Predicting the Fluid-Phase Behavior of Aqueous Solutions of ELP

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Predicting the Fluid Phase Behaviour of Aqueous Solutions of ELP (VPGVG) Sequences Using SAFT-VR Binwu Zhao, Tom Lindeboom, Steven W. Benner, George Jackson, Amparo Galindo, and Carol K Hall Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b02249 • Publication Date (Web): 08 Aug 2017 Downloaded from http://pubs.acs.org on August 14, 2017

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Predicting the Fluid Phase Behaviour of Aqueous Solutions of ELP (VPGVG) Sequences Using SAFT-VR Binwu Zhao†, Tom Lindeboom‡, Steven Benner†, George Jackson‡ , Amparo Galindo‡, and Carol K. Hall*† †

Department of Chemical and Biomolecular Engineering, North Carolina State University, Raleigh, North Carolina 27606, United States

‡

Department of Chemical Engineering, Imperial College London, London, SW7 2AZ, United Kingdom

Abstract The statistical associating fluid theory for potentials of variable range (SAFT-VR) is used to predict the fluid phase behaviour of elastin-like polypeptides (ELPs) sequences in aqueous solution with special focus on the loci of lower critical solution temperatures (LCST). A SAFTVR model for these solutions is developed following a coarse-graining approach combining information from atomistic simulations and from previous SAFT models for relevant systems previously reported. Constant-pressure temperature-composition phase diagrams for are determined for aqueous solutions of (VPGVG)n sequences + water with n =1 to 300. The SAFTVR equation of state lends itself to the straightforward calculation of phase boundaries, so that complete fluid-phase equilibria can be calculated efficiently. A broad range of thermodynamic conditions of temperature and pressure are considered, and regions of vapour-liquid and liquidliquid coexistence, including LCSTs are found. The calculated phase boundaries at low concentrations match those measured experimentally. The temperature-composition phase diagrams of the aqueous ELP solutions at low pressure (0.1 MPa) are similar to those of types V and VI phase behaviour in the classification of Scott and van Konynenburg. An analysis of the 1

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high-pressure phase behaviour confirms however that a closed-loop liquid-liquid immiscibility region, separate from the gas-liquid envelope, is present for aqueous solutions of (VPGVG)30; such a phase diagram is typical of type VI phase behavior. ELPs of shorter lengths exhibit both liquid-liquid and gas-liquid regions, both of which become less extensive as the chain length of the ELP is decreased. The strength of the hydrogen-bonding interaction is also found to affect the phase diagram of (VPGVG)30 system in that the liquid-liquid and gas-liquid region expand as the hydrogen-bonding strength is decreased and shrink as it is increased. The LCSTs of the mixtures are seen to decrease as the ELP chain length is increased.

2


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1 Introduction Elastin-like polypeptides (ELPs) are artificial polymer-like polypeptides that contain the motif (Val-Pro-Gly-Xaa-Gly)n, where Val, Pro, and Gly represent valine, proline and glycine, respectively, Xaa is a guest residue that can be any amino acid except proline, and n is the number of pentamer repeats along the chain 1. This unique motif is derived from the hydrophobic domain of tropolelastin, the precursor of elastin. In aqueous solution ELPs exhibit lower critical solution temperature (LCST) behavior

2,3

so that they are insoluble in water when the

temperature is above a transition temperature (Tt), resulting in coexisting peptide-rich and peptide-poor liquid phases, and are soluble otherwise 4. The LSCT is the lowest temperature of the liquid-liquid cloud-curve boundary corresponding to the locus of transition temperatures with composition at a given pressure. The precise location of the LCST and the transition temperatures in these systems depend on the molecular weight of the ELP 5-8, and on the identity 2

and composition 1 of the guest residues, all of which can be precisely controlled 9,10. There are

also reports relating to the effect on the cloud curves of aqueous ELP solutions of varying external conditions such as salt concentration

11-14

, pressure

15,16

, and pH

17,18

. This unique

functional feature of ELPs makes them ideal candidates for biomedical applications, as thermoresponsive gels 19, in recombinant protein purification 20-22, as drug delivery carriers

23-28

, and in

tissue engineering scaffolds 29,30. One goal of our current study is to predict the fluid-phase behaviour of ELP aqueous solutions over broad ranges of thermodynamic conditions. The complete fluid-phase diagrams of these systems have never been predicted and the minimum transition temperature(the LCST) of an ELP of arbitrary length, has not been reported previously. In fact, the overall fluid-phase 3


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diagrams for aqueous systems of chains of generic amino acids have not been described before. Even though the transition temperatures of aqueous solutions of ELPs at different conditions have been extensively reported 8,31,32, the concentrations of ELPs considered were very low33,34 , and are likely to be well below the concentration at which the LCST is located. It is therefore of interest to carry out predictive calculations of the complete binary fluid-phase equilibria of ELPs in water and to characterize the type of phase behaviour observed. Moreover, knowing the global phase diagram of aqueous solutions of ELPs could help establish the length of ELPs with particular transition temperatures of interest to aid design thermo-responsive systems with desired properties. In the classification of van Konynenburg and Scott

35,36

, phase diagrams of types IV, V,

and VI are characterized by a liquid-liquid immiscibility region with LCSTs and a lower critical end point (LCEP). In types IV and V the critical line emanates from the critical point of the volatile (light) component presenting a continuous change from vapour-liquid states to liquidliquid LSCTs, ending at a LCEP, associated with liquid-liquid demixing. Compared with type V phase diagram, type IV phase diagram has an additional region of liquid-liquid immiscibility bounded by an upper critical end point (UCEP) at low temperature. In Type VI phase behavior, liquid-liquid separation is exhibited in the low-temperature region of the phase diagram bounded above by an upper critical solution temperature (UCST) and below by an LCST. A closed-loop of liquid-liquid immiscibility is observed in the constant-pressure temperature-composition representation. Although the behaviour of Type IV, V and VI systems are all characterized by the presence of LCSTs, the physical origin giving rise to the LCSTs is very different in the different classes of mixtures 37-42.

4


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The consensus regarding the LCST behavior for many polymer solutions is that it is a consequence of the differences in compressibility of the solute (macromolecule) and solvent 43,44, which are accentuated as the solute (macromolecule) gets longer. This is often associated to types IV and V phase behavior. By contrast, the LCST behavior in low molecular weight mixtures is often related to the existence of directional forces, such as hydrogen bonds, in the mixture 39-42, often associated to type VI phase behavior. Most statistical mechanical studies have focus on one case or the other. In our study we consider a model ELP system, where both effects are operative. An examination of fluid-phase behavior of aqeuous solutions of ELPs can serve as a framework for unifying the two descriptions. In a previous study it was shown that the LCST behavior of ELPs can be attributed to directional forces in the solution, i.e., hydrogen bonds between the peptide and water as well as between the peptide and peptide7,45. The hydration properties of ELPs, i.e., the number of hydrogen bonds formed between peptide and water and the number of water molecules within the first hydration shell of the peptide, was found to undergo an abrupt change, as the temperature is increased from below to above Tt. In our current study, we explore not only the degree of hydrogen-bonding, but also the effect of compressibility associated with varying the length of the ELP. The statistical associating fluid theory (SAFT) has been used previously to predict the LCST behavior and fluid-phase equilibria of both a generic polymer-water system poly(ethylene glycol) (PEG)-water system

47

. The SAFT framework was used

46

46

and a real

to investigate

the liquid-liquid immiscibility and the LCST behavior of model polymer solutions. A mixture of hard spheres (to mimic water) and hard sphere chains (to mimic a polymer) was considered and no evidence of fluid-phase separation was found for these athermal (purely repulsive) systems was found regardless of the length of the polymer. When mean-field attractive forces between 5


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the water and polymer segments are incorporated into the model, vapour-liquid and liquid-liquid phase separation with LCST behaviour was observed, but only for chains of seven segments or longer. The incorporation of directional forces in the water-polymer interactions gave rise to closed-loop behavior, suggesting a mechanism different from that typically seen in non-aqueous polymer solutions. An extensively studied48-51 and modelled52-56 polymer system that exhibits liquid-liquid phase separation is water and PEG. The closed-loop LCST behavior of aqeuous solutions of PEG47 has been studied with the SAFT-VR approach57,58. In this model, both dispersion and hydrogen-bonding interactions were taken into account, resulting in excellent agreement with the experimentally determined phase behavior. The closed-loop immiscibility was found to become less extensive as the pressure was increased, finally disappearing when the pressure is higher than a critical value. We follow this approach to model the ELP-water binary mixture system in an effort to predict the phase diagrams of ELP (VPGVG)n sequences of different lengths n in aqueous solution. In our SAFT-VR description the molecules are treated as homonuclear chains of tangentially bonded spherical segments, with dispersion interactions incorporated via attractive square-well potentials and directional hydrogen bonding interactions mediated by short-range attractive sites. The interaction parameters employed in the SAFT calculations are obtained following several approaches and then scaled to be self-consistent: The interactions between water molecules (both dispersion forces and hydrogen bonding strength) are the same as those presented in reference

47

. The parameters for peptide-peptide dispersion forces are determined

from the trajectories of an explicit-solvent atomistic simulation of 16 (VPGVG) chains. A modified iterative Boltzmann inversion (IBI) scheme is used to match the radial distribution function from the coarse-grained simulation to that from the atomistic simulation. The 6


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parameters for peptide-water dispersion forces (unlike interactions) are calculated using the standard Lorentz-Berthelot a combining rules 47, 58, 59with an unlike interaction parameter, k12. Highlights of our results are as follows: We predict the fluid-phase equilibria for binary mixtures of ELP (VPGVG)n and water for n = 1 (427.5 g mol-1) to n = 300 (1.228 x 105 g mol-1) for the entire composition range (mass fractions, w, from 0 to 1). We are able to identify the LCSTs of ELP (VPGVG)n sequences of different lengths in aqueous solutions. The phase behaviour resulting from the SAFT-VR calculations are in good agreement with those measured experimentally by Meyer et al.

8

at low concentrations; we use one adjustable parameter, k12,

which varies slightly with the length of the ELP, to better describe the peptide-water unlike interactions. The T-w phase diagrams of ELPs at 0.1 MPa resemble those charaterizing type V and VI systems. We observe a closed-loop liquid-liquid immiscibility region which separates from the gas-liquid immiscibility region as the pressure is increased in the (VPGVG)30-water system; this is typical of type VI phase behavior. The gas-liquid two-phase region disappears at very high pressures (P > 30 MPa ), leaving only the liquid-liquid immiscibility region. The LCSTs of ELPs of different lengths is found to increase as the pressure is increased, and have a linear correlation with the pressure. The predicted dependence of the LCSTs of ELPs on their lengths can be described with a power law in the temperature and composition, conforming to the linear behaviour expected in a Schultz-Flory representation 60. Finally, we find that as the peptide-peptide and peptide-water hydrogen-bonding interaction strengths become weaker, the liquid-liquid immiscibility region of the T-w phase diagram widens, and the LCSTs decrease and eventually disappear. The resulting phase behaviour corresponds to type II or III, which exhibit characteristically large regions of liquid-liquid immiscibility. An increase in the value of the hydrogen-bonding interaction results in less extensive regions of liquid-liquid separation and 7


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phase diagrams resembling those of type I phase behaviour, where only vapour-liquid coexistence is observed.

2 SAFT-VR model for ELP + water mixtures Water In SAFT-VR, water (component 1) is represented with a four-site model as presented in references

47,61

and other studies

62-67

as shown in Figure 1. Each water molecule is represented

as a single square-well sphere with a core diameter of σ11, a square well depth of ε11, and square well range of λ11. Four off-center hydrogen bonding sites are placed on each spherical segment, two sites of type e and two sites of type H. The two e sites represent the electron lone pairs on the oxygen atom in a water molecule; the two H sites represent hydrogen atoms in a water molecule. Hydrogen bonding interaction will only be considered between e and H sites. In the perturbation theory of Wertheim 68-71, on which the SAFT equation of state is based, these e and H association sites are modelled as being completely attractive. It is assumed in the theory that due to the steric hindrance/ incompatibility of the small association sites attached to large repulsive cores each site can only bond to one other site at a time. Note that in a water molecule, the oxygen atom can be involved in two hydrogen bonds at the same time as there are two e sites available. The cut-off distance for hydrogen bonds, ,, , and the hydrogen bonding energy,

, , are the same as reported in previous work47. The values of the intermolecular parameters

for water are listed in Table 1.

8


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Figure 1. Model of water used in the SAFT-VR calculations. A water molecule is represented as a square-well sphere with four off-center square-well hydrogen-bonding sites, two of type e and two of type H. Sites of type e represent the electron lone pairs on the oxygen atom and sites of type H represent the hydrogen atoms in a water molecule. Only e-H hydrogen bonding is considered and multiple bonding at a given site is forbidden.

Table 1. Pure Component Intermolecular Parameters for the Water Model / ⁄K

σ11/Å 3.03420

250.000

λ11 1.78890

HB 11,eH / ⁄K 1400.00

,, /Å

2.10822

a

Parameters in the first three columns characterize the repulsion/dispersion forces of the water model: the diameter of the spherical core, σ11, the square-well depth, ε11 ( is the Boltzmann constant), and the range of the square well, λ11. Parameters in the last two columns describe the water-water hydrogen-bonding strength, , and the cut-off distance ,, for the waterwater hydrogen bonds.

Elastin-like Polypeptides (ELPs) The ELP (VPGVG)n chain (component 2) is modeled as a chain of uniformly-sized square-well segments with each sphere representing an amino acid. A number of hydrogenbonding sites are placed on each segment. Sites of type e are used to mimic the electronegative lone pairs on the oxygen atoms of the backbone C=O group and sites of type H correspond to the hydrogen atoms on the backbone N-H group. Hydrogen-bonding sites on the terminal amino acids (labeled as types e* and H*) are treated slightly differently in terms of the hydrogenbonding interaction energy with water molecules. In the body of the peptide chain, valine and 9


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glycine have two e sites and one H site each, and proline has two e sites but no H site. At the N terminal group, valine has two e* sites and two H* sites. At the C terminal group, glycine has three e* sites and two H* sites. The hard-core diameter, square-well depth, and square-well range of the segments forming the ELP chain are characterised by σ22, ε22 and λ22, respectively.

Peptide-peptide hydrogen bonds all have the same energy ,Η and cut off ,, . The values

of these parameters are listed in Table 2.

Figure 2. Model of the elastin-like polypeptides (ELP) with the VPGVG sequence (where V represents valine, P proline, and G glycine) used in the SAFT-VR calculations. All of the amino acids in the molecule with sequence (VPGVG)n are represented by the same-sized square-well spherical segments with well depth ε22 and range of λ22. The hydrogen-bonding interactions occur between the off-center square-well sites: types e (lone pairs on the oxygen in amino acids in the middle of the chain), types H (the hydrogen atoms in amino acids in the middle of the chain), e* and H* representing lone pairs on the oxygen and hydrogen atoms on the terminal of the chain, respectively. Valine and glycine both have two e sites and one H site; proline has only 10


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two e sites. Valine at the N terminal has two e* and two H* sites; glycine at the C terminal has three e* and two H* sites. Table 2. Pure Component Parameters for Elastin-Like Polypeptide (ELP) Model b σ22/Å

⁄ /K

λ22

⁄ ,

/K

,, /Å

4.29500

187.000

1.50000

800.000

3.20000

b

Parameters in the first three columns characterize the repulsion/dispersion forces of the ELP model, which are the diameter of the spherical core, σ22, the square-well depth, ε22 (with the Boltzmann constant), and the range of the square well, λ22. Parameters in the last two columns

describe the strength of the peptide-peptide hydrogen bonds, , = , ∗ = ,∗ = ,∗ ∗ ,

= ,, ∗ = ,,∗ = ,,∗ ∗ of the hydrogen-bonding interactions. and the range ,, The values for the dispersion interactions between amino acid beads are generated based on data obtained from atomistic simulations on a system containing 16 (VPGVG) chains and 6053 water molecules at 300 K. Explicit-solvent, atomistic simulations are first carried out on the system containing 16 (VPGVG) peptide molecules and water molecules using GROMACS 5.1.2 with the Amber 99SB force field 72. The peptides are initially in a linear extended conformations. These 16 molecules are first solvated in a box with 6053 SPC/E water molecules, making the closest distance between polypeptide and the edge of the box 12 Angstroms. A simulation is performed at 300 K and 0.1 MPa. The simulation consists of the following stages: first, a 50,000step minimization is performed on the system using the steepest descent algorithm. Subsequently, a short 100 ps NVT ensemble MD run with a box size 64 Å × 56 Å × 61 Å is conducted to achieve target temperature 300 K. Then an NPT ensemble MD run is performed on the system using the Berendsen thermostat mesh Ewald (PME)

74

73

to maintain the simulation temperature. Particle

summation is used to calculate the long-ranged electrostatic interactions.

The SHAKE 75 algorithm is used to constrain the bonds involving hydrogen atoms. The run time 11


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for simulations is 100 ns. The HB dynamics analysis is performed using GROMACS 5.1.2. The first 20 ns of the simulation are discarded, leaving the last 80 ns for analysis. The trajectory of the simulation is saved every 100 fs. We coarse grain the VPGVG sequence into five sites, one for each amino acid. Each coarse-grained amino acid has the same σ22, λ22, and ε22 values, but its mass is the sum of the masses of the atoms in that particular amino acid. We use the average atomistic radial distribution function, g(r), between each amino acid pair, i.e., between valine and glycine, valine and proline, proline and glycine, to calculate the values of σ22, λ22, and ε22. The value of σ22 is taken to be the first value of r in the radial distribution function at which g(r) is greater than or equal to one. The range of the square well, λ22, is taken to be 1.5 so as to represent closely the width of the g(r) peak 76. The interaction energy between the coarse-grained sites is determined using an iterative procedure that matches the radial distribution function from the coarse-grained simulation to that from the atomistic simulation. This method has been modified for application to coarse grained square-well/square-shoulder potentials, and is presented in detail in previous work77. The value of ε22 is initially set to zero, and the depth of the square-well/square-shoulder is adjusted until the coarse-grained simulation results in a radial distribution function that matches the atomistic radial distribution function in the region between σ22 and λ22. The values of the parameters σ22, λ22, and ε22 used in our current work are shown in the first three columns of Table 2.

We estimated the values for the hydrogen-bonding , / strengths between amino

acid pairs from parameter values reported for peptide-water interactions in earlierPC-SAFT (Cameretti et al.78) and RP-SAFT (Seyfi et al.

79

) studies and then scale these values to be

consistent with the SAFT-VR parameters of reference

47

. (Although we initially tried to obtain 12


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the peptide-peptide parameters from our atomistic simulations using the Chandler and Luzar method 80, as we did for the peptide-water interactions, the resulting force field led to states with chains that tended to aggregate during the simulations, resulting in unrealistically high values of the hydrogen-bonding strengths). Instead, we take as reference the value of the hydrogen

bonding strength , / , between peptide and peptide in the PC-SAFT investigation of

Cameretti et al.78, which they give as 1457 K. To find a value of , / that is consistent with

our SAFT – VR model, we multiply this value by the ratio, RPC of the water-water hydrogen bonding strengths in the SAFT-VR model

47

(1400 K) to that of the water-water hydrogen bond

in the PC-SAFT model of Cameretti et al. 78 (2425.67 K) , i.e., R PC =

1400 .00 = 0 .577 2425 .6714

(1),

to obtain , / = 1457 K × 0.577 = 840 K. The value of the hydrogen-bonding strength

between peptide beads in the RP-SAFT investigation of Seyfi et al

79

. is 838.9 K. Following a

similar procedure, the ratio RRP of the water-water hydrogen-bonding strengths in SAFT-VR 47 to that in RP-SAFT of Seyfi et al. 79 (~1597.76) is found as R RP =

1400 .00 = 0 . 87 1597 .7621

(2).

By scaling the peptide-peptide hydrogen bonding strength obtained from RP-SAFT with RRP, we

/ = 838K × 0.87 = 729 K. In this way, we get a rough range of hydrogen obtain a value ,

bonding strengths , / to be used in the SAFT coarse-grained model between amino acid

pairs that is between 729 K and 840 K. In our final model, , / was further refined to the

value of 800 K in order to give a better description of the experimental data of Meyer et al 8 for

13


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the T-w diagram of (VPGVG)30 + water in the low concentration region. The pure component parameters for our elastin-like polymer (ELP) model can be found in Table 2.

Water-peptide Interaction As well as the pure component parameters, unlike dispersion parameters also need to be specified in order to determine the thermodynamic properties and phase behaviour of binary mixtures; standard combining rules are used58. The unlike diameter is calculated as the arithmetic mean of the like parameters: σ12=(σ11+σ22)/2, which sets the closest distance of the hard spheres to the point at which they are tangent. Τhe range of the unlike water-ELPs square-well dispersion attractive interaction is determined using the arithmetic average λ12=(λ11σ11+λ22σ22)/(2σ12). Τhe unlike water-ELP square well dispersive attractive interaction energy is given by ε12=(1k12)(ε11ε22)1/2, where k12 is an adjustable parameter for the unlike interaction energy between water and ELPs employed to achieve a better description of the measured liquid-liquid equilibrium data and takes into account asymmetry and polar interactions

81

effects. The

combining rules for σ12 and λ12 do not contain any adjustable parameters: changing σ12 is unphysical in a system of hard spheres as it allows for overlap of hard spheres or hard repulsions larger than the average diameter; further modifying λ12 is not effective as it is highly correlated with the k12 (or equivalently ε12) as since they both modify the overall strength of the square-well interaction. The unlike interaction parameter, k12, is estimated by matching the (VPGVG)30 + water T-w phase diagram calculated with our SAFT-VR model to that measured experimentally by Meyer et al.8, resulting in a k12 for (VPGVG)30 of -0.367. The adjustable k12 parameter is further refined for each of the chain lengths of the ELPs considered by comparison with the experimental T-w phase boundaries in each case. Although the value of the k12 is determined 14


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using experimental data over a limited range of mass fractions, the SAFT-VR equation of state has been shown to successfully model and predict phase equilibria transferably

82-88

, including

polymer systems47, providing confidence in our predictions over a broad temperature, pressure and composition range. The resulting chain-length dependent k12 can be represented as k12 = −0.367 + 0.001× (

n − 30 ). 30

(3)

This corresponds to k12 = -0.367 for (VPGVG)30, -0.366 for (VPGVG)60, -0.365 for (VPGVG)90, and -0.364 for (VPGVG)120. Although this variation may appear small, the number of bead-bead interactions is large and the resulting phase behavior is very sensitive to the precise value of the k12 parameter46.

The peptide-water hydrogen-bonding strengths , / between the middle peptide

beads and water, and , / between the terminal peptide beads and water, are calculated

from the trajectories of atomistic simulations using the method of Chandler and Luzar 80. In this method, measurements of the hydrogen-bonding kinetics are used to determine the lifetime of the hydrogen bond; the hydrogen-bonding energy is then calculated by assuming that HB breakage is an Eyring process 89, which allows the Gibbs energy of activation of hydrogen bonding to be calculated from the bonding lifetime. The hydrogen-bonding strengths are then scaled to make them consistent with the SAFT-VR model of water47. The following procedure is used: The value of the water-water hydrogen-bonding interaction in the atomistic simulation is 998 K, while the value of the SAFT-VR model47 is 1400 K, thus the ratio, Rsimulation of the water-water hydrogen-bonding strengths is 1400 R = ~ 1.4, (4) simulation 998

15


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Using the same scaling factor for the hydrogen-bonding energy between the water and the

middle peptide bead obtained from atomistic simulation, a value , / ~1490.68 K is

obtained for the SAFT-VR model. In the case of the hydrogen-bonding energy between the water

and the end peptide bead, the value obtained is , / ~1653.14 K. The peptide-water unlike

interaction parameters used in the SAFT-VR model proposed are summarized in Table.3. Table 3. Unlike peptide-water SAFT-VR interaction parameters c k12 Eqn. 3

⁄ ,

⁄K

,, /Å

HB 12,eH∗ ⁄k ⁄K

,, ∗ /Å

1490.68

2.18800

1653.14

2.18800

c

The parameters characterizing the unlike interactions between peptide and water are the

dispersion adjustable parameter k12, the hydrogen-bonding energy , = , , and range

,, = ,, between site e and site H, , and the hydrogen-bonding energy , ∗ = * *

,∗ = ,∗ and range ,,∗ = ,∗ = ,∗ between H-e or e-H . 3 The SAFT Equation of State Statistical associating fluid theory for potential of variable ranges (SAFT-VR)57,58 is used in our current study to determine the thermodynamic properties for systems containing ELPs and water using the model outlined in the previous section. The Helmholtz free energy A of the associating fluid can be described as !"#$% '()( *$!) $++(* = + + + where N is the total number of molecules in the mixture, and T is the temperature of the system. The terms AIDEAL represents the ideal free energy of the mixture, and the others terms are residual contributions to the free energy due to: (1) the monomer-monomer repulsion/dispersion interactions (AMONO), which are treated following a Barker and Henderson high-temperature 16


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expansion up to second order

90-92

; (2) the formation of the polypeptide chain (ACHAIN), which

according to the first-order thermodynamic perturbation theory of Wertheim 68-71, 93 is a function of the number of segments in a given chain and of the natural logarithm of the contact pair distribution function; and (3) the site-site hydrogen-bonding interactions (AASSOC), which is also obtained from Wertheim’s first order thermodynamic perturbation theory in terms of the fractions of molecules not bonded at given association sites. More detailed expressions can be found in the original SAFT-VR papers57,58 and in reference 47. The other thermodynamic properties can be obtained from the Helmholtz free energy following standard thermodynamic relationships. The pressure is calculated from the partial derivative with respect to volume as , = −.⁄./),0 , and the chemical potential is calculated from the partial derivative with the number of particles of each species as 12 = −.⁄.32 4,0,)5 . The phase equilibria is then solved using the HELD algorithm.

94

, which

allows one to indentify all of the stable equilibrium phases.

4 Results and Discussion 4.1 Binary phase diagrams from calculations of SAFT match experimental observations The T-w phase diagrams including the lower critical solution temperatures (LCSTs) for aqueous solutions of (VPGVG)n with four different lengths, n = 30, 60, 90, and 120, at P = 0.1 MPa (1 bar) are calculated using the SAFT-VR equation of state with the models presented in the previous section. The regions of liquid-liquid rimmiscibility corresponding to these systems are shown in Figure 3 together with the available experimental data8. The LCSTs for all of the systems considered are found to be located at similar ELP mass fractions, close to w ~ 0.89; a 17


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mass fraction that is much larger than those considered in previous experiments6,8,31. On closer inspection it can be seen that the LCST is lower for the ELPS with longer chain length. This is consistent with previous reports of the LCSTs of aqueous solutions containing PEG48,49 , poly(Nisopropylacrylamide) (PNIPAM)5,95 , and PEO52,96. In all of these studies, the LCST of the solution is found to decrease as the chain length of the polymer is increased. At 0.1 MPa, the predictions of the LCSTs of the ELPs considered here are found to be 193.6 K for (VPGVG)30, 190.9 K for (VPGVG)60, 189.6 K for (VPGVG)90, and 188.6 K for (VPGVG)120, all of which are lower than the freezing point of water (273.15 K). There are several possible explanations for our finding that the calculated LCSTs are lower than the freezing point of water. Firstly, our unlike interaction parameter k12, which was determined by comparing the calculated transition temperature of (VPGVG)30 at very low ELP mass fraction to that measured experimentally, may be inaccurate; the large number of segmentsegment interactions mean that very small variations in k12 can have a large impact in the predicted bulk phase behaviour

42

. Secondly, the SAFT-VR model proposed here does not

incorporate the presence of salts, and the experiments were conducted in phosphate-buffered saline (PBS), which contains salt, and other components (a common composition of PBS is 8.0 g L-1 of NaCl, 0.2 g L-1 of KCl, 1.42 g L-1 of Na2HPO4, and 0.24 g L-1 of KH2PO4). Thirdly, the SAFT-VR treatment as implemented here does take into account the appearance of a solid phase, and so the calculated LCST may be metastable with respect to the transition to the solid phase. It should be noted however, that it is unlikely that a solution containing ~ 0.89 mass fraction of ELP would freeze at the same temperature as pure water. In fact one would expect solid phases to appear at significantly lower temperatures. As an indication, the melting and glass transition temperature of polyethylene glycol of molecular weight 6,000 g mol-1 are 333.5K and 256K 18


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respectively; there is a slight increase in melting point experiences with increasing polymer molecular weight whereas the glass transition temperature is found to decease

97

. The

experimental solubility of PEG4000 in water at 298.15K is found to be 68.2% by mass 98.

Figure 3. The temperature-mass fraction T-w fluid phase behaviour (cloud curves) of aqueous solutions of ELP (VPGVG)n sequences in the liquid-liquid region, at P = 0.1 MPa for four different ELP lengths (from top to bottom): n = 30 (black), 60 (red), 90 (blue), 120 (green)). The continuous curves correspond to SAFT-VR calculations and the symbols to experimental data 8. The minimum point of each curve is the LCST.

In the low-concentration region the cloud curves calculated with SAFT-VR are found to match very well with those obtained experimentally (Figure 4). The chain-length dependent k12 19


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(cf. equation (3)) allows one to obtain a good description of the temperature dependence of the systems for different chain lengths.

Figure 4. Temperature-mass fraction T-w cloud curves of aqueous solutions of (VPGVG)30 (black), (VPGVG)60 (red), (VPGVG)90 (blue) calculated with SAFT-VR (lines) compared to experimental data8 (symbols) at 0.1MPa. The low-mass fraction region is highlighted in this figure. Having confirmed the adequacy of the SAFT-VR model for the description of the liquidliquid phase boundaries of the ELPs + water systems of interest, it is useful to consider the overall fluid-phase behaviour of these mixtures, including the high-temperature gas-liquid region as well as the liquid-liquid region. In this way an analysis of the classification of the type of 20


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phase behavior 35 exhibited by these mixtures can be made. In some cases (type IV and type V phase behavior), the gas-liquid critical curve merges into a liquid-liquid critical line associated with the presence of LCSTs close to the critical point of the more volatile component (water in our case). By contrast, the gas-liquid critical line extends continuously between the critical points of the two components in type VI phase behavior, and a locus of LCSTs critical points is observed at low temperatures associated with the onset of a closed-loop of liquid-liquid immiscibility which is bound at high temperature by an UCST. The complete T-w fluid-phase diagrams of aqueous solutions of ELP (VPGVG)n sequences of various lengths, n = 10, 30, 60, and 300 at a pressure of 0.1 MPa are shown in Figure 5. The gas-liquid region can be seen to extend to higher temperatures as the chain lengths of the ELP is increased, as the peptides become less volatile, with correspondingly higher saturation temperatures. The shape of these phase diagrams resembles those found in type V and VI systems35 in which a liquid-liquid region bound by an LCST is observed below a gas-liquid region. Three phases (vapour-liquid-liquid) are found at coexistence at 373.2 K (represented by the dashed line), which is the boiling point of our SAFT-VR model of water at 0.1MPa. Type IV mixtures present an additional region of liquid-liquid phase separation with an UCEP at low temperature but such a region is not observed in our case. One could also interpret the T-w diagrams presented here as corresponding to type VI phase behavior

99

with a very wide liquid-

liquid phase-separation region spanning mass fractions of 0 to 1. Regions of higher pressure have to be considered to fully characterize the type of phase behaviour, as shown in the next section47.

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Figure 5. Temperature-mass fraction T-w fluid-phase behaviour of aqueous solutions of ELP (VPGVG)n at P = 0.1 MPa for lengths of n=10 (black line), 30 (red line), 60 (blue line), and 300 (green line). The dashed line represents the three-phase vapor-liquid-liquid equilibrium line. 4.2 The phase diagram of aqeuous solutions of ELP (VPGVG)30 is characterized as type VI To identify the type of fluid-phase behaviour exhibited by the ELP-water system, the T-w phase diagrams of the (VPGVG)30 - water system is determined at different pressures. This allows one to distinguish the type of phase diagrams among the type IV, V, and VI systems, as they can exhibit the same T-w topology in the low-pressure region near the three-phase line. As was mentioned in the introduction, ELP polymer solutions exhibit a behaviour which is somewhere in between those of type V phase systems with LCSTs driven by changes in the 22


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volume of mixing of the system, and those of mixtures dominated by hydrogen-bonding interactions which are characterized by type VI phase behaviour, with LCSTs driven by changes in the enthalpy of mixing47. An advantage of theoretical (equation of state) methods such as SAFT-VR is that the thermodynamic properties and phase behavior of complex mixtures can be evaluated over a broad range of condistions in a matter of seconds. In Figure 6 the temperature-mass fraction fluid phase diagrams for the (VPGVG)30-water system at six different pressures (0.1, 10, 20, 30, 50, and 150 MPa) are presented. At low pressure ~ 0.1 MPa, the liquid-liquid immiscibility and vapour-liquid region can be seen to meet at a vapour-liquid-liquid three-phase coexistence temperature. As the pressure is increased to 10 MPa, the liquid-liquid and vapour-liquid regions detach, and a closed-loop of liquid-liquid immiscibility with an UCST at a mass fraction of 0.092 is seen, corresponding to type VI phase behaviour. As the pressure is further increased to 20 MPa, the gas-liquid region becomes smaller and moves towards higher temperatures. The gas-liquid region can be seen to shrink further as the pressure is increased to 30 MPa and has finally disappeared altogether from the phase diagram at 50 MPa. It is interesting to note that at 50 MPa a new liqui-liquid phase separation region emerges at low ELP weight fractions. The nature of the new coexisting phases can be assessed by analyzing the packing fraction (the fraction of the volume of a system occupied by its constituent particles). For our binary mixture of water and ELP the packing fraction 7 can be expressed as 7=

8 : ; < + : ; < 5, 6

where :> is the number density of segments of diameter ;>> , 1 corresponding to water and 2 to the ELP monomer. From an examination of the packing fraction of the coexisting phasesin Figure 7, 23


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it is apparent that the new phases formed, though very dense, are still liquid-like (with packing fractions 7 ~ 0.485). Three-phase liquid-liquid-liquid equilibrium can be seen at 328.5K. The coexisting phases at the lower temperatures ~ 250 K with packing fractions 7 > 0.5 are most probably metastable relative to solid phases. At the even higher pressure of 150 MPa the emergent liquid-liquid equilibrium region can be seen to become more exntensive with the triple line shifted upwards to 385.3K. This interesting liquid phase behavior at pressures above 50 MPa is atypical from what is usually observed with type VI phase behavior and will be the subject of a separate paper.

Figure 6 Temperature-mass fraction T-w isobars of the fluid-phase behaviour of aqueous solutions of (VPGVG)30 at pressures of 0.1, 10, 20, 30, 50, and 150 MPa. The dashed lines depict three phase equilibrium lines.

24


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Figure 7. Temperature-packing fraction T-7 isobars of the fluid-phase behaviour of aqueous solutions of (VPGVG)30 at constant pressures of 0.1, 50 and 150 MPa. 4.3 The phase behavior of aqueous solutions of short-chain ELP We have also calculated the fluid-phase behaviour of ELP (VPGVG)n sequences at 0.1 MPa for some short chain ELPs. The fluid phase T-w diagrams for ELP (VPGVG)n sequences at P = 0.1 MPa with n = 1, 2, 3, and 5 are shown in Figures 8 (a) to (d). We find that LCST behavior is present for all chain lengths of the biopolymers. The vapor-liquid area close to pure water is seen to decrease with increasing chain length, and the three phase line approaches 373.2K at longer chain lengths. The phase diagrams shown in Figure 8 (e) correspond to constant pressure T-w slices characteristic of those found in types V and VI systems99, as mentioned earlier. We have not carried out calculations at high pressure in this case, although it is likely that the phase behaviour corresponds to a type VI phase diagram as found for the longer ELP (VPGVG)30 in the previous section. The region of liquid-liquid immiscibility is seen to shrink as the ELP chain length is decreased, but it is still present for the shortest chain considered (n=1). Nuhn and Klok

34

have

reported the absence of LCST for (GVGVP)n sequences when the chain length is less than 4 (20

25


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amino acids in the chain), although it should be noted that our reported LCSTs and the reported experimental data are in different regions of concentration.

Figure 8. Temperature-mass fraction T-w isobars of the fluid-phase behaviour of at P = 0.1 MPa of aqueous solutions of ELP (VPGVG)n for chain lengths of n = 5 (a), 3 (b), 2 (c) and 1 (d). The dashed lines depict the three phase vapor-liquid-liquid equilibrium lines. 4.4 The changes of liquid-liquid immiscibility region with hydrogen-bonding strength It is also of interest to explore the kinds of phase diagrams that are obtained as the strength of the hydrogen bonding is changed in the proposed SAFT-VR model. We first consider a simultaneous decrease of both the peptide-peptide and peptide-water hydrogen bonding (HB) 26


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by scaling them by a value less than 1.0 and calculating the corresponding T-w phase diagrams of aqueous solutions of (VPGVG)30. The phase behaviour obtained for scaling factors of 0.6 HB, 0.8 HB, 0.9 HB, and 1.0 HB are shown in Figure 9 (a). The liquid-liquid immiscibility region is seen to become more extensive and move to lower temperatures as the peptide-peptide and peptide-water hydrogen bonding strengths are decreased. For a value of the hydrogen bonding strengths of 0.8 HB, the LCST decreases below 100 K and disappears. Furthermore, in the case when the hydrogen bonding strength is 0.6HB, the mass fractions of the coexisting liquid phases are close to 0 and 1 and no LCST is seen. Conversely, when the peptide-peptide and peptide-water hydrogen bonding strengths are increased (cf. Figure 9 (b)) the region of liquid-liquid immiscibility is seen to decrease and the LCST moves to higher temperatures. For a value of the hydrogen bonding strength of 1.2 HB, the liquid-liquid immiscibility region disappears altogether, leaving only the vapour-liquid phase separation region. At 1.05 HB a second small liquid-liquid immiscibility region is seen, similar to that observed at 50 MPa for 1.0 HB (as can be seen in Figure 7). Tentatively, the calculated T-w phase diagrams would suggest that the types of phase behaviour exhibited by these model systems change from a type VI 99 phase behaviour to type II or III phase behaviour, with broader liquid-liquid immiscibility and no LSCT,

when the

hydrogen bonding strength is decreased. A corresponding increase in the hydrogen bond strength lead to the disappearance of any regions of liquid-liquid separation, leading to phase diagrams of type I phase behaviour with no liquid-liquid separation.

27


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Figure 9. Temperature-mass fraction T-w isobars of the fluid-phase behaviour at P = 0.1 MPa of aqueous solutions of (VPGVG)30: (a) hydrogen bonding strengths of 1.0 HB (black), 0.9 HB (red), 0.8 HB (blue) and 0.6 HB (green): (b) hydrogen bonding strengths of 1.0 HB (black), 1.05 HB (red), 1.07 HB (blue) and 1.2H B (green) . The dashed lines represent the coexistence of three phases. 28


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4.6 The dependence of the LCSTs of ELPs on the chain lengths The LCSTs calculated from SAFT can be represented with a power law dependence on the chain length of the ELP as seen in Figure 10. The values of LCSTs obtained for aqueous solutions of ELP (VPGVG)n with n = 20, 30, 60, 90, 120 and 300 fitted against the chain length gives the following relationship TLCST = 208.87 × n-0.022 with R2 = 0.995, see Figure 10(a). This is different than the length dependence of the transition temperatures (Tt) of ELPs at very low mole fractions ~ 25 µM reported experimentally by Meyer et al. 8 and also found in earlier simulations 7

where the power law function was , Tt (°C) = constant × n-0.64±0.1, or equivalently Tt (K) =

constant × n-0.081±0.1. An explanation for this difference could be that both the experiments and simulations focussed on systems with very low ELP mass fraction and the transition temperature of an ELP is much higher than the LCST. The transition temperatures of ELPs of different lengths at molar concentration ~ 25 µM (this value was chosen to be the same as experiments and corresponds to mass fractions of 0.000205 for (VPGVG)20, 0.000308 for (VPGVG)30, 0.000615 for (VPGVG)60, 0.000922 for (VPGVG)90, and 0.001229 for (VPGVG)120) calculated with the SAFT-VR model are plotted against the ELP lengths and subsequently fitted with a power law function, Tt (K) = constant × n-0.048, as shown in Figure 11. A Shultz-Flory 60 plot for our aqueous solutions of ELP is presented In Figure 10 (b). We obtain the following linear ???

relationship: 0

@ABC

= −3.809 H

√JK

+ JKL + 5.491 when chain lengths from n = 1 to 300 are

considered, where 5n represents the number of beads or segments in the ELP (VPGVG)n. This result is consistent with the study of the critical point behavior of polymer-solvent mixtures carried out by Shultz and Flory60 who employed a lattice based model 100-103 to predict and verify 29


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this linear relationship for polymer-solvent systems. Note that Shultz and Flory60 specifically use the following proportionality

0@ABC

∝H

√O

+

O

L, where is the ratio of the molar volume of the

polymer to that of the solvent which, in the Flory-Huggins framework, is identical to the number of polymer segments (when the solvent consists of a single segment). We have used the number of polymer segments in Figure 10 (b) for consistency with Figure 10 (a). When we use the ratio ???

of the molar volumes we obtain the following relationship: 0 with = /Q,RST

UVWX , ,//Q,YST UVWX , ,

@ABC

= −7.230 H

√

+ OL + 5.509, O

. On the Shultz-Flory plot (Figure 12 (b)), the

extrapolation to zero in the x-axis gives the critical temperature for the infinitely long polymer system, which is equivalent to the theta temperature (Z), the temperature at which the solventpolymer system behaves ideally. The Z temperature demarcates the regions in which the solvent behaves as a good solvent, with favorable interactions between the polymer monomers and solvent, and poor solvent, with monomer-monomer interactions preferred over monomer-solvent interactions 104. In our aqueous solutions of ELP Z is found to be ~180.34K.

30


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31


Langmuir

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Figure 10. (a) The LCSTs of aqueous solutions of ELP (VPGVG)n for chain lengths of n = 1 to 300 calculated from SAFT-VR (circles) plotted against their lengths n at 0.1 MPa pressure. The correlation (dashed curve) was obtained from the LCSTs of n = 20, 30, 60, 90, 120, 300. (b) The Shultz-Flory diagram depicting a linear relationship60 between 1000/TLCST and H

√JK

+ JKL,

where the LCSTs are calculated from the SAFT equation (circles). The dashed line represents the correlation.

Figure 11. The transition temperatures of aqueous solutions of ELP (VPGVG)n sequences for chain lengths of n =20, 30, 60, 90, and 120 calculated from SAFT-VR (open circles) and interpolated experimental from Meyer et al. 8 (filled circles) plotted against their lengths at 0.1 MPa pressure at very low concentration ~ 25µM, which corresponds to mass fractions of 0.000205 for (VPGVG)20, 0.000308 for (VPGVG)30, 0.000615 for (VPGVG)60, 0.000922 for (VPGVG)90, and 0.001229 for (VPGVG)120. The dashed curve represents the correlation.

32


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5 Conclusions The statistical associating fluid theory for potentials of variable range (SAFT-VR) is used to predict the fluid-phase behaviour of aqueous solutions of ELP (VPGVG)n sequences with n =30, 60, 90, and 120. A molecular model of the mixture is developed taking into account the water-water, water-peptide, and peptide- peptide repulsion/dispersion forces and hydrogenbonding strengths. The intermolecular parameters for water (both the dispersion forces and the hydrogen bonding strengths) are the same as those used in the investigation of PEG-water systems47. The peptide-peptide dispersion forces are calculated from atomistic simulation using a modified iterative Boltzmann inversion method appropriate for square well/ square shoulder potentials. The unlike dispersion interactions between the peptide segments and water are adjusted using a length-dependent unlike interaction parameter k12 by matching the temperatureweight fraction phase diagram to the experimental data ELP (VPGVG)n at n = 30, 60, 90, and 120. The phase behaviour is very sensitive to the value of k12 given the large number of unlike interactions in the system42 and it is therefore essential to adjust this value to experimental data. The hydrogen bonding interaction strengths between peptide and water are determined from atomistic simulation using Chandler and Luzar’s method consistent with that used in reference

47

80

and are subsequently scaled to be

The hydrogen-bonding strength between peptide

segments is estimated by using parameter values from peptide-water PC-SAFT 78 and RP-SAFT 79

investigations and then scaling these values to be consistent with reference 47. This procedure

enables the parametrization of a SAFT-VR model for the (VPGVG)n polypeptide using limited experimental data, and can also be adopted to study other polymer-solvent systems particularly when experimental data is scarce. 33


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The lower critical solution temperatures (LCSTs) of aqueous solution of ELPs of length n = 1 to 300 are predicted. The value of the LCST is found to decrease as the chain length of the ELP is made longer, in agreement with the trends for other LCST systems, such as PEG-water 47 and PNIPAAM-water 5. The tempearture-mass fraction diagrams calculated with SAFT-VR approach are in good correspondence with the corresponding experimental measurements

8

at

low mass fractions for ELP (VPGVG)n sequences with n =30, 60, 90, and 120. The phase diagrams of ELPs at 0.1 MPa pressure resemble those of type V and VI systems, and an examination of the phase behaviour with increasing pressure in aqueous solutions of (VPGVG)30 suggests that these systems exhibit type VI phase behavior: a closed-loop liquid-liquid immiscibility region is found that is separate from the vapour-liquid region. Interestingly at pressures above 50 MPa, a second liquid-liquid equilibrium is found with a corresponding three phase liquid-liquid-liquid equilibrium line; this highly unusual behavior deviates from the typical Type VI phase behavior and will be the subject of a future publication. The phase diagrams of ELPs with short lengths are found to exhibit both liquid-liquid and vapour-liquid coexistence regions, which become less extensive as the chain length of the ELP is made shorter. We also examined the effect of the hydrogen-bonding strength on the phase diagram of aqeuous solutions of (VPGVG)30. The vapour-liquid and liquid-liquid regions become more extensive as the hydrogen-bonding strength is weakened and shrink as the hydrogen-bonding strength is increased. A linear relationship H

√JK

60

is observed between the inverse temperature 1000/TLCST and

+ JKL where TLCST is the LCST of the ELP solution, and n is the number of repeat units in

(VPGVG)n , consistent with the behavior of polymer-solvent mixtures predicted by Shultz and Flory 60 using a lattice based model. 34


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Acknowledgement This work was supported in part by the NSF’s Research Triangle MRSEC (DMR-1121007) and the North Carolina State University College of Engineering. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1053575. TL would like to thank Imperial College London for the award of a President’s PhD Scholarship. We also acknowledge support from the Engineering and Physical Sciences Council (EPSRC) of the UK (grants EP/E016340, EP/J014958/1, and EP/J003840/1) for financial support to the Molecular Systems Engineering (MSE) group.

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6 References (1)

(2)

(3) (4) (5)

(6)

(7)

(8)

(9)

(10) (11)

(12)

(13)

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A A IDEAL A MONO A CHAIN A ASSOC = + + + NkBT NkBT NkBT NkBT NkBT (VPGVG)300 (VPGVG)60 (VPGVG)30 (VPGVG)10

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