Information

What is membrane-partitioning free energy? Can it be simulated?

What is membrane-partitioning free energy? Can it be simulated?


We are searching data for your request:

Forums and discussions:
Manuals and reference books:
Data from registers:
Wait the end of the search in all databases.
Upon completion, a link will appear to access the found materials.

Firstly, is there a strict definition of the "membrane-partitioning free energy"? It is banded around in membrane biology, but I have never seen it strictly defined. The only non-scholarly site that google shows is this Q&A on quora, and the answer there even has ambiguity in what is defined as partitioning in the membrane. This seems wholly unclear.

Furthermore, is it possible to study free energy changes upon partitioning in GROMACS, or with any other molecular dynamics simulations? If not, what are the methods I would need to use to determine the free energy of partitioning?

Links and citations for further reading on this topic are encouraged.


In the biological context, membrane-partitioning is usually referring to the stage in which the transmembrane-destined region of a protein moves from interacting with the water, to interact with the interface of the membrane.

In the diagram below showing a four step thermodynamic cycle, the partitioning free energy can be referred to as-is ΔGwiu in terms of free energy where w is water, i is the interface, and u is unfolded. The image comes from the same paper that introduced the famous Wimley and White octanol-interface scale from 1999.

Note that in a more recent 2015 article, they comment that the partitioning phase ΔGwiu is generally the only experimentally accessible step. With that in mind, simulating ΔGwif where f is the folded helix becomes necessary. This 2014 Nature Comms paper use folding-partitioning molecular dynamics simulations to estimate the free energies that are experimentally inaccessible. They used Gomacs 4.5. In fact, a study using a simulation from 2005 suggests that folding isn't necessarily required for insertion of the helices.


Note that all the information here is behind a paywall. Feel free to ask me for clarification or expansion in the comments.


Free-Energy Analysis of Peptide Binding in Lipid Membrane Using All-Atom Molecular Dynamics Simulation Combined with Theory of Solutions

Publication History

  • Received 18 August 2017
  • Revised 14 December 2017
  • Published online 10 January 2018
  • Published in issue 5 April 2018
Article Views
Altmetric
Citations

Article Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.

Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.

The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated.


Chemical Thermodynamics

Some reactions are spontaneous because they give off energy in the form of heat (H < 0). Others are spontaneous because they lead to an increase in the disorder of the system (S > 0). Calculations of H and S can be used to probe the driving force behind a particular reaction.

Calculate H and S for the following reaction and decide in which direction each of these factors will drive the reaction.

What happens when one of the potential driving forces behind a chemical reaction is favorable and the other is not? We can answer this question by defining a new quantity known as the Gibbs free energy (G) of the system, which reflects the balance between these forces.

The Gibbs free energy of a system at any moment in time is defined as the enthalpy of the system minus the product of the temperature times the entropy of the system.

The Gibbs free energy of the system is a state function because it is defined in terms of thermodynamic properties that are state functions. The change in the Gibbs free energy of the system that occurs during a reaction is therefore equal to the change in the enthalpy of the system minus the change in the product of the temperature times the entropy of the system.

If the reaction is run at constant temperature, this equation can be written as follows.

The change in the free energy of a system that occurs during a reaction can be measured under any set of conditions. If the data are collected under standard-state conditions, the result is the standard-state free energy of reaction (G o ).

G o = H o - TS o

The beauty of the equation defining the free energy of a system is its ability to determine the relative importance of the enthalpy and entropy terms as driving forces behind a particular reaction. The change in the free energy of the system that occurs during a reaction measures the balance between the two driving forces that determine whether a reaction is spontaneous. As we have seen, the enthalpy and entropy terms have different sign conventions.

The entropy term is therefore subtracted from the enthalpy term when calculating G o for a reaction.

Because of the way the free energy of the system is defined, G o is negative for any reaction for which H o is negative and S o is positive. G o is therefore negative for any reaction that is favored by both the enthalpy and entropy terms. We can therefore conclude that any reaction for which G o is negative should be favorable, or spontaneous.

Conversely, G o is positive for any reaction for which H o is positive and S o is negative. Any reaction for which G o is positive is therefore unfavorable.

Reactions are classified as either exothermic (H < 0) or endothermic (H > 0) on the basis of whether they give off or absorb heat. Reactions can also be classified as exergonic (G < 0) or endergonic (G > 0) on the basis of whether the free energy of the system decreases or increases during the reaction.

When a reaction is favored by both enthalpy (H o < 0) and entropy (S o > 0), there is no need to calculate the value of G o to decide whether the reaction should proceed. The same can be said for reactions favored by neither enthalpy (H o > 0) nor entropy (S o < 0). Free energy calculations become important for reactions favored by only one of these factors.

Calculate H and S for the following reaction:

Use the results of this calculation to determine the value of G o for this reaction at 25 o C, and explain why NH4NO3 spontaneously dissolves is water at room temperature.

The balance between the contributions from the enthalpy and entropy terms to the free energy of a reaction depends on the temperature at which the reaction is run.

Use the values of H and S calculated in Practice Problem 5 to predict whether the following reaction is spontaneous at 25C:

The equation used to define free energy suggests that the entropy term will become more important as the temperature increases.

G o = H o - TS o

Since the entropy term is unfavorable, the reaction should become less favorable as the temperature increases.

Predict whether the following reaction is still spontaneous at 500C:

Assume that the values of H o and S used in Practice Problem 7 are still valid at this temperature.

G o for a reaction can be calculated from tabulated standard-state free energy data. Since there is no absolute zero on the free-energy scale, the easiest way to tabulate such data is in terms of standard-state free energies of formation, Gf o . As might be expected, the standard-state free energy of formation of a substance is the difference between the free energy of the substance and the free energies of its elements in their thermodynamically most stable states at 1 atm, all measurements being made under standard-state conditions.

We are now ready to ask the obvious question: What does the value of G o tell us about the following reaction?

By definition, the value of G o for a reaction measures the difference between the free energies of the reactants and products when all components of the reaction are present at standard-state conditions.

G o therefore describes this reaction only when all three components are present at 1 atm pressure.

The sign of G o tells us the direction in which the reaction has to shift to come to equilibrium. The fact that G o is negative for this reaction at 25 o C means that a system under standard-state conditions at this temperature would have to shift to the right, converting some of the reactants into products, before it can reach equilibrium. The magnitude of G o for a reaction tells us how far the standard state is from equilibrium. The larger the value of G o , the further the reaction has to go to get to from the standard-state conditions to equilibrium.

Assume, for example, that we start with the following reaction under standard-state conditions, as shown in the figure below.

The value of G at that moment in time will be equal to the standard-state free energy for this reaction, G o .

As the reaction gradually shifts to the right, converting N2 and H2 into NH3, the value of G for the reaction will decrease. If we could find some way to harness the tendency of this reaction to come to equilibrium, we could get the reaction to do work. The free energy of a reaction at any moment in time is therefore said to be a measure of the energy available to do work.

When a reaction leaves the standard state because of a change in the ratio of the concentrations of the products to the reactants, we have to describe the system in terms of non-standard-state free energies of reaction. The difference between G o and G for a reaction is important. There is only one value of G o for a reaction at a given temperature, but there are an infinite number of possible values of G.

The figure below shows the relationship between G for the following reaction and the logarithm to the base e of the reaction quotient for the reaction between N2 and H2 to form NH3.

Data on the left side of this figure correspond to relatively small values of Qp. They therefore describe systems in which there is far more reactant than product. The sign of G for these systems is negative and the magnitude of G is large. The system is therefore relatively far from equilibrium and the reaction must shift to the right to reach equilibrium.

Data on the far right side of this figure describe systems in which there is more product than reactant. The sign of G is now positive and the magnitude of G is moderately large. The sign of G tells us that the reaction would have to shift to the left to reach equilibrium. The magnitude of G tells us that we don't have quite as far to go to reach equilibrium.

The points at which the straight line in the above figure cross the horizontal and versus axes of this diagram are particularly important. The straight line crosses the vertical axis when the reaction quotient for the system is equal to 1. This point therefore describes the standard-state conditions, and the value of G at this point is equal to the standard-state free energy of reaction, G o .

The point at which the straight line crosses the horizontal axis describes a system for which G is equal to zero. Because there is no driving force behind the reaction, the system must be at equilibrium.

The relationship between the free energy of reaction at any moment in time (G) and the standard-state free energy of reaction (G o ) is described by the following equation.

G = G o + RT ln Q

In this equation, R is the ideal gas constant in units of J/mol-K, T is the temperature in kelvin, ln represents a logarithm to the base e, and Q is the reaction quotient at that moment in time.

As we have seen, the driving force behind a chemical reaction is zero (G = 0) when the reaction is at equilibrium (Q = K).

0 = G o + RT ln K

We can therefore solve this equation for the relationship between G o and K.

G o = - RT ln K

This equation allows us to calculate the equilibrium constant for any reaction from the standard-state free energy of reaction, or vice versa.

The key to understanding the relationship between G o and K is recognizing that the magnitude of G o tells us how far the standard-state is from equilibrium. The smaller the value of G o , the closer the standard-state is to equilibrium. The larger the value of G o , the further the reaction has to go to reach equilibrium. The relationship between G o and the equilibrium constant for a chemical reaction is illustrated by the data in the table below.

Values of G o and K for Common Reactions at 25 o C

Reaction G o (kJ) K
2 SO3(g) 2 SO2(g) + O2(g) 141.7 1.4 x 10 -25
H2O(l) H + (aq) + OH - (aq) 79.9 1.0 x 10 -14
AgCl(s) + H2O Ag + (aq) + Cl - (aq) 55.6 1.8 x 10 -10
HOAc(aq) + H2O H + (aq) + OAc - (aq) 27.1 1.8 x 10 -5
N2(g) + 3 H2(g) 2 NH3(g) -32.9 5.8 x 10 5
HCl(aq) + H2O H + (aq) + Cl - (aq) -34.2 1 x 10 6
Cu 2+ (aq) + 4 NH3(aq) Cu(NH3)4 2+ (aq) -76.0 2.1 x 10 13
Zn(s) + Cu 2+ (aq) Zn 2+ (aq) + Cu(s) -211.8 1.4 x 10 37

Use the value of G o obtained in Practice Problem 7 to calculate the equilibrium constant for the following reaction at 25C:

The equilibrium constant for a reaction can be expressed in two ways: Kc and Kp. We can write equilibrium constant expressions in terms of the partial pressures of the reactants and products, or in terms of their concentrations in units of moles per liter.

For gas-phase reactions the equilibrium constant obtained from G o is based on the partial pressures of the gases (Kp). For reactions in solution, the equilibrium constant that comes from the calculation is based on concentrations (Kc).

Use the following standard-state free energy of formation data to calculate the acid-dissociation equilibrium constant (Ka) at for formic acid:

Compound Gf o (kJ/mol)

Equilibrium constants are not strictly constant because they change with temperature. We are now ready to understand why.

The standard-state free energy of reaction is a measure of how far the standard-state is from equilibrium.

G o = - RT ln K

But the magnitude of G o depends on the temperature of the reaction.

G o = H o - TS o

As a result, the equilibrium constant must depend on the temperature of the reaction.

A good example of this phenomenon is the reaction in which NO2 dimerizes to form N2O4.

This reaction is favored by enthalpy because it forms a new bond, which makes the system more stable. The reaction is not favored by entropy because it leads to a decrease in the disorder of the system.

NO2 is a brown gas and N2O4 is colorless. We can therefore monitor the extent to which NO2 dimerizes to form N2O4 by examining the intensity of the brown color in a sealed tube of this gas. What should happen to the equilibrium between NO2 and N2O4 as the temperature is lowered?

For the sake of argument, let's assume that there is no significant change in either H o or S o as the system is cooled. The contribution to the free energy of the reaction from the enthalpy term is therefore constant, but the contribution from the entropy term becomes smaller as the temperature is lowered.

G o = H o - TS o

As the tube is cooled, and the entropy term becomes less important, the net effect is a shift in the equilibrium toward the right. The figure below shows what happens to the intensity of the brown color when a sealed tube containing NO2 gas is immersed in liquid nitrogen. There is a drastic decrease in the amount of NO2 in the tube as it is cooled to -196 o C.

Use values of H o and S o for the following reaction at 25C to estimate the equilibrium constant for this reaction at the temperature of boiling water (100C), ice(0C), a dry ice-acetone bath (-78C), and liquid nitrogen (-196C):

The value of G for a reaction at any moment in time tells us two things. The sign of G tells us in what direction the reaction has to shift to reach equilibrium. The magnitude of G tells us how far the reaction is from equilibrium at that moment.

The potential of an electrochemical cell is a measure of how far an oxidation-reduction reaction is from equilibrium. The Nernst equation describes the relationship between the cell potential at any moment in time and the standard-state cell potential.

Let's rearrange this equation as follows.

nFE = nFE o - RT ln Q

We can now compare it with the equation used to describe the relationship between the free energy of reaction at any moment in time and the standard-state free energy of reaction.

G = G o + RT ln Q

These equations are similar because the Nernst equation is a special case of the more general free energy relationship. We can convert one of these equations to the other by taking advantage of the following relationships between the free energy of a reaction and the cell potential of the reaction when it is run as an electrochemical cell.

Use the relationship between G o and E o for an electrochemical reaction to derive the relationship between the standard-state cell potential and the equilibrium constant for the reaction.


Free energy

Our editors will review what you’ve submitted and determine whether to revise the article.

Free energy, in thermodynamics, energy-like property or state function of a system in thermodynamic equilibrium. Free energy has the dimensions of energy, and its value is determined by the state of the system and not by its history. Free energy is used to determine how systems change and how much work they can produce. It is expressed in two forms: the Helmholtz free energy F, sometimes called the work function, and the Gibbs free energy G. If U is the internal energy of a system, PV the pressure-volume product, and TS the temperature-entropy product (T being the temperature above absolute zero), then F = UTS and G = U + PVTS. The latter equation can also be written in the form G = HTS, where H = U + PV is the enthalpy. Free energy is an extensive property, meaning that its magnitude depends on the amount of a substance in a given thermodynamic state.

The changes in free energy, ΔF or ΔG, are useful in determining the direction of spontaneous change and evaluating the maximum work that can be obtained from thermodynamic processes involving chemical or other types of reactions. In a reversible process the maximum useful work that can be obtained from a system under constant temperature and constant volume is equal to the (negative) change in the Helmholtz free energy, −ΔF = −ΔU + TΔS, and the maximum useful work under constant temperature and constant pressure (other than work done against the atmosphere) is equal to the (negative) change in the Gibbs free energy, −ΔG = −ΔH + TΔS. In each case, the TΔS entropy term represents the heat absorbed by the system from a heat reservoir at temperature T under conditions where the system does maximum work. By conservation of energy, the total work done also includes the decrease in internal energy U or enthalpy H as the case may be. For example, the energy for the maximum electrical work done by a battery as it discharges comes both from the decrease in its internal energy due to chemical reactions and from the heat TΔS it absorbs in order to keep its temperature constant, which is the ideal maximum heat that can be absorbed. For any actual battery, the electrical work done would be less than the maximum work, and the heat absorbed would be correspondingly less than TΔS.

Changes in free energy can be used to judge whether changes of state can occur spontaneously. Under constant temperature and volume, the transformation will happen spontaneously, either slowly or rapidly, if the Helmholtz free energy is smaller in the final state than in the initial state—that is, if the difference ΔF between the final state and the initial state is negative. Under constant temperature and pressure, the transformation of state will occur spontaneously if the change in the Gibbs free energy, ΔG, is negative.

Phase transitions provide instructive examples, as when ice melts to form water at 0.01 °C (T = 273.16 K), with the solid and liquid phases in equilibrium. Then ΔH = 79.71 calories per gram is the latent heat of fusion, and by definition ΔS = ΔH / T = 0.292 calories per gram∙K is the entropy change. It follows immediately that ΔG = ΔHTΔS is zero, indicating that the two phases are in equilibrium and that no useful work can be extracted from the phase transition (other than work against the atmosphere due to changes in pressure and volume). Furthermore, ΔG is negative for T > 273.16 K, indicating that the direction of spontaneous change is from ice to water, and ΔG is positive for T < 273.16 K, where the reverse reaction of freezing takes place.


Refining the treatment of membrane proteins by coarse-grained models

Correspondence to: Arieh Warshel, Department of Chemistry, University of Southern California, SGM 401, 3620 McClintock Avenue, Los Angeles, CA 90089-1062. E-mail: [email protected] Search for more papers by this author

Department of Chemistry, University of Southern California, Los Angeles, California, 90089-1062

Department of Chemistry, University of Southern California, Los Angeles, California, 90089-1062

Department of Chemistry, University of Southern California, Los Angeles, California, 90089-1062

Department of Chemistry, University of Southern California, Los Angeles, California, 90089-1062

Correspondence to: Arieh Warshel, Department of Chemistry, University of Southern California, SGM 401, 3620 McClintock Avenue, Los Angeles, CA 90089-1062. E-mail: [email protected] Search for more papers by this author

ABSTRACT

Obtaining a quantitative description of the membrane proteins stability is crucial for understanding many biological processes. However the advance in this direction has remained a major challenge for both experimental studies and molecular modeling. One of the possible directions is the use of coarse-grained models but such models must be carefully calibrated and validated. Here we use a recent progress in benchmark studies on the energetics of amino acid residue and peptide membrane insertion and membrane protein stability in refining our previously developed coarse-grained model (Vicatos et al., Proteins 201482:1168). Our refined model parameters were fitted and/or tested to reproduce water/membrane partitioning energetics of amino acid side chains and a couple of model peptides. This new model provides a reasonable agreement with experiment for absolute folding free energies of several β-barrel membrane proteins as well as effects of point mutations on a relative stability for one of those proteins, OmpLA. The consideration and ranking of different rotameric states for a mutated residue was found to be essential to achieve satisfactory agreement with the reference data. Proteins 2016 84:92–117. © 2015 Wiley Periodicals, Inc.

Additional Supporting Information may be found in the online version of this article.

Please note: The publisher is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.


Results

Comparative Ionized Drug Membrane Partitioning

First, we studied membrane partitioning of SotC and compared it to the partitioning of CisC and MoxZ, each drug form representing the dominant protonation state in aqueous solution at the physiological pH. We studied their translocation across POPC membranes using US MD simulations, which allow for more efficient sampling of energetically unfavorable drug distributions across a lipid membrane compared to conventional unbiased MD simulations. US works by restraining drug positions at different values of z across the membrane using a harmonic potential. Thus, we can compute free energy for drug positions along the bilayer normal, with z = 0 corresponding to membrane center.

When all 3 drugs are located near z = 0 (see Figure 1A), we observed substantial membrane deformations, where they are coordinated by water molecules and lipid headgroups from one (for CisC) or both (for SotC and especially for MoxZ) membrane interfaces. Not surprisingly, such membrane deformations lead to substantial energetic penalties for ionized drugs to move across the membrane with the peak values at z = 0: around 18 kcal/mol for MoxZ, 10 kcal/mol for SotC and just 5 kcal/mol for CisC. Interestingly, such differences in peak free energy values correlate with computed MM drug dipole moments, which are 41.3 Debye for MoxZ, 15.5 Debye for SotC and 6.8 Debye for CisC for the same drug molecule “standard” positions and orientation (as defined by Gaussian software). For MoxZ, extensive membrane deformation exhibited by both leaflets are due to the positively charged ammonium and negatively charged carboxylate moieties at opposite ends of the molecule (Figure 1C). For SotC, a cationic secondary ammonium and polar sulfonamide groups can also attract water molecules or lipid headgroups. Both SotC and MoxZ can stretch along the membrane normal to interact with both bilayer interfaces. However, the situation is different for CisC, which also has several polar functional groups and a positively charged tertiary ammonium functionality at the center of the molecule, but it is floppier than those drugs and seems to be attracted to one membrane interface (see Figure 1). Also, CisC has a pronounced binding trough of around 𢄣 kcal/mol at 14 ≤ |z| ≤ 17 Å. This suggests, that unlike SotC and MoxZ it will accumulate at water—membrane interface. The presence of the binding trough will also inadvertently increase a barrier a drug will need to overcome to cross a membrane from ߥ to 8 kcal/mol (see Figure 1B). These calculations suggest fairly high but surprisingly different energetic costs to cross the membrane for this collection of ionized molecules.

Figure 1. Ionized drug translocation across a POPC membrane. (A) Representative snapshots from a central (z = 0) umbrella sampling windows for cationic d-sotalol (SotC), cationic cisapride (CisC), and zwitterionic moxifloxacin (MoxZ). Drug molecules along with lipid P atoms (orange), K + (purple) and Cl − (cyan) ions are shown in a space-filling representation. Other elements are colored as follows: C—gray, H—white, O—red, N𠅋lue, S—yellow. Water molecules are shown as tubes and lipid tails as wireframes. (B) PMF profiles for POPC membrane crossing for 3 drugs shown in (A). Error bars represent measures of asymmetry. (C) Chemical structures of those drugs drawn using MarvinSketch program.

Models of d-Sotalol

We performed a more detailed analysis of different protonation states of d-sotalol, focusing on the energetics of its membrane crossing. Like many other drugs in aqueous solution, d-sotalol can exist in several protonation states depending on solution pH and other factors, such as proximity to specific protein residues. Data from the literature indicate that aqueous pKa-values for d-sotalol are 8.3 and 9.8 attributed to deprotonation of sulfonamide and secondary ammonium functionalities, respectively (Foster and Carr, 1992 Hancu et al., 2014). This indicates that at physiological pH 7.4, SotC is the predominant form (around 89%), while deprotonation of the sulfonamide functionality leads to a second dominant SotZ form (around 11%). At more basic pH, the secondary ammonium functionality will deprotonate as well, leading to a negatively charged, anionic form SotA (Figure 2).

Figure 2. Protonation states of d-sotalol. Chemical structures were drawn using ChemDraw program. Asterisk (*) indicates chiral C atom.

However, there is yet another possibility, in which deprotonation of secondary ammonium group occurs first, leading to a neutral d-sotalol form (SotN). In fact, there is likely an equilibrium, and possibly interconversion, between SotN and SotZ forms, in which either one is favored depending on the polarity of the surrounding medium. We expect that a substantially less polar SotN form would be favored in the hydrophobic environment of the lipid membrane interior based on our MoxZ simulations discussed above, whereas a more polar SotZ might be favored in aqueous solution. Unfortunately, there are no experimental data to address this issue for d-sotalol. We performed a series of implicit solvent QM calculations, which seem to indicate slight preference for SotN even in bulk water (see Supplementary text for more information), but their accuracy is very uncertain. However, a recent experimental study using a combination of potentiometric titration and spectrophotometry measurements has suggested around 90% of zwitterionic and 10% of neutral form of moxifloxacin is present at physiological pH range, and that only a neutral form contributes to drug partitioning into a non-polar environment of lipid membranes or 1-octanol often used as a membrane mimetic (Langlois et al., 2005). This suggests that a neutral form of a drug is likely the one to undergo an unassisted membrane translocation.

Since we are particularly interested in lipophilic access of cardiotoxic drugs known to block hERG, we have developed standard CHARMM (Klauda et al., 2010) compatible models of d-sotalol in charged (SotC) and neutral (SotN) forms. The QM and MM dipole moments for those d-sotalol forms and drug—water interactions probed for the model optimizations are shown in Figures 3A,B for SotN and SotC, respectively. Optimized CHARMM charges (Table S3) provide good agreement with QM target dipole values. The optimized MM dipole moments point in same direction (ρ° difference in angle between QM and MM for both SotC and SotN) and are each within 20% difference in magnitude (SotN 6%, and SotC 14%). The water interaction distances were all within 0.4 Å of QM target values (see Tables S4, S5). The dipole moment is significantly higher for SotC (17.64 Debye), than for SotN (5.98 Debye), as is to be expected for charged vs. neutral species and in agreement with QM-values. Interaction energies with water were also in good agreement with QM-values with root mean square (RMS) and maximum errors of 0.8 and 1.5 kcal/mol for SotN (Table S5) as well as 1.6 and 3.0 kcal/mol (see Table S4) for SotC, respectively. No internal (bond, angle, dihedral angle) parameters needed to be optimized for SotC, whereas for SotN there was a high penalty score for the C2-N1-C3 bond angle (shown by blue arrow in Figure 3C), and optimization yielded a difference of 0.86° (i.e., ρ° as required) between MM and QM values. Also for SotN, 7 dihedral angle parameter optimizations yielded marked improvement over CGENFF initial guesses (illustrated in Figure 3C for SotN C8-C3-N1-C2 dihedral angle highlighted in pink, with all the dihedral scan profiles shown in Figure S2), with optimized torsional energy minima within ߠ.5 kcal/mol of QM values. For comparison, raw CGENFF dihedral parameters with high penalties yielded QM energy minima differences sometimes as high ߢ kcal/mol. These optimized parameters represent a significant improvement over initial guesses and should yield more accurate computed energetics from MD simulations.

Figure 3. CHARMM force field parameter optimization of d-sotalol. The QM (blue arrow) and MM (red arrow) dipole moments for neutral, SotN (A), and charged, SotC (B), forms of d-sotalol are compared, and their QM optimized water interactions are shown by dashed blue lines. A sample dihedral angle C8-C3-N1-C2 optimization (bonds are highlighted in purple on SotN molecule) is shown in (C), with reference QM computed energy scan in blue, non-optimized CGENFF energy scan in green, and optimized MM energy scan in red, demonstrating marked improvement. Asterisk (*) indicates chiral C atom (C1).

At this time, we were not able to develop empirical models of the SotZ and SotA forms of the drug (Figure 2), since a negatively charged sulfonamide nitrogen atom type does not exist in either CHARMM biomolecular, or generalized (CGENFF) force fields. The fraction of these forms in aqueous solution or other media is uncertain, but based on a very high free energy barrier for zwitterionic moxifloxacin translocation (Figure 1 and discussion above) as well as the very large dipole moments for SotZ and SotA (see Table S1 and Supplementary text), we do not expect them to contribute substantially to the passive diffusion of d-sotalol across a lipid membrane, or the lipophilic access of this drug to hERG or other protein targets.

We should also mention that sotalol has a chiral center at C1 atom (shown by an asterisk in Figures 2, 3C), and in this study we only focused on S-enantiomer, d-sotalol. However, the developed force field parameters can be also used for R-enantiomer, l-sotalol, which will be also considered in our subsequent studies.

D-Sotalol Solvation and Orientation across the Membrane

We used our SotC and SotN models to investigate their interactions with a lipid membrane as they move across using US MD simulations. For those simulations we applied extensive sampling, especially important for hindered drug reorientation in the membrane interior (see Supplementary text for more information). We also performed those simulations with two popular biomolecular modeling packages, NAMD and CHARMM, with the former being more computationally efficient on our GPU (Graphics Processing Unit) cluster. However, CHARMM allows using P21 symmetry to take into account likely changes in the areas of top and bottom bilayer leaflets as a drug moves through the membrane by shuffling lipid molecules between them as it happens. We established that the lipid membrane properties of our simulated systems are in agreement with experimental data in this case (See Supplemental text).

We then started to investigate membrane𠅍rug interactions, first, by looking at equilibrated system snapshots at the membrane center (z = 0 Å) and water/membrane interfacial region |z| = 14 Å, corresponding to free energy minimum for SotN (see Figure 4). It can clearly be seen that both charged and neutral drug molecules can adapt different orientations with respect to the membrane normal and can be solvated by both water molecules and lipid head groups even deep in the membrane interior for SotC in agreement with our CHARMM multiple-drug simulations shown in Figure 1 and discussed above. Interestingly, that in NAMD simulation snapshots shown in Figure 4, we observed that SotC while held around membrane center (z = 0) can adopt different long-lasting (see below) orientations “grabbing” water molecules and lipid head groups from either top or bottom membrane interface, but did not observe them making interfacial connections to both leaflets, as was observed in our CHARMM simulations (Figure 1).

Figure 4. Representative snapshots of charged (SotC) and neutral (SotN) d-sotalol moving across a POPC membrane from umbrella sampling MD simulations. Reference d-sotalol center of mass (COM) z positions with respect to membrane COM are shown on the top. See Figure 1 caption for molecular representation and coloring information. Two structures for z = 0 for each drug represent final system snapshots from two independent simulations with a different initial drug orientation (see Supplementary text for more information).

Next, we performed a quantitative analysis of drug solvation shown in Figure 5. While SotC and SotN are found in bulk water regions, for |z| > 25 Å (ߥ Å beyond phosphate groups), they are solvated by ߥ.5 and 5 water molecules, respectively. We defined the interfacial region as 15 < |z| < 25 Å, where 15 Å boundary was established based on an experimentally determined POPC hydrophobic thickness of 28.8 ± 0.6 Å (Kucerka et al., 2011). The water coordination remains the same as in bulk, until the drug reaches inside the core of the membrane, where we observe a bigger drop in the number of water molecules solvating SotN. In the center of the bilayer, at z = 0 Å, almost no water molecules are found coordinating the neutral drug, while at least 1.2 water molecules continue to coordinate the charged species. Additionally, when SotC is found at the interface or the hydrophobic core of the membrane, it is coordinated by lipid phosphate and carbonyl groups, while SotN remains virtually uncoordinated by these functional groups in the membrane core and has a similar coordination by carbonyl O and smaller by phosphate O atoms in the interfacial region (Figure 5).

Figure 5. Analysis of d-sotalol solvation from umbrella sampling MD simulations. Solvation numbers of water or lipid oxygen atoms within 4.25 Å cutoff distance from non-hydrogen atoms of SotC or SotN were computed based on integrated radial distribution function (RDF) profiles. See Figure S6 for a few representative RDF profiles. Error bars shown in all the graphs are computed from profile asymmetries.

Such solvation results in the preferential orientation of both SotC and SotN with respect to bilayer normal (coinciding with the z axis) as shown in Figure 6. There is no preferred orientation of both drugs in bulk water as expected, which is exemplified by average θ being around 90° and order parameter being 0 (see Figure 6 and top right panels in Figures S7, S8 for time series). There is a strong preference for N1 … S vector of both drugs to be aligned with the z axis in the outer interfacial region i.e., at 20 < |z| < 25 Å, whereas there is some tendency for drugs to lie perpendicular to the membrane normal i.e., in the membrane plane (with order parameter S < 0) in the inner interfacial and outer core regions at 10 < |z| < 20 Å (see Figures S7, S8 for time series). In the inner core region (|z| < 10 Å) the drugs again become aligned or anti-aligned with the z-axis. Interestingly, the orientation of SotN and SotC in the inner interfacial and core regions seem to be opposite—with SotC favoring parallel orientation and SotN𠅊ntiparallel with the membrane normal for the drug positions with the negative z-values (Figure 6). This results from different relative affinities of SotC and SotN functional groups: the cationic ammonium group in SotC strongly attracts water molecules and lipid head groups, whereas its deprotonation makes its sulfonamide functionality a better attractor leading to this functional group re-orientation to be closer to the membrane interface. These interactions lead to hindered rotation (see Figures S7, S8) on the time scale of MD simulations we performed here (10� ns for each drug z) leading to difficulties sampling thermodynamics of drug—membrane interactions discussed below (see Supplemental text for more details).

Figure 6. Analysis of d-sotalol tumbling during umbrella sampling MD simulations. (A) Average polar angle θ distribution for N1 … S d-sotalol vector with respect to the z axis for charged (SotC, blue) and neutral (SotN, red) drug moving across POPC membrane. (B) Corresponding order parameter profiles for this vector with respect to the z axis. Error bars shown in all the graphs are computed from profile asymmetries. See Figures S7, S8 for a few representative θ(N1 … S) time series.

D-Sotalol Energetics and Protonation across the Membrane

We computed free energy profiles for SotC and SotN moving across a POPC membranes based on analysis of drug position fluctuations around restrained z positions in US MD simulations as described above. Those profiles are shown in Figure 7A for both NAMD and CHARMM simulations. For SotN, differences between NAMD and CHARMM free energies are within uncertainties (shown as error bars in Figure 7A), obtained as measures of profile asymmetries (see Figure S9 and Supplemental text). However, for SotC the free energy barrier is ߣ kcal/mol smaller for CHARMM (11.2 ± 1.1 kcal/mol) compared to NAMD (14.4 ± 0.1 kcal/mol). Such free energy decrease along with a flat free energy profile for |z| < 3 Å can be due to P21 point group transformations used in CHARMM simulations. This is also in line with interfacial connections to both bilayer interfaces seen in these simulations (see Figure 1 and discussion above). However, relatively large asymmetries of up to ߢ kcal/mol (Figure S9) preclude us from an unambiguous assignment of this difference.

Figure 7. Analysis of d-sotalol thermodynamics from umbrella sampling MD simulations. (A) Free energy or potential of mean force (PMF) profiles for charged (SotC, blue and cyan) and neutral (SotN, red and orange) d-sotalol moving across a POPC membrane. In CHARMM simulations (cyan for SotC and orange for SotN) P21 symmetry was used. See text for more details. (B) d-sotalol pKa shifts computed from PMFs in (A).

If we compare SotC and SotN free energy profiles shown in Figure 7A, we will see differences such as substantially higher central peak for SotC, e.g., 14.4 vs. 5.4 kcal/mol for SotN from NAMD simulations, as well as presence of a deep interfacial minimum of 𢄢.8 kcal/mol for SotN at |z| = 14 Å, similar to one seen for cationic cisapride (Figure 1 and discussion above). Such minimum indicates a substantial neutral drug accumulation at the water-membrane interface. Interestingly, there is practically no such minimum for SotC, although, a shallow ~-1 kcal/mol trough can be seen on a not-symmetrized PMF profile in Figure S9. The substantial difference in peak heights for SotC and SotN is not unexpected, however, and was also observed for basic amino acid side chains in our previous simulations (Li et al., 2008, 2013). It can be explained by different molecular mechanisms governing SotC and SotN permeation: substantial membrane deformations for the former and nearly complete drug dehydration for the latter (Vorobyov et al., 2010, 2014 Li et al., 2012, 2013). Based on free energy difference between charged and neutral drug forms we can also approximate pKa shift and thus preferred protonation form of a drug across the membrane:

where kB is Boltzmann constant, T�solute temperature and ΔW (z) are position-specific free energies for charged and neutral d-sotalol. Corresponding ΔpKa profiles are shown in Figure 7B and indicate rapid downward ΔpKa shifts soon after the drug gets into contact with membrane. Near the membrane center ΔpKa reaches about 𢄦.5 for NAMD and 𢄤.5 for CHARMM based calculations, with the latter estimate being smaller due to a ߣ kcal/mol smaller SotC free energy barrier discussed above. Qualitatively, both results are similar and indicate rapid drug deprotonation soon after a drug starts moving across a membrane. In fact, considering its first aqueous pKa of 8.3, even getting as close as 20 Å to the membrane center will already lead to drug deprotonation. However, it should be noted that we have not considered a possible role of a zwitterionic d-sotalol form, SotZ, in this equilibrium.

D-Sotalol Water-Membrane Partitioning and Permeations: Connection to Experiments

Next, we need to attempt connecting our findings to experimental observables such as water—membrane partitioning coefficient K and permeability rate P. All the relevant data are summarized in Table 1. There is an experimental estimate for water𠅍imyristoylphosphatidylcholine (DMPC) membrane K′(wat → mem) of 2.50 obtained at 303 K (Redman-Furey and Antinore, 1991). This is an apparent value, which takes into account a pH-dependent fraction of membrane-active drug species at those conditions. However, since we know that only SotN is expected to accumulate in the membrane we can compute an intrinsic K-value at experimental pH = 7.2 using drug aqueous pKa = 8.37 and Henderson-Hasselbach equation to obtain K(wat → mem) = 2.50 * 10 (8.37𢄧.20) = 37.0. And corresponding partitioning free energy is ΔG(wat → mem) = −RT ln K(wat → mem) = 𢄢.17 kcal/mol. These estimates, again, do not take into account a presence of SotZ form in the drug protonation equilibrium, which will likely further increase K-value and decrease corresponding ΔG. Nevertheless, we can compare experimental estimates with values we computed from NAMD US free energy profiles using Equations (3) and (4). Estimated K(wat → mem) and ΔG(wat → mem) values for SotN of 13.4 ± 8.6 and 𢄡.6 ± 0.4 kcal/mol (see also Table 1), respectively, are in good agreement with experiment also considering a different lipid (POPC vs. DMPC) and temperature (310 vs. 303 K) used in simulations and experiment. Estimates from CHARMM simulations (Table S6) are similar, within an error of NAMD values. As expected, SotC does not accumulate in the membrane, with K(wat → mem) and ΔG(wat → mem) of 0.69 ± 0.36 and 0.23 ± 0.0.28 kcal/mol, respectively (Table 1).

Table 1. Water-membrane partitioning and permeability data from umbrella sampling MD simulations for charged (SotC) and neutral (SotN) d-sotalol translocation across a POPC membrane using NAMD.

MD simulations of water-membrane partitioning are a good test of the drug model accuracy, and can predict how much drug accumulates in the membrane compared to bulk water. However, it does not consider the kinetics of drug movement across a membrane, which is also essential for predicting its pharmacology and toxicology. Permeability rates, P, provide corresponding estimates and are measured experimentally using different cell lines such as caco-2 or artificial membrane systems such as PAMPA (Parallel Artificial Membrane Permeability Assay) (Bermejo et al., 2004). Experimental estimates for d-sotalol P are available from a recent study (Liu et al., 2012) with a PAMPA P-value of 3.2 × 10 𢄧 cm/s. A direct comparison between experimental and computed P values is known to be challenging, with many complicating factors precluding direct quantitative assessment of absolute values (Carpenter et al., 2014 Di Meo et al., 2016 Bennion et al., 2017). Nevertheless, we computed P estimates for both SotC and SotN using Equation (8) as was done in our previous study (Vorobyov et al., 2014) based on free energy and diffusion coefficient profiles. The latter, shown in Figure 8, were obtained based on correlation times and mean fluctuations of drug COM in z direction using Equation (5) as was also done previously (Vorobyov et al., 2014). The computed diffusion coefficient profiles indicate a rapid 10-fold drop of diffusion coefficients for both SotC and SotN as drug molecules start interacting with lipid membranes, similar to many previous observations (Carpenter et al., 2014 Vorobyov et al., 2014). Interestingly, diffusion coefficients for SotC and SotN are similar, both in water and in the membrane interior (Figure 8 and Table 1), despite difference in net charge and very different drug—membrane interactions. Computed P-values, presented in Table 1 as log P of 𢄨.57 for SotC, and 𢄤.43 for SotN encompass an experimental estimate of 𢄦.50. Based on those values alone, we cannot comment on accuracy of our prediction, and comparison with values for other drug molecules (desirably, with similar functionalities) as was done in a recent study (Bennion et al., 2017) would be the best. What our computed values indicate though, that a neutral drug is about 4-orders of magnitude more permeable compared to a cationic one, and that both values are within few orders of magnitude of an experimentally observed permeability.

Figure 8. Analysis of d-sotalol diffusion from umbrella sampling MD simulations. Diffusion coefficient profiles are computed as described in the text. Error bars shown are computed from profile asymmetries.

D-Sotalol—Membrane Interactions: Effect of Anionic Lipids

Thus far, we only considered d-sotalol partitioning across a POPC membrane using US MD simulations for a single drug molecule. However, we also tested if lipid membrane composition affects drug—lipid interactions. In fact, cardiomyocyte lipid membrane is known to host multiple lipid types: in addition to dominant zwitterionic phosphatidylcholine and phosphatidylethanolamine, it also has a substantial fraction of anionic lipids—phosphatidylserine, phosphatidylinositol and phosphatidic acid [6�% in human (Post et al., 1995) or 17�% in feline cardiac cells (Leskova and Kryzhanovsky, 2011) based on total phospholipid content]. Anionic lipids are expected to increase membrane binding affinity for cationic drug forms and other cations, as was evidenced by our previous study where we saw increase in the interfacial binding for a positively charged arginine side chain analog, methyl guanidinium, in the presence of an anionic lipid phosphatidylglycerol (Vorobyov and Allen, 2011). The possible effect of anionic lipids on neutral drug binding is less clear and is worth testing as well. Therefore, we performed simulations of both SotC and SotN in lipid membranes containing 15% POPS and 85% POPC, respectively, and compared the results to corresponding drug simulations with pure POPC membranes.

We used 500 or 1000 ns long unbiased MD simulations with multiple (15�) drug molecules initially placed in bulk aqueous solution, corresponding to 繀 mM drug concentration. Most SotN molecules become bound to the lipid membrane within 200 ns for the simulation with pure POPC and around 400 ns with a POPC/POPS mixture (see Figure S11). The equilibrium aqueous concentration of SotN drops to ߨ mM for POPC/POPS and ߥ mM for a POPC only system. For systems containing SotC, most drug molecules remain in aqueous solution throughout the simulations with only ߤ (out of 15) interacting with membrane regardless of the lipid composition (Figure S11). Equilibrated systems are shown in Figure 9C demonstrating substantial membrane binding of SotN but not of SotC. Drug probability distributions from those simulations, computed based on simulation data after equilibration (which was achieved in 200 or 400 ns), are shown in Figure 9A. These data confirm the picture demonstrating substantial interfacial binding for SotN with well-defined probability maxima around |z| = 15 Å for both POPC and POPC/POPS systems. No interfacial binding was detected for systems containing SotC (Figure 9A). In the cationic sotalol system with POPS present, there is a slightly increased accumulation of the drug density in |z| range of 15� Å compared to a system with POPC only. This can be due to expected attraction between anionic lipid head groups of POPS and positively charged SotC moieties. However, the effect is small and is thus unlikely to be physiologically significant in this case.

Figure 9. Analysis of d-sotalol partitioning in the presence of anionic lipids from unbiased MD simulations. (A) Probability density and (B) free energy or potential of mean force (PMF) profiles for charged (SotC) and neutral (SotN) d-sotalol moving across a 100% POPC lipid bilayer (cyan and yellow for SotC and SotN, respectively) or a bilayer composed of an 85% POPC and 15% POPS lipid mixture (green and magenta for SotC and SotN, respectively). (C) Molecular snapshots of equilibrated SotN+POPC, SotC+POPC, SotC+POPC/POPS systems after 500 ns, and SotN+POPC/POPS system after 1000 ns of unbiased MD simulations on the Anton 2 supercomputer. P atoms of POPC and POPS lipids are shown as orange and green balls, respectively. See Figure 1 caption for other molecular representation and coloring information.

The probability distributions shown in Figure 9A can be converted to free energy profiles as ΔG(z) = –kBT ln ρ(z), where ρ is probability density, kB is Boltzmann constant, and T is the absolute temperature (see also analogous Equation 4 above). Those profiles are shown in Figure 9B and are in general agreement with those from US MD simulations shown in Figure 7A previously. As expected, we did not observe SotC located near the membrane center during 500 ns of unbiased MD simulations, and therefore free energy profiles are not defined in this region. However, we observe that the slope of the profile is steeper in the presence of POPS, suggesting a higher translocation barrier and hence slower translocation in this case. SotN molecules were distributed throughout the membrane, and thus we could compute complete free energy profiles including central peaks. Interestingly, there are shallower interfacial binding troughs (by 0.5𠄰.6 kcal/mol at |z| = 14� Å), higher central peak (by ߡ.1 kcal/mol) and thus larger translocation barriers in the presence of POPS, indicating less favorable membrane binding and slower translocation rates for SotN. Upon comparison of SotN free energy profiles from US and unbiased MD simulations, shown in Figure 7A, 9B, respectively, we observed a substantially smaller central free energy peak (by 3.7 kcal/mol) and shallower interfacial binding (by 0.6 kcal/mol) in unbiased simulations. There are several factors which can contribute to such differences, including multiple drug molecules, larger membrane patch, and presence of applied electric field in unbiased MD simulations, all of which can possibly lead to smaller permeation barriers. A detailed elucidation of these and other factors is beyond the scope of this study and will be investigated in our subsequent works.


Characterization of Lipid-Protein Interactions and Lipid-Mediated Modulation of Membrane Protein Function through Molecular Simulation

The cellular membrane constitutes one of the most fundamental compartments of a living cell, where key processes such as selective transport of material and exchange of information between the cell and its environment are mediated by proteins that are closely associated with the membrane. The heterogeneity of lipid composition of biological membranes and the effect of lipid molecules on the structure, dynamics, and function of membrane proteins are now widely recognized. Characterization of these functionally important lipid-protein interactions with experimental techniques is however still prohibitively challenging. Molecular dynamics (MD) simulations offer a powerful complementary approach with sufficient temporal and spatial resolutions to gain atomic-level structural information and energetics on lipid-protein interactions. In this review, we aim to provide a broad survey of MD simulations focusing on exploring lipid-protein interactions and characterizing lipid-modulated protein structure and dynamics that have been successful in providing novel insight into the mechanism of membrane protein function.

Figures

Proteins engage with lipids in…

Proteins engage with lipids in diverse modes, many of which have functional significance.…

Experimental techniques that yield information…

Experimental techniques that yield information on protein-lipid interactions. (A) Electron crystallography showing lipid-mediated…

Representative structures of membrane proteins…

Representative structures of membrane proteins (blue) resolved experimentally with various types of lipids…

Examples of common resolutions/representations used…

Examples of common resolutions/representations used in the investigation of lipid-protein interactions. The upper…

Scope of methods in describing…

Scope of methods in describing the dynamics of chemical and biological processes. Effective…

Early simulations of lipid bilayers.…

Early simulations of lipid bilayers. (A) Snapshot of a united-atom (UA), unsolvated model…

Illustration of a key improvement…

Illustration of a key improvement to simulations of lipid bilayers resulting from changes…

(A) CG representations of common…

(A) CG representations of common lipids in MARTINI, overlaid on the corresponding AA…

Spontaneous binding and insertion of…

Spontaneous binding and insertion of the factor VII GLA domain to anionic membranes…

Methods for assembling proteins in…

Methods for assembling proteins in membranes. Proteins, lipid head groups and lipid tails…

Representative membrane channels covered in…

Representative membrane channels covered in Section 3.1. The channels shown from left to…

Representative conformations ( α ,…

Representative conformations ( α , β , γ , δ , and ε…

PIP 2 molecules access different…

PIP 2 molecules access different regions of the KCNQ2 channel depending on protein…

Membrane partitioning and the facilitated…

Membrane partitioning and the facilitated binding of anesthetics to the modulation sites of…

Free energy landscape of PIP…

Free energy landscape of PIP 2 -Kir2.2 interaction. (A) Replica exchange umbrella sampling…

Lipid exchange between the membrane-exposed…

Lipid exchange between the membrane-exposed pockets of MscS and the bilayer upon gating.…

Bilayer stretch induced TREK-2 conformational…

Bilayer stretch induced TREK-2 conformational change between the two major states. (A) A…

Direct involvement of phospholipids in…

Direct involvement of phospholipids in ion translocation across the membrane, mediated by intimate…

Representative membrane transporters covered in…

Representative membrane transporters covered in Section 3.2. The transporters shown from left to…

Lipid entry into the lumen…

Lipid entry into the lumen of Pgp in its inward-facing and outward-facing states.…

The role of lipids in the H + transfer reactions of the H…

Representative integral membrane receptors covered…

Representative integral membrane receptors covered in Section 3.3. The receptors shown from left…

Lipid-modulated structural dynamics of membrane…

Lipid-modulated structural dynamics of membrane receptors. (A) Normalized probability distribution of cholesterol around…

Proposed mechanism of integrin inside-out…

Proposed mechanism of integrin inside-out activation by talin. The figure illustrates the proposed…

Binding of EGFR kinase to…

Binding of EGFR kinase to the anionic membrane. (A) Electrostatic potential surface of…

Representative peripheral proteins discussed in…

Representative peripheral proteins discussed in Section 4. Ras proteins are key regulators in…

Results of spontaneous bilayer formation…

Results of spontaneous bilayer formation and protein-membrane association from CG simulations of nine…

Spontaneous membrane binding of CYP3A4.…

Spontaneous membrane binding of CYP3A4. (Top) Snapshots taken at different time points in…

Membrane bound conformation of phospholipases,…

Membrane bound conformation of phospholipases, and the critical reactions they catalyze. Membrane binding…

Binding modes of (A) GDP-bound…

Binding modes of (A) GDP-bound and (B) GTP-bound G-domain of H-Ras observed by…

MD simulations revealing membrane curvatures…

MD simulations revealing membrane curvatures induced by the N-BAR domain. (A) Snapshots from…

Membrane budding caused by α…

Membrane budding caused by α synuclein. (A) Top-down view of the spoke starting…

Example of a refined voltage…

Example of a refined voltage sensor (VS)/VsTX1 complex structure, showing a t =…

Model of membrane-mediated binding of…

Model of membrane-mediated binding of ProTx-II to Na + channels. (A) ProTx-II surface…

Representative conformations of A β…

Representative conformations of A β tetramer and tetramer-membrane interactions. The images represent the…

(A) Distribution of lipid-protein distances…

(A) Distribution of lipid-protein distances between lipids with polyunsaturated fatty acid (PUFA) and…

Marburg VP40 undergoing substantial conformational…

Marburg VP40 undergoing substantial conformational rearrangements upon binding to the membrane. (A-C) Snapshots…

Snapshots from a 4 μ…

Snapshots from a 4 μ s CG simulation of the assembly of an…

Special lipids modulating protein structure…

Special lipids modulating protein structure and function. Sphingomyelin, phosphatidylinositol 4,5-bisphosphate (PIP2), phosphatidylglycerol (PG),…

Cholesterol modulation of human β…

Cholesterol modulation of human β 2AR characterized by MD simulations. (A) Cholesterol binding…

Lipid headgroup density profiles around…

Lipid headgroup density profiles around the protein for (A) all phospholipids, (B) PE,…

Putative PS binding sites for…

Putative PS binding sites for coagulation factor X GLA domain (FX-GLA). Ca 2+…

CG simulations describe the diffusion…

CG simulations describe the diffusion of CDL in a mixed POPC/CDL bilayer and…

CDL mediated formation of respiratory…

CDL mediated formation of respiratory supercomplexes. (A) View of the CDL-containing system after…

CDL interaction with the mitochondrial…

CDL interaction with the mitochondrial ADP/ATP carrier (AAC). (A) The time-averaged probability density…

PIP-PH domain interaction examined by…

PIP-PH domain interaction examined by free energy and multiscale methods. (A) Free energy…

Computational modeling of OprH in…

Computational modeling of OprH in an LPS bilayer. Chemical structures of lipid A,…

Membrane thickness profiles near ten…

Membrane thickness profiles near ten different membrane proteins from CG simulations. For each…

Leaflet asymmetry of lipid mobility…

Leaflet asymmetry of lipid mobility near NanC and OmpF. Ratio of diffusion coefficients…


How Enthalpy, Entropy and Gibbs Free Energy are Interrelated ?

In a process carried out at constant volume (e.g., in a sealed tube), the heat content of a system is equal to internal energy (E), as no PV (pressure volume) work is done. But, in a constant pressure process, the system also expends energy in doing PV work.

There­fore, the total heat content of a system at constant pressure is equivalent to the internal energy (E) plus the PV (pressure volume product) energy. This is called as enthalpy and is represented by the symbol H. Thus, enthalpy may be defined by the equation,

In simpler words, enthalpy is the total heat content of a system. It reflects the num­ber and kinds of chemical bonds in the reactants and products. Like internal energy, enthalpy is also a function of state and therefore, it is not pos­sible to quantify the absolute enthalpy. However, a change in enthalpy (∆H) accompany­ing a process can be measured accurately. Thus,

(where p = products r = reactants)

The unit of ∆H is joules/mole (or calories/mole)

The reactions which are accompanied by release of heat energy are called as exother­mic reactions. In such cases, there is negative change in enthalpy (-∆H) from reactants to products. For example, in combustion of glucose to CO2 + H2O, large amount of heat is released. Therefore, this is an exothermic reaction with -∆H. melting of ice into liquid water and its subsequent vaporization into water vapours absorb considerable heat from the surroundings, therefore this is an endothermic reaction with + ∆H.

Entropy (S):

Entropy is a quantitative expression for the randomness or disorder in a system and is represented by the symbol S. Entropy has already been discussed in quite some detail while describing Second Law of Thermodynamics earlier. According to this law, ‘the entropy of the universe tends towards a maximum’.

Any change in entropy or disorder accompanying a process from start to finish, is represented by ∆S. When the products of a reaction are less complex or more disordered than the reactants, the reaction is said to proceed with gain in entropy (+∆S) or vice versa (-∆S). In all spontaneous reactions such as oxidation of glucose or melting of ice, the ∆S is positive.

Gibbs Free Energy (G):

Free energy is the component of the total energy of a system that is available to do work at constant temperature and pressure and is represented by the symbol G. It is called as Gibbs free energy in honour of Josiah Willard Gibbs (1839-1903), an American math­ematician and physical chemist who developed the theory of chemical thermodynamics in 1870s at Yale University and also the concept of free energy.

Since Gibbs free energy is also a thermodynamic quantity, it is not possible to quantify its absolute value. However, a change in Gibbs free energy (∆G) accompanying a process can be measured accurately. The unit of Gibbs free energy is joules/mole (or calories/ mole).

Gibbs free energy (G) can be defined by combining the enthalpy (H), entropy (S), along with the Kelvin temperature (T) as shown in the following equation,

As with enthalpy (H) and entropy (S), we cannot quantify absolute free energy but only differences in free energy (i.e., ∆G), the above equation becomes,

∆G is the quantity that is used to describe whether a process is spontaneous or not. Processes with a negative free energy change (-∆G) are energetically feasible and are capable of occurring spontaneously.

Because the free energy of the products is less than those of reactants, reactions with negative ∆G (< 0) are also known as exergonic reactions or energy yielding reactions. Oxidation of glucose to CO2 + H2O, is an example of exergonic reaction which has negative free energy change (-∆G). Similarly, hydrolysis of ATP mol­ecules is an exergonic reaction.

On the other hand, processes with a positive free energy change (+∆G) are not ener­getically feasible and will not proceed without an input of energy. Reactions with a posi­tive ∆G (i.e., ∆G > 0) are known as endergonic reactions or energy consuming reactions. Conversion of glucose + Pi into glucose-6-phosphate accompanies with a positive free en­ergy change, and thus, is an endergonic reaction.

The syntheses of macromolecules such as proteins and nucleic acids from their simple monomeric components also require input of energy and are therefore, endergonic with +∆G. Synthesis of ATP during oxidative phos­phorylation whose apparent ∆G is as high as 67 kJ mol -1 , is also an endergonic process.

In biological systems, very often thermodynamically un-favourable energy requiring endergonic reactions couple them to other reactions that liberate free energy (exergonic re­actions), so that the overall process is exergonic.

The free energy change (∆G) of a chemical reaction is a function of its displacement from equilibrium. “The farther a reaction is poised away from equilibrium, the more free energy is available as the reaction proceeds towards equilibrium”. When a reaction is at equilibrium, ∆G is zero and no further work can be done.

The magnitude of free energy changes is mostly a function of the particular set of conditions for that reaction. Therefore, free energy changes in chemical reactions are compared under standard reaction conditions. The standard free energy change (∆G°’) represents free energy change of a reaction that occurs at pH 7 and 25°C under conditions when both reactants and products are at unit concentration i.e., 1M.

The actual free energy change (∆G) and standard free energy change (∆G°’) are two different quantities that will not necessarily match each other. The AG°’ is a constant that is characteristic for each specific reaction and is directly related to equilibrium constant (Keq).


Nut used pecan

Mass of the burned pecan 1.3 g
Temperature change of 100mL of water 19 degrees C
Calories required to produce temperature change in 100mL of water 1900 calories
Calories per gram contained in the pecan 1357.1

  1. What is the relationship between matter and energy? The more the matter the more the energy.
  2. What do we call stored energy and where is energy stored in compounds such as glucose? We call it glycogen, and its stored in the bonds.
  3. Discuss what happened to the energy stored in the nut? It was released by the heat.
  4. Why was the mass of the less after burning? The oils in the nut were evaporated.
  5. How do our bodies make use of this process? They break down the glucose to form energy known as glycogen.

Errors could have occurred if all the oils were not all evaporated during the process of burning of the pecan. Also if you didn’t use the correct amount of water this could have caused an inaccurate measurement.

Discussion and Conclusion:

The temperature of the can with 100mL of water in it changed from the energy stored in the pecan. The temperature of the water started out being 22 degrees C and as the pecan burned it released the energy and heated the water to 41 ©. Also the mass of the pecan before it was burned was 1.4g and after burning was .1g. One calorie equals the heat required to change the temperature of 1 gram of water 1degree C. In this experiment, the temperature change was 19degrees C which meant 1900 calories were produced to change the temperature of the water. With the mass of the nut before burning and the amount of calories required to change the temperature gave me the information to find that my pecan had 1357.1 calories in it.


&lambda-Repressor

&lambda-repressor is another fast-folding protein we have studied using molecular dynamics simulations. The &lambda-repressor is more than twice the size of villin headpiece and processes a more complicated native topology, namely a five-helix bundle. Therefore, it is considerably more challenging and computationally intense to simulate than the previous cases (WW domain and villin headpiece). Despite its moderately large size, the experimentally known folding time for some of its mutants is less than 15 microseconds. Simulation of the &lambda-repressor in explicit solvent involves a system containing more than 74,000 atoms, and requires 10-20 microseconds to observe complete folding events.

Enhanced sampling simulation

We focused on a fast-folding mutant of the &lambda-repressor, &lambda-HG, with a folding time of 15 microseconds in temperature-jump experiments. By means of a recently developed tempering method (see Zhang and Ma, J. Chem. Phys. 132:244101 (2010)), we observed reversible folding and unfolding of &lambda-repressor in a 10-microsecond trajectory (see movie at right). The folded state is ranked as the most populated cluster without any prior knowledge of the crystal structure in the subsequent cluster analysis. Moreover, the pathway that leads to complete folding of the protein can be followed based on this cluster analysis. Due to the enhanced sampling method used in the current study, the folding pathways observed may not be the most probable ones. Nevertheless, they represent one of the many physical pathways on the folding landscape. In addition to accelerating the search in conformational space, the enhanced sampling method also covers a broad range of temperatures in the simulation, permitting the calculation of the temperature dependence of certain structural characteristics given enough sampling (see figure below). These results highlight the potential of this enhanced sampling method and the accuracy of the underlying physical model (force field) in studying a relatively large helical protein. The simulations also revealed that the folding of &lambda-repressor is not a simple two-state process as proposed for most fast-folding proteins.

Temperature dependence of the histogram distribution of the radius of gyration and the root-mean-square-deviation.

Constant temperature simulation

Pressure jump simulation

Transition state passage observed in a time window of 2 microseconds. Click to view a folding trajectory.

2 microsecond measured by temperature-jump experiments on near-downhill folding &lambda-repressor mutants. It also agrees with the transition-state passage time observed in slower folding proteins by single-molecule spectroscopy. In summary, our simulations provide the atomic level explanation for pressure-jump induced slow and fast folding pathways. They also shed light on how proteins pass through transition states (see figure below) and how long it takes for the proteins to pass through transition states in the process of folding.

Molecular rearrangements observed in the simulation when the protein passes through transition states in the process of folding. At each time point, the folded residues are colored blue.


Watch the video: AIChE 2020; Discovering Molecules with Selective Membrane Partitioning.. (November 2022).