N/2 N/2 ds [f(s)(log f(s) 1) (2.256) µf(s) + 3 ( ) 2 dr0 (s) (2.257) 2b 2 ds } N/2 + ds f(s)f(s ) e r 0(s) r 0 (s ) b/λ D. (2.258) r 0 (s) r 0 (s ) b N/2 This involves as a new length scale, the electrostatic blob size One blob thus contains ξ = b (uf) 1/3. (2.259) g = (ξ/b) 2 (2.260) monomers a chain consists of n b = N/g = N(uf 2 ) 2/3 (2.261) such electrostatic blobs. For a chain of finite length the blob size varies having the smallest (denoted by ξ 0 ) in the center and reaches ξ at the ends.

7 74 MACROMOLECULES With this, the result for the minimization turns out to be as before except for a rescaling of the Bjerrum and the Debye length l B = 3ξ 0 b l B (2.262) λ D = 3ξ 0 b λ D. (2.263) 2.6 Proteins Proteins (see figure 2.31 for an example ) are the machines and building blocks of living cells. They are polymers of the 20 naturally occuring amino acids listed in table 2.3. Recall from figure 2.2 that all amino acids have a COO and a NHHH part or a COOH carboxyl and NHH amino part. In addition, there is a side chain usually labeled R. The configuration of the side chain is called rotamer. This is due to the fact that the tetrahedral geometry stays the same and the main degree of freedom is rotation about the carbon bonds. In figure 2.29 is shown the amino acid Analine and its geometry. To form a protein, amino acids are bonded together in sequence and fold into a protein. Each protein has a unique three-dimensional structure. It was shown [46] that a protein in its natural environment folds into, i.e. vibrates around, a unique three dimensional structure, the native conformation, independent of the starting conformation. There are four levels of architecture in proteins Primary structure: The sequence of peptide-bonded amino acids (as in the example: RSDAEPHYLPQLRKDILEVICKYVQIDPEMVTVQLEQKDG- DISILELNVTLPEAEELK). This is determined by protein synthesis. Secondary structure: The regular, recurring arrangement in space of adjacent amino acid residues in a polypeptide chain. Two main types of secondary structures have been found in proteins, namely the α-helices and β-sheets. The α-helix-complex has already been studied in a previous section. In a β-sheet, two or more polypeptide chains run alongside of each other and are linked in a regular manner by hydrogen bonds between the

8 2.6 PROTEINS 75 Table 2.3: List of the 20 amino acids. The single letter code is used when comparing and aligning sequences of proteins amino acids 3-letter code single letter code Alanine Ala A Cysteine Cys C Aspartic AciD Asp D Glutamic Acid Glu E Phenylalanine Phe F Glycine Gly G Histidine His H Isoleucine Ile I Lysine Lys K Leucine Leu L Methionine Met M AsparagiNe Asn N Proline Pro P Glutamine Gln Q ARginine Arg R Serine Ser S Threonine Thr T Valine Val V Tryptophan Trp W TYrosine Tyr Y

9 76 MACROMOLECULES Figure 2.29: The amino acid Alanine. Note that the bond directions for carbon are the same as from the centroid of a tetrahedron to the vertices.

10 2.6 PROTEINS 77 Figure 2.30: β-sheet. The protein thioredoxin contains a five-stranded beta sheet comprised of three parallel strands and three antiparallel strands. The entire protein is shown as a cartoon with the beta strands (three parallel strands and three antiparallel strands) colored red and alpha helices colored yellow. main chain C=O and N-H groups. Hence, all hydrogen bonds in a β-sheet are between different segments of polypeptide. An example of one strand of a β-sheet is shown in figure A third type of secondary structure are loops. A loop is a section of the sequence that connects the other two kinds of secondary structures. Tertiary structure: The spatial arrangement among all amino acids in a polypeptide. The twisted shape is slightly flexible, and the chain folds upon itself. Quaternary structure: The spatial relationship of polypeptides or subunits. Several proteins interact and form complexes. From the point of view of polymer physics the protein is simply a polymer consisting of a long chain of amino acid residues, i.e. a polypeptides. An important protein which exists in both monomeric or globular (G-actin) and polymeric or filamentary (F-actin) forms is actin. The filaments can form a network of entangled and crosslinked filaments and is the basis for the cytoskeletal network.

11 78 MACROMOLECULES Figure 2.31: MinE protein showing α-helices and β-sheets Protein folding The long-standing question is: how do proteins fold? A protein folds due to the angles φ and ψ between the carbon atom of a residue and the neighboring atoms, i.e. N and CO, in the peptide bond -N-C-(CO)-. These angles can assume only a few values independently of each other. Denaturants such as urea added to the system caused proteins that are folded in the native conformation to loose tertiary structure and revert to a random coiled state. After removal of the denaturants, the protein folds back into the native conformation. The protein folding problem entails the mathematical prediction of (tertiary, 3- dimensional) protein structure given the (primary, linear) structure defined by the sequence of amino acids of the protein. With some exceptions, proteins fold spontaneously. What we want to have is a theoretical model that accurately predicts the folding and properties of the fold. The problem lies in the fact that a variety of globally different structures have very low energies, but within a few k B T of each other. Hence, we would need a very good energy function for possible predictions and the ensuing dynamics are glassy as we have seen before. What we would like to predict is for example the number of observable thermodynamic states

12 2.6 PROTEINS 79 the rate of folding the effect of specific mutations on the folding rate Folding is an interesting problem because it involves mathematical modeling and numerical analysis. It is a extremely challenging task which has not been satisfactorily solved to date. Here we can only give a very brief introduction into some current methods. Basically, we need to distinguish between continuous and discrete models. Within continuous space models, a crucial problem is of course the large number of degrees of freedom. The configuration space is an n dimensional space, where n = 3 number of atoms in molecule. For example, the bacteriorhodopsin has 3576 atoms and hence we have coordinates! This results in the Levinthal s paradox [47]: The 3-D structure of a protein is determined by the dihedral angles. These angles have a few preferred values that correspond to the local minima of torsion energy around each rotation bond. We only have to consider about 10 conformations per AA in a polypeptide chain. This means that we have to examine at least as many as 10 N conformations for a protein with N amino acids. Assuming that a protein can sample of the order of structures per second, would take this protein about seconds or years to examine all the possible conformations. This is longer than the age of the universe. Indeed, the problem of finding the minimum energy configuration is NP-complete under a variety of models. Consequently, it is still impossible to determine the minimum energy structure for larger proteins based on the knowledge of only their sequence. Since, for the foreseeable future it remains doubtful, that we find a satisfying solution for the molecular mechanics of the folding pathway, starting from the random coil conformation to the folded pattern that will emerge. The standard approach is to investigate models that are reduced in complexity. These can be discrete protein models on a lattice to reduce the conformational degrees of freedom or on the other end of the spectrum the reduction to paths in a random energy landscape model. We have already touched on the energy landscape models and will here focus on molecular modeling and lattice models.

13 80 MACROMOLECULES Numerical approaches Molecular Modeling To be able to predict the folded structure, we crucially depend on an energy function. The energy function of all the parameters are used to describe the protein structure. The task is then to find values of the parameters which minimize this function. Molecular mechanics describes the energy of a molecule in terms of a simple function which accounts for distortion from ideal bond distances and angles, as well as and for nonbonded van der Waals and Coulombic interactions. Thus, such force field methods ignore the electronic motions to calculate the energy of a system. To model macromolecular systems the CHARMM potential (Chemistry at HARvard Macromolecular Mechanics) [50, 51], AMBER and GROMOS (GROningen MOLecular Simulation System) force fields are often used. They are empirical force field parametrizations that consists in general of six terms: where V ({R}) = c i (l i l 0 ) 2 (2.264) bonds + c α (θ α θ 0 ) 2 (2.265) bond angles + c β (τ β τ 0 ) 2 (2.266) improper torsion angles + tri(ω) (2.267) dihedral angles Q i Q j + (2.268) ɛr ij charged pairs ( ) Ri + R j + c w Φ (2.269) r ij unbond pairs r ij = R i R j. (2.270)

14 2.6 PROTEINS 81 Here ɛ is the dielectric constant and Q i are the partial charges. The term tri refers to a linear combination of trigonometric functions and and multiples of ω. The term Φ refers to a Lennard-Jones potential. The parameters c etc. are usually fitted and derived from first principles. The approach taken by the Molecular Dynamics and the Langevin Dynamics method discussed in the next section is to solve the equations of motion resulting from a force field, such as the one above, numerically. Molecular Dynamics simulations The starting point for the Molecular Dynamics (MD) simulatuion [52, 53, 54, 55, 56, 57, 58] is thus a well-defined force field. Using Hamiltonian, Lagrangian or Newton s equations of motion these are approximated by suitable schemes Ψ such that they can be solved numerically. Let us be more general for the moment and use generalized coordinates p and q with the Hamiltonian instead of the usual coordinates used above H(q, p) = 1 2 pt M 1 p + V (q). (2.271) From this Hamiltonian we get the equations of motion d dt q = q H (2.272) d dt p = p H. (2.273) An important consideration for any numerical integration scheme is that we want to conserve as many quantities during a numerical evaluation as possible that are conserved due to symmetries etc. This leads us to the concept of symplectic methods. Symplectic methods preserve certain abstract invariants of Hamiltonian systems [59, 60, 61] and are stable for linear systems for sufficiently small values of the stepsize. We denote the trajectory that we want to generate by Γ Γ(t) = ( ) q p Then this trajectory obeys the equation of motion. (2.274)

15 82 MACROMOLECULES where I denotes the unit matrix. d Γ = J H(Γ) (2.275) dt ( ) 0 I J =, (2.276) I 0 We compute an observable A along the trajectories and hence average along the states we find along the path < A >= 1 n obs A(Γ ν (t)). (2.277) n obs ν=1 Here n obs is the number observations we took, i.e., how many interations we took in the numerical integration of the equations of motion. Let ρ 0 (Γ) denote the probability density at time t = 0: Γ(0) = ρ 0 (Γ) and let ρ(γ, t) denote the probability density for Γ(t). Then we have the Liouville theorem for the trajectories. ρ(γ, 0) = ρ 0 (2.278) ρ t + (ρj H) = 0 (2.279) This states that the y flow in phase space is that of an incompressible fluid. If ρ t = 0, then and with this ρ(γ) = H J ρ = 0 (2.280) e H(Γ)/k BT e H(Γ)/k BT dγ. (2.281) Now let Ψ be a numerical integrator, i.e. motion a discretization of the equations of then the phase space volume needs to be conserved Γ n+1 = Ψ(Γ n ), (2.282)

16 2.6 PROTEINS 83 The integrator is sympletic, if det Γ Ψ(Γ) = 1. (2.283) ( Γ ψ(γ) T J( Γ ψ(γ) = J (2.284) i.e. phase space volume and the energy is conserved. If, and only if, Ψ is symplectic, there exists a shadow Hamiltonian H t (Γ), such that for times n t e c/ t. H t (Γ n ) H t (Γ 0 ) = O(e c/ t ) (2.285) The most straightforward discretization of the equations of motion that involve differentials comes from the Taylor expansion. The idea is to base the algorithm on a discrete version of the differential operator. With suitable assumptions we can expand the variable r in a Taylor series r(t + ) = r(t) + n 1 i=1 where R n gives the error involved in the approximation. Using the forward t + and the backward difference t i i! r(i) (t) + R n. (2.286) r(t + ) = r(t) + v(t) + F (t) 2m 2 + d3 r 3 + R dt 3 4 3! (2.287) r(t ) = r(t) v(t) + F (t) 2m 2 d3 r 3 + R dt 3 4 3! (2.288) (2.289) If we add the two equations, we obtain r(t + ) = 2r(t) r(t ) + F (t) 2m 2 + R 4 (2.290) If we substract the two equations we obtain r(t + ) + r(t ) = 2v(t) + R 3 (2.291)

17 84 MACROMOLECULES Figure 2.32: Free energy landscape and hence an estimator for the velocity v(t) = r(t + ) + r(t ) 2 + R 2. (2.292) The estimator for the position and the velocity together comprise what is known as the Verlet algorithm [58]. The Verlet algorithm is a second-order method that is indeed symplectic. So, much hinges on the simulation step-size, since this determines the time-scales, that we can cover. As we have seen above the choice of stepsize is dominated by stability demands and not by accuracy demands. What we also need to stress is that we actually need to simulate rare events, namely the folding which becomes more and more glassy as time progresses. Langevin Dynamics simulations Another approach to understand how the primary structure gives rise to the tertiary structure is to model the protein chemically as realistic as possible and take the interaction of atoms with the environment into account only through their stochastic

18 2.6 PROTEINS 85 influence. The water molecules and all other possible solvent molecules are not explicitly taken into account. Thus we start off with a Langevin equation [48] Mẍ + ηẋ + V = ξ(t), (2.293) where M is the mass matrix, η the damping matrix and ξ a normalized white noise resulting from a Wiener process. Because of the independence of the coordinates, the Langevin equation (2.293) only depends on the covariance matrix of the noise ξ(t) = DẆ (t) (here D is the diffusion matrix and W a Wiener process) DD T =< ξξ T dt >. (2.294) The fluctuation-dissipation theorem relates the diffusion matrix to the damping matrix and the temperature DD T = 2k B T η. (2.295) Because of the lack of knowledge of the detailed damping a further reduction in complexity is usual taken by setting the damping matrix proportional to the mass matrix with a damping constant γ. We then get η = γm (2.296) D = (2k B T γm) 1/2. (2.297) Let E be the energy barrier which the molecule has to take to get to a new state. Then the activation energy can be related to the mean frequency of transition f by an Arrhenius law, i.e., the rate increases exponentially with the absolute temperature where h is the Plank constant. f = k BT h ( exp E ) k B T (2.298) We shall now assume that we are at low temperatures, where the motions of the involved atoms are small. The protein is further assumed in a local minimum x min

19 86 MACROMOLECULES and we are interested in the high frequency modes. The highest frequencies are of the order of sec (roughly the C-H bond vibrations). For simplicity we shall also assume that the eigenmodes are non-degenerate. Then we can expand the potential up to second order V (x) V (x min ) (x x min) T V (x xmin ) (2.299) where V = 2 V (x min ). Ignoring in the limit of low temperature and high frequency the damping and the random force term we obtain with the general solution Mẍ + V (x x min ) = 0 (2.300) x = x min + l u l exp(iω l t) (2.301) Here ω l are the frequencies and u l are the normal modes. Monte Carlo Method The dynamical Monte-Carlo-Method, to generate a sample from an ensemble can be specified as follows: Determine the matrix of transition probabilities from state to the other P = {p xy } = {p(x y)}, (2.302) that satisfies the following transition probabilities (i) x, y S n 0 : p (0) xy > 0 (ii) y S : x π xp xy = π y. A sufficient, but by no means necessary condition to satisfy (ii) is the condition of detailed balance for die for the distribution π, (ii ) x, y S : π x p xy = π y p yx.