Electrostatically Tuned Self-Assembly of Branched Amphiphilic

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Electrostatically Tuned Self-Assembly of Branched Amphiphilic Peptides Christina Ting, Amalie L. Frischknecht, Mark J. Stevens, and Erik D. Spoerke J. Phys. Chem. B, Just Accepted Manuscript • Publication Date (Web): 19 Jun 2014 Downloaded from http://pubs.acs.org on June 28, 2014

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The Journal of Physical Chemistry

Electrostatically Tuned Self-Assembly of Branched Amphiphilic Peptides Christina L. Ting,∗,† Amalie L. Frischknecht,†,‡ Mark J. Stevens,†,‡ and Erik D. Spoerke¶ Computational Materials and Data Science, Sandia National Laboratories, Albuquerque, New Mexico 87185, United States, Center for Integrated Nanotechnologies, Sandia National Laboratories Albuquerque, New Mexico 87185, United States, and Electronic, Optical, and Nano Materials, Sandia National Laboratories, Albuquerque, New Mexico 87185, United States E-mail: [email protected]

∗

To whom correspondence should be addressed Computational Materials and Data Science, Sandia National Laboratories, Albuquerque, New Mexico 87185, United States ‡ Center for Integrated Nanotechnologies, Sandia National Laboratories Albuquerque, New Mexico 87185, United States ¶ Electronic, Optical, and Nano Materials, Sandia National Laboratories, Albuquerque, New Mexico 87185, United States †

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Abstract Electrostatics plays an important role in the self-assembly of amphiphilic peptides. To develop a molecular understanding of the role of the electrostatic interactions, we develop a coarse-grained model peptide and apply self-consistent field theory to investigate the peptide assembly into a variety of aggregate nanostructures. We find that the presence and distribution of charged groups on the hydrophilic branches of the peptide can modify the molecular configuration from extended to collapsed. This change in molecular configuration influences the packing into spherical micelles, cylindrical micelles (nanofibers), or planar bilayers. The effects of charge distribution therefore has important implications for the design and utility of functional materials based on peptides.

Keywords supramolecular assembly, self-consistent field theory, nanostructures, micelles

Introduction The self-assembly of amphiphilic peptides is an active field of research with the potential of impacting critical technological fields such as molecular sensing, catalysis, and biomedical applications. 1,2 In addition, these molecules provide a precise model system to study the fundamental science behind supramolecular self-assembly. Driving forces such as electrostatics, hydrogen bonding, hydrophobicity, and solvation influence the molecular conformation and subsequent primary and secondary assembly of the peptides into structures such as cylindrical nanofibers, spherical micelles, sheet-like morphologies, and ribbons. 3–8 Herein, we explore the role of the electrostatics. In a recent experimental study, Gough et al. 9 observed that the relative positions of charged amino acids in branched amphiphilic peptides were correlated with whether the 2


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peptides assembled into nanofibers or into sheet-like structures. Understanding how charge distribution in these molecules affects the supramolecular assembly therefore stands to impact the molecular design and ultimate utility of peptide nanomaterials. Towards this end, simulations have previously been used to provide some important insights into the electrostatic interactions that govern the self-assembly of peptides. In particular, Tsonchev et al. 10–12 have used Monte Carlo and molecular dynamics simulations on coarse-grained models to explore the self-assembly of peptide amphiphiles with a dipole in the hydrophilic group. Depending on the orientation of the dipole, their model predicts spherical or cylindrical micelles. Fu et al. 13 have also used molecular dynamics to explore the role of the electrostatic interactions. The authors proposed three kinetic mechanisms for self-assembly, depending on temperature and strength of electrostatic interactions. From these previous simulations, the importance of the electrostatic interactions is apparent. Although simulations can provide valuable information about assembly dynamics, it can be difficult to obtain thermodynamic information in these complex systems when long ranged electrostatic interactions are of interest. Furthermore, directly simulating the large sheetlike morphologies is a significant challenge. Self-consistent field theory (SCFT) has emerged as a powerful theoretical tool to study equilibrium thermodynamics of other molecular systems, including surfactants and block-copolymers. 14–20 In the present work, we use SCFT to complement the prior simulation work, taking a free energy-based approach to explore the cooperative effects of the hydrophobic and electrostatic interactions on the self-assembly of peptide molecules at the intra-aggregate level. We focus on geometric packing considerations governed by a combination of the structural asymmetry of the hydrophobic/hydrophilic volume fraction as well as the charge asymmetry. Insight into this question is an important component in the supramolecular assembly of charged systems. In what follows, we begin by introducing a model peptide to study the effects of these interactions. We then vary the presence and distribution of charged groups on the model peptide and calculate the density profiles and free energies for aggregation into spherical micelles, cylindrical micelles, and

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peptide molecules. Here the bond energy takes the general form N −1 3kB T X h({r}) = (ra+1 − ra )2 , 2 2b a=1

(1)

where kB T is the thermal energy, N is the number of monomers in the chain, and b = 0.5 nm is the bond length. We note that b in our model is on the order of the peptide persistence length, 0.4 nm. 21 The essential contributions to the model are the chain connectivity of the peptides, the short-ranged pairwise interactions between monomer species, the long-ranged electrostatic interactions among charged species, and the excluded volume effects. We work in the grand canonical ensemble, where the numbers of molecules are determined from the respective chemical potentials µα=P,S,± , which are obtained from the homogeneous bulk phase. Following the usual SCF derivation, 22 we obtain the final form for the field-theoretic Hamiltonian: e µP e µS e µ± G=− ZP [ξA , ξB , ψ] − ZS [ξS ] − Z± [ψ] vP vS v± Z ±φA± κJ ǫ 2 2 + dr χJK φJ φK + |∇φJ | − ξJ φJ + ψ − |∇ψ| . 2 vA 2

(2)

In this expression, φJ , ξJ , and ψ denote the monomer volume fraction, its conjugate potential, and the electrostatic potential fields, respectively, where it is understood that these denote the mean fields. χJK and κJ capture the strength of local and nonlocal interactions, respectively, 23 and ǫ is the spatially varying dielectric, which depends on the local volume fraction of species. The double indices denote a sum over monomer species JK ∈ {AB, BS, SA}, and we have omitted the r dependence for conciseness. The interaction parameters are meant to capture the general amphiphilic nature of the peptides. Setting χAB = χBS = 35, χSA = 0 and κA = κB = 1, κS = 0, we are able to capture the concentration range at which aggregates are observed experimentally by Gough et al. We note that the molecular solubility is less sensitive to the values for κJ , but since we are dealing with relatively short chains, we

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include it to capture the nonlocal effects at shorter length scales. The first line in eq. 2 contains the partition functions for a single molecule in its respective field(s), defined as

ZP [ξA , ξB , ψ] =

Z

NA +1 NA +1 NB 2vB ξB , drqA1 qA2 qB e

(3)

ZS [ξS ] =

Z

dr exp{−vS ξS },

(4)

Z± [ψ] =

Z

dr exp ∓ψe − ub± ,

(5)

for the peptides, solvents, and ions, respectively. The form of ZS is straightforward; ZP and Z± require some brief comments. Firstly, we have introduced the chain propagators in eq. 3 to build up the single chain statistics of the peptide one branch at a time, where qji denotes the chain propagator for monomer i of branch j. The extra exponential factor e2vB ξB corrects for over-counting the monomer when the propagators are joined at the branch point. Secondly, eq. 5 accounts for the Born self-energy of the ions in a spatially varying dielectric medium, 24 defined as ub± = e2 /(8πa± ǫ), where a± = 0.2 nm is the Born radius. 25 The self-consistent field equations are obtained by applying the mean-field approximation to eq. 2. Additional details on the theory and method can be found in the Appendix.

Results and discussion In this work, we are interested in how structural and charge asymmetry of branched amphiphilic peptides can be used to control the molecular shape and hence the self-assembly into spherical micelles, nanofibers, and lamellar sheets. In what follows, we have not attempted to obtain a complete description of the solution thermodynamics of surfactant-like molecules; this problem has already been addressed using SCFT for other similar systems. 14,16,17,26 Instead, to address the molecular shape and packing question, we focus on the intra-aggregate interactions. We work in the dilute regime so that only isolated aggregates are considered.

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Furthermore, we have fixed the position of the aggregates, which is equivalent to neglecting the translational entropy of the aggregates. Infinitely long cylinders and bilayers have negligible kinetic energy and hence this is a good approximation; for spherical micelles, the translational entropy will depend on the volume fraction of micelles. However, for the problem at hand, in which we are comparing identical peptides except for a change in the placement of charged groups, the difference in the micellar entropic terms in the free energy should be negligible. The SCFT solutions in the grand canonical ensemble directly give the preferred packing density and free energy of an aggregate. Therefore, by working in the appropriate geometry, coexistence curves can be mapped out by comparing the excess free energy per molecule in the different aggregates: ∆g = (G − Gsoln )/nagg .

(6)

Here nagg is the excess number of molecules in the aggregate, G is the total free energy of the system (eq. 2), and Gsoln is the free energy of the homogenous bulk solution. In fig. 2 we plot the two-dimensional phase diagram as a function of hydrophobic tail length NB and equilibrium bulk peptide concentration CP (molecules/nm3 ). Before we study the effects of the charge asymmetry, it is useful to understand how the structural asymmetry arising from the hydrophilic/hydrophobic volume fraction affects the self-assembly. Therefore, we first consider the neutral peptide (dotted). The sphere-cylinder and cylinder-bilayer coexistence curves are determined from the equality of the excess free energy per molecule ∆g in the different aggregate geometries. The coexistence curve between the homogeneous bulk solution and the spherical micelle is determined by the condition ∆g = 0, which also defines the critical micelle concentration. 26 Increasing NB at fixed concentration shifts the preferred aggregate structure from spherical micelles, to cylindrical micelles, to planar bilayers. These transitions can be understood from packing arguments. More specifically, increasing the length of the hydrophobic tail changes the geometry of the individual peptide molecule from more conical to more cylindri7




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0.6 0.4 0.2 0.0 !10

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0.02 0.00 !10 0.00

!5

10

!5

0r 0r

5

10 0.030

5

10

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10

5

10

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10

0.10

0.10 0.08

b)

Φ

0.08 0.06 0.06 0.04 0.04 0.02

Ρ$

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0.030 c) 0.025 0.030 0.025 0.020 0.025 0.020 0.015 0.020 0.015 0.010 0.015 0.010 0.005 0.010 0.005 0.000 !10 0.005 0.000 10 0.000 10

r

0 0r (nm) 0r

!5 !5

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r Figure 3: Radial density profiles for the cylindrical micelle comprised of neutral (dotted), symmetric-horizontal (solid) peptides. Top to bottom: (a) volume fraction of hydrophilic φA (light blue) and hydrophobic φB (grey) monomers; (b) select hydrophilic monomers for i = 1 (red) and i = NA (dark blue); (c) anions (green) and cations (purple). esults for the symmetric-vertical peptide directly coincide with the neutral peptide and are not explicitly shown. Parameters here are: CP = 1.11 × 10−4 , NB = 9, and ρb± = 15mM.

phase diagram shows regions for which the system shifts from spherical micelle to homogeneous bulk solution, from cylindrical micelle to spherical micelle, and from planar bilayer to cylindrical micelle, labeled 1–3, respectively. Region 3 is notable in that it supports recent observations that symmetric-horizontal peptides will form nanofibers and symmetric-vertical peptides will form bilayers under the same experimental conditions. 9 To better understand these results, we take a closer look at the molecular structure of the aggregates. The radial density profiles for different components in the cylindrical micelle are shown in fig. 3 for the neutral (dotted) and for the symmetric-horizontal (solid) peptides. The density profiles for the symmetric-vertical peptide directly coincide with the density profiles for the neutral peptide and are not separately included in the figures. Results for

9




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profiles of the neutral and symmetric vertical peptides are identical. More specifically, the charged monomers at i = NA (blue) are effectively a zwitterion pair and those at i = 1 (red) can easily come together to form a neutral zwitterion pair without a significant loss in the conformational entropy of the hydrophilic segments. The resulting conformation is shown schematically in fig. 4a, where we emphasize that this extended configuration is identical to the neutral peptide. In contrast, the symmetric-horizontal peptide is not self-neutralizing in the extended conformation of the neutral peptide, since monomers with the same index will now contain like charges. To minimize these unfavorable electrostatic interactions, the peptide collapses in on itself to form charged pairs, as suggested by the density profiles in fig. 3b. Here it can be seen that the red and blue densities for the symmetric-horizontal peptide (solid) merge to a common peak position. In particular, the i = 1 monomer has shifted from it’s original extended position at r = 2.8 nm (dotted red) to r = 1.7 nm (solid red). The neutralizing rearrangement of the symmetric-horizontal peptide molecule is shown schematically in fig. 4b. The electrostatically-driven rearrangement of monomers in the symmetric-horizontal peptide has the effect of collapsing the entire molecule into a more compact conformation so that its molecular shape becomes more conical. This results in an increase in the area per peptide within the aggregate, which increases the free energy ∆g. It is therefore this change in molecular shape that is responsible for the change in the phase diagram at a given concentration CP and hydrophobic tail length NB . Specifically, regions 1-3 in fig. 2 correspond to changes from spherical micelle to homogenous bulk solution, from cylindrical micelle to spherical micelle, and from bilayer to cylindrical micelle, respectively. Ultimately, these results indicate that the horizontal-symmetric distribution of charges has a distinct effect on the self-assembly. −1 exp{±ψ− In fig. 3c, we plot the concentrations of added salt ions, defined by ρ± = eµ± v±

ub± }. In all cases, the decrease in ion concentration inside the micelle is due to the penalty for the ions to enter the lower dielectric medium in the aggregate, where we have defined

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ǫA , ǫB ≈ 10 for the peptide 28 and ǫS = 80 for the solvents. This result is an effect captured by the Born self-energy of the ions in the Born-augmented Poisson-Boltzmann equation (see appendix). For the neutral and symmetric-vertical peptides, there is a single dotted curve for both the cations and anions. Here, the two hydrophilic branches of the peptide are fully self-neutralizing and there is no charge separation of free ions. In contrast, for the symmetrichorizontal peptide there is an excess of cations (solid purple) at r = 4 nm and an excess of anions (solid green) at r = 1.7 nm. To understand this result, we note that in fig. 3b, the profiles for the charged monomers of the symmetric-horizontal peptide do not overlap identically. In particular, relative to the i = NA (solid blue) monomer, the i = 1 monomer (solid red) has a lower peak and a broader distribution with a tail that extends to larger r. Although the peptide would like to satisfy the electrostatic interactions, it is difficult to fit all the end monomers in the interfacial region near the hydrophobic core. This result can be attributed to both the the excluded volume and the loss in configurational entropy when constricting the ends of the chain. Thus, there is not a perfect overlap in the densities of the positive and negative monomers, which leads to the observed charge separation for the symmetric-horizontal peptide. Finally, we have also explored the effects of salt concentration ρb± , since the range of electrostatic interactions will depend on the salt concentration through the debye length. In our aqueous solution of amphiphilic molecules and monovalent ions, the mean-field PoissonBoltzmann theory is a good description of the electrostatic effects. In the case of divalent ions (not explored here), short range ionic correlations must be taken into account. By increasing the salt concentration, we find that the electrostatic interactions are screened and the coexistence curves for the symmetric-horizontal peptide (solid lines in fig. 2) shift back towards the coexistence curves for the neutral peptide case (dotted lines in fig. 2). This effect provides an additional parameter from which to control the self-assembly of charged amphiphilic peptides into a variety of nanostructures.

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Conclusions We have used SCFT to explore the free energies associated with the self-assembly of a branched amphiphilic peptide. We have demonstrated the important role of charge asymmetry in the assembly of peptide nanostructures. In particular, we showed that the effective molecular shape, as defined by the hydrophobic volume fraction, can be further tuned by the distribution of charged groups on the peptide. This in turn affects the packing into spherical micelles, cylindrical micelles, or planar bilayers. Whereas symmetric-vertical peptides are self-neutralizing and produce the same results as neutral peptides, symmetric horizontal peptides must rearrange and collapse the hydrophilic branch to form charged pairs; see fig. 4. The result is a more conical molecule that increases the assembly free energy and shifts the coexistence curves in the phase diagram towards higher concentrations. In particular, we find a region (labeled 3 in fig. 2) for which symmetric-horizontal peptides assemble into cylindrical micelles, whereas symmetric-vertical peptides assemble into bilayers. These results are in agreement with recent experimental observations 9 and show that the presence and distribution of electrostatic interactions play an important role in the assembly. Recognition of these important interactions will have significant implications for the design and utility of the next generation of functional supramolecular materials. Recently, Leung et al. 29,30 demonstrated that the competition between the ionization state and the hydrophobic tail length of amphiphilic molecules may be used to control the crystallinity of bilayer membranes, which may enable controlled encapsulation and release of molecules from within the membrane. More specific to the work here, branched peptides have been explored for use in a number of biomedical applications, 1 where the branched architectures are designed to influence interactions between nanofiber surfaces and biological agents or secondary materials. 31–33 Since the conformation of the branched peptides and the aggregate morphology will have a significant impact on the bioaccessibility of these interactions, understanding how charge distribution may affect molecular assembly should influence molecular design. Finally, although the present study is based on the assembly of branched amphiphilic pep13



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tides, the model is general and may also be applied to study the assembly of other functional materials based on lipids, polymers, or other small molecules.

Appendix We begin with a particle-based Hamiltonian for our system of amphiphilic peptides in explicit solvent containing ions:

H=

nP X

hi ({r}i )

(7)

i=1

+

Z

1 ρˆc (r)ˆ ρc (r′ ) ′ ˆ ′ ˆ drdr φJ (r)uJK (r, r )φK (r ) + . 8π ǫ(r)|r − r′ | ′

The first term accounts for the chain connectivity of the nP peptides (see eq. 1). The second term accounts for the short-ranged interaction energy, where the φˆJ are the instantaneous volume fractions, generally defined as ˆ =v φ(r)

n X N X

δ(r − ria )

(8)

i=1 a=1

for n molecules, each with N monomers of volume v. The pairwise interaction potential, uJK (r, r′ ), is assumed it to be short-ranged so that one may perform a gradient expansion to quadratic order as

uJK (r, r′ ) =

(9)

1 uJK (r, r) + ∇u · (r′ − r) + ∇∇u : (r′ − r)(r′ − r). 2 The linear-order terms vanish by symmetry upon spatial integration. The zeroth-order terms give to rise to the local terms and the second-order terms correspond to the square-gradient terms, where, for simplicity, we ignore the cross terms. The third term in eq. 7 is the Coulomb energy of the system, which accounts for the long-ranged electrostatic interactions from the 14


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total charge density of all charged species:

ρˆc (r) = ±

1 ˆ φA (r) ± eˆ ρ± (r). vA ±

(10)

The grand canonical partition function Ξ is obtained by summing over all molecular degrees of freedom, including the position of each solvent and ion, as well as the position and configuration of each peptide: ∞ X

Ξ=

nα∈{P,S,±}

×

Y

e µα n α n !v nα =0 α α

Z Y nP i=0

d{r}i

nS Y

drj

j=0

n± Y

drk

k=0

δ[1 − φÂ (r) − φˆB (r) − φˆS (r)]e−H .

(11)

r

Here, the delta functional accounts for the incompressibility (excluded volume) at all positions within the system volume. To convert the above particle-based model into a fieldtheoretic model, the microscopic density operators are converted into scalar density fields by inserting the identity Z

h i DφJ δ φJ (r) − φˆJ (r) F [φˆJ (r)] = F [φJ (r)].

(12)

The delta function is in turn expressed using the Fourier representation δ[φJ (r) − φˆJ (r)] = Z Z h i ˆ DξJ exp i drξJ (r) φJ (r) − φJ (r) .

(13)

These operations decouple the interactions among molecules and replace them with interactions between single molecules and effective fluctuating fields. The field-theoretic parR tition function can now be generically written Ξ = Dω exp(−F [ω]), where F [ω] is an

effective Hamiltonian and ω is the multidimensional field variable. In general, the field-

theoretic partition function cannot be evaluated in closed form. The mean-field approxima15



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tion amounts to assuming that a single field configuration ω ∗ dominates the functional so that Ξ ≈ exp(F [ω ∗ ]), where F[ω ∗ ] is given by eq. 2 and ω ∗ is obtained by requiring that eq. 2 is stationary with respect to variations in the fields. Here and in what follows, we have recognized the imaginary nature of the potential field variables at the saddle point and have redefined the conjugate chemical potential fields iξ → ξ, and the electrostatic potential field −iψ → ψ. Variation with respect to the volume fraction fields φA and φB gives

ξA = ξS + χSA (φS − φA ) + (χAB − χBS )φB (∇ψ)2 e 2 ρ± −κA ∆φA − (ǫA − ǫS ) , + 2 8πa± ǫ2 ξB = ξS + χBS (φS − φB ) + (χAB − χSA )φA e 2 ρ± (∇ψ)2 . + −κB ∆φB − (ǫB − ǫS ) 2 8πa± ǫ2

(14)

(15)

−1 exp{±ψ − e2 /(8πa± ǫ)} is the ion number distribution. In these expressions, ρ± = eµ± v±

Variation with respect to the conjugate potential fields ξA , ξB , and ξS gives NA h i v A e µP X NA +1−i+1 vA ξA +ciA1 ψ NA +1−i+1 vA ξA +ciA2 ψ i i , q q˜ e + qA2 qÃ2 e φA = vP i=1 A1 A1

(16)

NB v B e µP X φB = q i qÑB −i+1 evB ξB , vP i=1 B B

(17)

φS =eµS e−vS ξS ,

(18)

Eq. 18 can be trivially solved to yield ξS = −vS−1 log(1 − φA − φB ), where we have defined µS ≡ 0 and used the incompressibility condition φA + φB + φS = 1 to eliminate φS . Finally, variation with respect to the electrostatic potential field ψ gives the Born–energy augmented

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Poisson-Boltzmann equation: 24

1 −∇ · (ǫ∇ψ) = ± φA± ± (eρ± ). vA

(19)

In eqs. 16 and 17, the chain propagator accounts for the chain connectivity and the Boltzmann weight of each monomer in its respective self-consistent potential field (eqs. 14 and 15). If the monomer is charged, there will be an additional contribution from the electrostatic potential field: −cij ψ, where cij = 0, ±1 is the charge on that monomer. Because of the lack of inversion symmetry of the peptides, we have also introduced the complementary chain propagator q˜ji , which propagates from the branch point to the end monomers. Once again, the extra exponential factor corrects for over counting the monomer when the propagators are joined. Note that φA± can be similarly obtained from eq. 16, except that the summation is restricted to the respective charged monomers. Calculation of the chain propagator for the discrete Gaussian chain begins with the initial condition 1

qj1 = e−(vξj +cj ψ)

(20)

for placing the end monomer and NA +1 NB 1 qB , = evξB qA2 qÃ1

(21)

NA +1 NB 1 qB , qÃ2 = evξB qA1

(22)

NA +1 NA +1 qA2 , q˜B1 = evξB qA1

(23)

for starting at the branch point. Additional details for the chain propagator calculation can be found in ref. 34, where we use a semi-implicit Crank-Nicolson scheme. 22 Eqs. 14–19 are solved, together with the chain propagators, iteratively until convergence. To obtain the electrostatic potential, we use the method described in ref. 35.

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Acknowledgement CLT and ALF were supported by the Harry S. Truman Fellowship in National Security Science and Engineering and the Laboratory Directed Research and Development program. MJS and EDS were supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award KC0203010. Sandia National Laboratories is a multi-program laboratory operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Company, for the US Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

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(33) Lin, B. F.; Megley, K. A.; Viswanathan, N.; Krogstad, D. V.; Drews, L. B.; Kade, M. J.; Qian, Y.; Tirrell, M. V. pH-Responsive Branched Peptide Amphiphile Hydrogel Designed for Applications in Regenerative Medicine with Potential as Injectable Tissue Scaffolds. J. Mater. Chem. 2012, 22, 19447–19454. (34) Ting, C. L.; Wang, Z.-G. Interactions of a Charged Nanoparticle with a Lipid Membrane: Implications for Gene Delivery. Biophys. J. 2011, 100, 1288–1297. (35) Luty, B. A.; Davis, M. E.; McCammon, J. A. Solving the Finite-Difference Non-Linear Poisson-Boltzmann Equation. J. Comput. Chem. 1992, 13, 1114–1118.

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