Understanding how proteins interact with each other is a major goal of biological science and one of its greatest challenges. Whether it is for molecular trafficking, signal transduction, structural support or storage, proteins play a vital role in the function of a cellular membrane . Protein interactions have been studied for many years, both experimentally and theoretically .
Figure 1: Hydrophobic mismatch and spontaneous membrane curvature may lead to proteins arranging themselves with a finite spacing. The foreground displays the physiological schematic, where the red represents the hydrophilic part of the protein and the purple the hydrophobic. The background demonstrates the free energy landscape for the placement of proteins in the membrane as a result of the aforementioned factors .
The fluid mosaic model (Singer and Nicolson, 1972) suggested that a cellular membrane is a fluid structure surrounded by two layers of lipids with free-moving proteins within . This model also states that any long-range interactions that arise from proteins are strictly random . Current theoretical and experimental work suggests that proteins interactions are by no means random. In fact, some proteins are coupled to each other such that the isomerisation of a monomer protein leads to the successful isomerisation of two other monomer proteins within a trimer set [19, 20]. There are many parameters that must be accounted for when analyzing protein interactions; bending stiffness and curvature of the membrane, hydrophobic mismatch, tilt and splay of integral proteins, etc [2-12]. Though there is a great deal of difficulty involved in obtaining concrete and thoroughly comprehensive results, the study of biological membranes is motivated by applications in nanotechnology, bio-inspired materials science, as well as the biomedical field . Trans-membrane proteins and membrane-associated proteins are, for instance, the first to be attacked in many infectious diseases .
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Here I provide a review of theoretical models of protein-protein and protein-ligand interactions within a membrane, supplemented with experimental methods. It is my goal to inspire further research into protein interactions; a field that may greatly contribute to our current understanding of cellular functions.
Inclusions in lipid membranes, such as membrane proteins, cause deformations in the bilayer. In order to minimize the exposure of nonpolar parts of the inclusion to the aqueous environment, the bilayer thickness adjusts to match the thickness of the hydrophobic region of the inclusion ("hydrophobic matching") . Interactions between inclusions may arise from direct interactions between the proteins, such as electrostatic, steric, and van der Waals interactions. Indirect forces include membrane-induced interactions, arising from the perturbation of the bilayer structure . Electrostatic interactions are repulsive between like inclusions and decay exponentially with the distance. Van der Waals interactions are always attractive . MarÄela was the first to study the lipid-mediated interaction through theoretical work on the mean field theory of chain orientational order in lipid membranesÂ . I will, in the following, discuss selected recent papers that I consider to be particularly enlightening and representative of the theoretical work addressing protein interactions in membranes.
Goulian et al. studied the long range, indirect interactions between proteins within a membrane. The authors utilized a Hamiltonian function to create a microscopic model for a homogeneous membrane for three scenarios of varied temperature and protein coupling strength. Each scenario revealed that the potential energy is inversely proportional to the fourth power of L, the protein separation . For asymmetrical proteins at low temperatures, their interactions can be either attractive or repulsive whereas symmetrical, rigid proteins will only have attractive interactions due to the entropic properties of the protein . The long-range forces for proteins with a large separation, L, are stronger than the direct forces, especially local Van der Waal forces, which Goulian et al. suggested may induce aggregation of membrane proteins. Therefore, Goulian et al. concluded that protein interactions dominate all other aforementioned direct forces within a membrane and generalized the force as being only dependant on the protein separation . However, their model only takes into account the bending energy of the membrane and the protein separation. As such, this theory is sufficient in identifying the importance of proteins for energy and force contributions within the membrane, but further research is required to understand the other phenomenological parameters that contribute to the protein's force domination within a membrane.
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Phenomenological models, which employ phenomenological parameters, such as interfacial tension, stretching and bending elastic moduli of the membranes, view membranes as a compressible and stretchable unit undergoing hydrophobic mismatch . A model of such a membrane is illustrated in Figure 2 a), depicting the size of the hydrophobic part of the inclusion, l0, and the equilibrium thickness of the bilayer, h, such that the hydrophobic mismatch h0=(l0-h)/2. One such model assumes that the membrane does not have any random shape alterations. That is, the membrane is symmetrical, as depicted in Figure 2, and that the protein is coupled to the elastic curvature of the membrane. Furthermore, the bilayer thickness is coupled to its density; the monolayer does not swell .
From this, the free energy per amphiphile of the monolayer can then be written as [1, 4, 5]:
The first term is the free energy of a flat monolayer given by , where is the surface tension between the aqueous media and the hydrophobic amphiphile tails, and G(u) a compression-expansion term of the amphiphiles [1, 4]. The thickness of the membrane, u(r), and the area per amphiphile molecule, al(r), are functions of the distance with respect to the inclusion, i.e., u(r) and al(r). Thickness and area are related by an incompressibility condition. The other terms stem from bending of the monolayer indicated by the local monolayer curvature [1, 4]. K is the bending stiffness per molecule so that represents the energy related to bending the leaflet. The second term stems from the spontaneous curvature of the monolayer. We can obtain the spontaneous curvature per molecule from /K [1, 4]
Figure 2: a) Profile of the membrane with inclusions. Hydrophobic matching leads to a distortion of the monolayer leaflets which is characterized by the thickness of the membrane as a function of the distance from the center of the inclusion (u(r)), h is the equilibrium thickness of the membrane, and lo is the size of the hydrophobic region of the inclusion. b) Regular 2D arrangement of membrane inclusions with equilibrium spacing L0. Depending on the chemical structure and membrane composition, lipid bilayers may have positive spontaneous curvature c) or negative spontaneous curvatures d) and e)) .
Using equation (1), the membrane perturbation profile and the membrane-induced interactions between an array of inclusions embedded in a two-dimensional membrane were calculated [1,4,6,7].
Figure 3: Free energy as a function of protein spacing. a) demonstrates the effect of spontaneous membrane curvature (adapted from Aranda-Espinoza, et al.(2006)) and b) shows the effect of protein size (adapted from Lagüe, et al. (2000)) .
Figure 3 a) depicts the free energy as a function of the distance between the inclusions for a chosen set of parameters corresponding to the example sketched in Figure 2 a). In the case of vanishing spontaneous curvature, the global energy minimum is obtained at r = 0, which favours aggregation [1, 4, 6, 7]. A metastable ordered state with a finite separation between the inclusions may exist, separated from the aggregated state by an energy barrier. Aggregation becomes unfavourable for nonzero spontaneous curvature and the energy becomes minimal at a finite spacing, Lo [1, 4, 6, 7]Â. The asymptotic limit r â†’ âˆž represents the total energy gain or loss of the membrane by the incorporation of a single inclusion. Only for negative spontaneous curvature is this process energetically favourable and the inclusion will not be rejected [1, 4, 6, 7]. While the compression-expansion term and the bending term would always favour an unperturbed membrane and aggregation of inclusions, the spontaneous curvature may favour incorporation and regular arrangement of inclusions at a certain distance [1, 4, 6, 7]. It was further observed that the spontaneous curvature of the monolayer determines the shape of the membrane deformation profile. The elastic properties of the membrane, such as compressibility and bending energy, set the perturbation length, i.e., the distance at which the membrane returns to its undistorted equilibrium thickness [1, 4, 6, 7].
In a different approach, Kralchevsky et al. utilized a "squeezing model" to study the theory of protein interactions by examining capillary forces. The squeezing model depicts proteins as cylindrical shapes which induce stretching or shrinking of the surrounding membrane, similar to that of Figure 2 a), b) and c) [8, 9]. This model is different from the previous one because it does not only take into account the elastic properties; it includes the two-dimensional hydrodynamics from floatation and immersion of the proteins within a fluid membrane and uses a pressure tensor. Furthermore, Kralchevsky et al. also worked with spherical membranes in addition to the previous planar model. By taking the change in tensor as a function of position, they can determine the pressure and displacement vector (in x, y and z direction) of the theoretical membrane [8, 9].
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Using parameters of the bacteriorhodopsin protein (almost perfectly cylindrical), the interaction energy, Î”Î© (scaled by the Boltzmann constant, k, and temperature, T), due to lateral capillary force versus protein spacing, L (scaled by rc= 1.5 nm) has been plotted in Figure 4 a) for negative curvature and Figure 4 b) for positive curvature [8, 9]. There is a great deal of discrepancy between the results of Figure 3 and Figure 4. To begin, Figure 4 does not plot the interaction energy between protein separation 0.0 and 1.0. If we assume that the force is zero when the proteins are in contact (L= 0), then we can predict that the energy in Figure 4 should decrease down to its minimum and L0, the energy barrier, would be found somewhere between 0 < L < 1.0. Furthermore, we do not see a local maximum in the energy of Figure 4 before the energy steadies off (when the proteins adopt a finite spacing). This model neglects the dynamic properties of the bacteriorhodopsin to adopt a finite spacing in a lattice; perhaps the floatation and immersion factor within the model produces this discrepancy. We see a similarity in these two methods in that that negative spontaneous curvature is favoured since the interaction energy is lower for negative spontaneous curvature (Figure 4 a)) as opposed to positive spontaneous curvature (Figure 4 b)) for L >1.0. This method is particularly interesting because the capillary force was calculated for a specific protein, bacteriorhodopsin, instead of proteins in general [8,9].
Figure 4: Plot of the capillary interaction energy, Î”Î©, as a function of the interprotein separation, L. Figure 4a) depicts the energy for negative spontaneous curvature while Figure 4 b) is for positive spontaneous curvature. The paramaters from bacteriorhodopsin have been given: rc = 1.5 nm, lo = 3.0 nm; Î»= 2 x lo6 N/m2, Ïƒ0 = 35 mN/m and Bo= -3.2 x l0-11 N [8, 9].
The previous papers have quantitatively and qualitatively described the energy of interactions between integral proteins. I have outlined how the shrinking or compression of the lateral membrane (negative and positive curvature) is induced by the protein interactions. I would like to take a moment discuss how the shape of the membrane as an entire unit is affected by the presence of proteins. Biscari et al. theorised about protein interactions from the spontaneous rigidity of a membrane between two parallel rod-shaped proteins. In a vector representation, I introduce two proteins in a closed membrane and denote Ï• as the angle of the tangent from membrane curve Î³ (contact angle) . Positive Ï• value denotes a protein that bends the membrane inwards whilst a negative Ï• value suggests that the protein bends the membrane outwards. These contact angles introduce a potential energy in the membrane from the bending of the membrane . Specifically, the membrane is bent due to the attractive (Figure 5 a) and c)) or repulsive (Figure 5 b) and d)) interactions between the proteins. By modeling the vesicle as having two proteins, separated by a fixed distance, an Euler-Lagrange equation is used in association with the membrane's potential energy to equate the static mediated force . This static mediated force is equal to the negative change in free energy as the protein separation is altered.
There exist four equilibrium states in this model as depicted in Figure 5 . The first, Figure 5 a), depicts an antipodal equilibrium configuration; a symmetrical model where the spacing between the two proteins is half of the total length of the vesicle such that the lateral tensions of the proteins are equal. In Figure 5 b), the proteins form a contact equilibrium configuration; the protein separation is so small that they act as a single protein. Similar to Figure 5 a), a parallel equilibrium configuration is illustrated in Figure 5 c); Ï•1 and Ï•2 have the same sign and the protein spacing is not exactly half of the vesicle. Finally, Figure 5 d) illustrates a symmetric equilibrium position; Ï•1 and Ï•2 of opposite signs sign and the protein spacing is not exactly half of the vesicle .
Figure 5: Equilibrium Shapes. A) Antipodal equilibrium configuration, b) Contact equilibrium configuration, c) Parallel equilibrium configuration and d) Asymmetric equilibrium position .
To summarize the results of their research, they conclude that protein interactions within this model are repulsive if both Ï•1 and Ï•2 are positive; the protein interaction is attractive if Î³1 and Î³2 are of opposite sign and their sums add to a positive value . Furthermore, they concluded that there is no distinction between long range and short range forces for two proteins within a closed geometry (such as that of Figure 5) . This theoretical paper is significantly different from those previously mentioned. Biscari et al. model proteins within a closed geometry, not a planar membrane. Therefore, they did not include the lateral forces between proteins in one direction. Instead, they were concerned about how a rather circular membrane would be affected by the inclusion of two proteins . Though their work adequately describes how a membrane changes its shape, and subsequently the potential energy, due to the interaction of integral proteins, it is a rather unrealistic model. A membrane would consist of more than two proteins, and even so, the interactions from neighbouring proteins from other membranes are neglected, not to mention the numerous phenomenological parameters.
Though it would be quite difficult, a comprehensive analysis on protein interactions which, as much as I have been able to gather, includes phenomenological parameters, direct and indirect forces, capillary forces, elastic and inelastic properties and defined variables for specific proteins within a membrane of closed geometry should be investigated. It has been the goal of this paper to provide theoretical models that provide insight into protein interactions. By incorporating these concepts into a study, one could achieve a comprehensive understanding of protein-protein interactions.
Nevertheless, there is a distinct similarity between some of the papers that model proteins within a planar structure. As seen in Figure 3, distance Lo is the protein spacing at which the local minimum in interaction energy occurs. This is the distance at which protein aggregation occurs. This is not specifically implied by the work of Kralchevksy et al. but the floatation and immersion properties of the proteins may account for this. Figure 3 and Figure 5 both illustrate a metastable state (where the energy becomes constant) at which the proteins adopt a finite spacing. Furthermore, the attractive and repulsive properties of proteins have been discussed in terms of spacing and membrane curvature.
Computer Supported Theories
The aforementioned theoretical determine the energy of protein interactions from the stretching and curvature properties of the surrounding membrane among other properties. To contribute further to this approach, I want to describe the director model  by introducing a director field, , as depicted in Figure 2 a).
The director field allows for a three dimensional analysis of the membrane and incorporates the chain stretching , the splay energy the tilt energy of the monomer and the twist of the lipid monomers to define the free energy of the monolayer as :
The director model is used to compute the director field and the other model constants through Euler equations and boundary conditions. Bohinic et al. concludes that the free energy of the bilayer depends on a function of hydrophobic mismatch . There is a strong attraction between proteins for a negative hydrophobic mismatch (membrane curves inwards). Subsequently, the protein interaction is weak for a positive hydrophobic mismatch (membrane curves outwards) . This is in agreement with the theoretical models, where the protein attraction is greater for a negative spontaneous curvature as opposed to the positive spontaneous curvature. Computing the director field, a protein spacing of Lo= 10 Å (with defined parameters h0=14 Å, k= 10kBT, K= 0.2kBT/Å2, K'= 5kBT, and Kt =0.1kBT/Å2) was determined as the point separating an attractive from a repulsive force. Any spacing less than this Lo would suggest protein aggregation whereas proteins adopt a finite spacing for distances greater than Lo. In comparison, this model is more comprehensive than that by Pincus et al. from Equation 1 in that it incorporates the tilt and splay of the integral proteins.
The results of the director model are similar to that of the chain packing [11, 12]. May et al. use the chain packing model to quantify the free elastic energy of a molecular layer as a result of protein interaction and protein spacing. This model assumes that the membrane is perfectly planar and incompressible. From this, the densities of the lipid headgroups in the membrane are converted into the probability of finding rigid, cylindrical proteins in a certain conformation (tilt, position, etc) . This probability is integrated with the initial conditions of the system as well as a Boltzmann constant to determine the entropy and energetic contributions of the system. The free energy, F, of the monolayer is calculated as F= E-TS, where T is the temperature of the system, S is the computed entropy from the lipid headgroups and E is the computed energetic contribution . The entropy is computed as the loss from the impenetrability of the inclusion wall . The calculated free energy of the chain packing model and director model as a function of protein spacing agree with the theoretical models from Figure 3. It is interesting to note that this model is significantly different than those previously mentioned, yet it produced similar results. May et al. approach proteins in terms of probability of position and do not take in account the properties of the membrane. As such, the calculated energy of protein interactions is a direct result of the proteins themselves.
Similar results are obtained by statistical mechanical integral equation theory [13-15]. Lague's model used lipid bilayer membranes (LBMs) made of dipalmitoyl phosphatidylcholine (DPPC) molecules to study non-specific lipid-mediated protein-protein interactions in a pure lipid bilayer [13-15]. The model combines mean-field theory and results from an atomic simulation (Pratt and Chandler, 1977) [13-15]. This model requires the input of a lateral density-density response function of the hydrocarbon core (obtained from molecular dynamic simulations of protein-free lipid bilayer) to produce the perturbed density of hydrocarbon chains around protein inclusion and lipid-mediated potential of mean force (PMF) between two proteins [13-15]. The results follow the same trend as in Figure 3 with the exception of minor fluctuations. The theory was also applied to other phospholipids, such as POPC, DMPC and DOPC, to investigate the effect of lipid diversity. Qualitatively the same results as for DPPC were reported [1, 13-15].
Furthermore, Monte Carlo simulations and mean-field calculations (as a guide for determining model parameters) to study protein interactions ,the phase equilibria and the aggregation of small integral membrane proteins, in dipalmitoyl phosphatidylcholine bilayers . This work suggests that protein concentration, temperature, lipid-protein interactions and direct protein-protein interactions control the lateral distribution of proteins in the lipid membrane plane . In addition, van der Waal forces contribute to the strength of the protein-lipid interactions, which in turn provides the strength for protein aggregation .
Coupled molecular motions can also be modeled using molecular dynamics computer simulations, and inter-protein motions in a carboxymyoglobin protein crystal were reported [1, 17]. Here, collective motions between the proteins were found and the phonon spectrum of the excitations was determined .
Computer simulated theories have several advantages over analytic theories. First, we are able to view protein interactions on a microscopic level (tilt and splay). As such, we are able to account for more variables within the study. The calculated results appear to be more accurate; Lague et al. obtained minor fluctuations in their calculated energies and were able to make calculations for specific proteins. Furthermore, computer simulated models are successful in determining specifically how the proteins contribute an interaction energy, not just from the hydrophobic mismatch of the membrane but from their location and position. The computer simulated and analytic theories do not conflict; they produce results that generally agree with Figure 3.
Protein-protein interactions can be experimentally observed and quantified by examining the atomic and molecular motions in the membranes . While emphasis has been on high-throughput screening techniques, such as mass spectroscopy, modern techniques are also capable of directly accessing molecular interactions in biological or biomimetic systems . Atomic and molecular motions in membranes and proteins can be classified as local, self-correlated, and collective, pair-correlated dynamics . In biology, any dynamics will most likely show a mixed behaviour of particles moving in local potentials but with a more or less pronounced coherent character. Very few approaches and techniques are capable of directly accessing collective molecular motions and molecular interactions because motions in lipid bilayers, for instance, range from long wavelength undulation and bending modes, with typical relaxation times on the order of nanoseconds and lateral length scales of several hundred lipid molecules (i.e. tens of nanometers), to short wavelength density fluctuations in the picosecond range and nearest neighbour length scales [1, 22-34]. Inelastic X-ray can directly access collective dynamics in membranes and proteins by measuring the corresponding spectrum. Phonon-like excitations of proteins in hydrated protein powder were reported from X-ray scattering experiments using synchrotron X-ray radiation [1,18]. Figure 6 depicts different experimental methods and the time, scattering vector, length and energy at which they are used .
Figure 6: Length and time scales, and corresponding energy and momentum transfers, for spectroscopic techniques covering a range of dynamics from the microscopic to the macroscopic. Light scattering techniques include Raman, Brillouin, and dynamic light scattering (DLS). Inelastic X-ray and neutron scattering access dynamics on Å and nm length scales. Dielectric spectroscopy probes the length scale of an elementary molecular electric dipole, which can be estimated through the C-O bond length (~ 140 picometer). High speed AFM has a spatial resolution of Çº to nm. The area enclosedin the dashed line box is the dynamical range accessible by computer simulations .
Rheinstädter et al. subjected bacteriorhodopsin, a light activated proton pump made of 7-transmembrane Î±-helices, in purple membranes (PM) to coherent inelastic neutron scattering to find evidence of long-range protein interactions and protein coupling for inter-protein communication . The magnitude of the coupling between proteins was determined by measuring the spectrum of the acoustic phonons in the 2D protein lattice . They utilized an IN12 cold-triple-axis spectrometer at the Institut Laue-Langevin in Grenoble, France which allowed for the simultaneous measurements of diffraction and inelastic scattering . A vacuum box prevented air scattering while the spectrometer was calibrated to kf= 1.25 Å-1, Î”q= 0.005 Å-1 and Î”Ä§Ï‰= 25 ÂµeV. 200 mg of deuterated (H2O removed) PM was centrifuged with water and spread to a 40 x 30 mm aluminum holder where it was dried to a ratio of 0.5 g of water/g of membrane . The sample was placed over a silica gel in a desiccator to dry. The gel was replaced by water such that lamellar spacing dz= 65 Å at 303 K . The model is depicted in Figure 7 a). The basic hexagonal translations are indicated by arrows. The interaction between protein trimers is contained within the springs with an effective (longitudinal) spring constant k [1, 19]. The calculated longitudinal spectrum Cl(q, Ï‰) is defined by Cl(q, Ï‰) = (Ï‰ 2/q2)S(q, Ï‰), and is shown in Figure 7 b) [1, 19]. The statistical average in the plane of the membrane leads to a superposition of the different phonon branches which start and end at the hexagonal Bragg peaks (at Ä§Ï‰ = 0). The data points are the excitation positions as determined from the inelastic neutron triple-axis experiment .
Figure 7: a) BR trimers are arranged on a hexagonal lattice of lattice constant a = 62 Å. The interaction between the protein trimers is depicted as springs with effective spring constant k. b) The calculated excitation spectrum Cl(q, Ï‰) in the range of the experimental data. Data points mark the positions of excitations. The horizontal line at Ä§Ï‰ =0:45 meV marks the position of a possible optical phonon mode, not included in the calculations. (adapted from Rheinstädter et al., 2009).
Rheinstädter et al. calculated the protein-protein spring constant k as approximately 54 N/m. The amplitude of this mode of vibration can be estimated from the equipartition theorem to
= 0.1 Å, and the interaction force between two neighbouring trimers to
= 0.5 nN [1, 19]. To put this into perspective, the protein interaction force falls between the weaker van der Waals forces and the stronger C-C bonds in graphite [1, 19]. Therefore, the protein-protein forces are significant and are possibly responsible for diffusive motions within membranes [1, 19]. To establish a clear relationship with protein function, protein dynamics of activated proteins (such as those activated in the photo cycle) will be studied using recently developed laser-neutron pump-probe experiments [1, 19, 21].
Additional experimental work on protein trimers of bacteriorhodopsin in purple membranes was conducted by VoÃ¯tchovsky et al. using atomic force microscopy (AFM) . They constructed a home-built high speed atomic force microscopy device to simultaneously study the monomeric, trimeric and membrane levels of bacteriorhodopsin in purple membranes of Halobacterium salinarum . Their device has approximately a 5 Å spatial resolution capable of recording 50 images/s; it has resonance frequency of approximately 600 kHz and a spring constant 0.2N m-1 . The bacteriorhodopsin aggregated as it were in Rheinstädter et al., Figure 7 a). A green laser was shot at the protein trimers to induce isomerisation through a photocycle. The excitations occurred near a cantilever that made dynamic measurements as seen in Figure 8 a) .
Figure 8: a) Experimental setup of the high speed AFM experiment. b) Conformation change of the bacteriorhodopsin protein during the photocyles. The photo induced tilt of helicies f and g opens a channel in the cytoplasmic part of the protein. c) High speed AFM image of the PM surface in the activated state after photoisomerization of the proteins. d) Dark state image of the PM surface. (Adapted from VoÃ¯tchovsky et al.) .
When one monomer within the trimer isomerised (became excited), the other two monomers successfully isomerised within 50 ms of each other . Figure 8 c) illustrates the conformational change that occurred upon excitation in comparison to a dark state in Figure 8 d). Furthermore, there were no simultaneous excitations; only one monomer was excited at any given time within the trimer set. Figure 9 depicts the excitations (a)) and relaxation time (b)) during the experiment . As such, the protein trimer acted in a cooperative dynamic. As Figure 8 c) depicts, their conformational change occurred from the movement of the excited monomer; a coherent movement [1, 20]. The relaxation for the proteins to return to their initial positions was determined to be Ï„Â =Â 120Â ms (see Figure 9 b)) [1, 20]. This process was associated with the membrane's elastic properties and the work necessary for the photo-induced tilt of the helices inside of a protein was estimated to WÂ =Â 5.4Â·10-20 J and the force constant of the monomer-monomer interaction can be estimated from the AFM results to N/m [1, 20].
Figure 9: a) excitation of the monomers within a trimer set. b) relaxation time after successful isomerisation .
Clearly, both the inelastic neutron scattering and high-speed AFM experimental methods provide evidence for protein cooperation. It is interesting, and coincidental, that both experiments were performed on bacteriorhodopsin in purple membranes around the same time. In comparison, the inelastic neutron scattering method yielded a force constant of k =54 N/m between trimers whereas the high-speed AFM estimated the constant to be kM= 0.2 N/m [1, 19, 20]. This discrepancy is possibly due to the time scales at which measurements were made. As depicted in Figure 6, neutron scattering measurements occur in units of picoseconds to nanoseconds whereas AFM measures in milliseconds . As such, I find that the neutron scattering methods is more accurate in obtaining protein-interaction measurements, but the AFM method is capable of visualizing the actual interactions that occur. Furthermore, the elastic behaviour and properties of the PM membrane and also the proteins may strongly depend on the time scale at which they are observed. The membrane may appear much stiffer when studied at high frequencies . It can be speculated that molecular reorientations, which occur on pico-microsecond time scales, may relax, also due to diffusion, between two AFM images and therefore lead to different, softer elastic constants . This finding may also be relevant if one wants to compare the experimental findings to the previously discussed theories .
Discussion and Conclusive Remarks
I have presented several theoretical papers that address microscopic, phenomenological and computer based approaches for studying protein interactions. The membrane`s bending, stretching, etc, properties play a vital role in understanding how the proteins orient themselves within the membrane. Protein aggregation is an important concept that can potentially be beneficial to medical sciences for drug production. Proteins aggregate when their separation has overcome any direct or indirect local forces; denoted position Lo. Above this distance, proteins have been theorized to position themselves at a finite spacing to supplement their given energy level.
It is difficult to adequately experiment on protein interactions because there are many properties that must be accounted for. Nonetheless, recent work by Rheinstädter et al. and VoÃ¯tchovsky et al. has been successful in documenting evidence for protein-protein interactions. They have both calculated the coupling factor between monomer proteins in a trimer set for bacteriorhodopsin in purple membranes on different time-scales. Furthermore, VoÃ¯tchovsky et al. have illustrated the direct interaction between these proteins by excitation (isomerisation) through their atomic force microscopy device. There is a limited amount of research papers available to discuss the techniques involved in quantifying and identifying protein-protein interactions. This supports the purpose of my research paper; we must further the investigations in this field because a fundamental understanding of protein interactions can enhance our knowledge on biological systems, especially in the fields of medicine.
I would like to take this time to thank Dr. Maikel Rheinstädter and Clare Armstrong for their continuous support and guidance this year. Your insightful meetings and e-mails have been greatly appreciated. Thank you for all the time and effort that went into my research and editing. I hope that our possible publication furthers your respective careers and research in biophysics.