├── pygbe2-CPC ├── figs │ ├── .DS_Store │ ├── Figure1.pdf │ ├── Figure10.pdf │ ├── Figure2.pdf │ ├── Figure3.pdf │ ├── Figure4.pdf │ ├── Figure5.pdf │ ├── Figure7.pdf │ ├── Figure8.pdf │ ├── Figure9.pdf │ ├── 1PGB_time.pdf │ ├── iterations.pdf │ ├── total_time.pdf │ ├── 1PGB_iterations.pdf │ ├── convergence_isolated.pdf │ └── molecule_surface_stern.pdf ├── conclusion.tex ├── model.tex ├── introduction.tex ├── discussion.tex ├── energy.tex ├── methods.tex ├── CooperBarba2014.tex ├── CooperBarba2014.bbl ├── results.tex ├── bem.tex ├── analytical_solution.tex └── elsarticle.cls ├── pygbe-orientation ├── figs │ ├── supp_1.pdf │ ├── supp_2.pdf │ ├── supp_3.pdf │ ├── Figure1.pdf │ ├── Figure10a.pdf │ ├── Figure10b.pdf │ ├── Figure2.pdf │ ├── Figure3.pdf │ ├── Figure4.pdf │ ├── Figure5a.pdf │ ├── Figure5b.pdf │ ├── Figure5c.pdf │ ├── Figure5d.pdf │ ├── Figure6a.pdf │ ├── Figure6b.pdf │ ├── Figure7.pdf │ ├── Figure8a.pdf │ ├── Figure8b.pdf │ ├── Figure8c.pdf │ ├── Figure8d.pdf │ ├── Figure8e.pdf │ ├── Figure8f.pdf │ ├── Figure8g.pdf │ ├── Figure8h.pdf │ ├── Figure9a.pdf │ ├── Figure9b.pdf │ ├── Figure9c.pdf │ ├── Figure9d.pdf │ ├── Figure9e.pdf │ ├── Figure9f.pdf │ ├── Figure9g.pdf │ └── Figure9h.pdf ├── supplementary_materialNotes.bib ├── energy.tex ├── methods.tex ├── conclusion.tex ├── model.tex ├── prot_orientation.tex ├── bem.tex ├── introduction.tex ├── supplementary_material.tex ├── CooperBarba-orientation.tex ├── results.tex └── discussion.tex ├── .gitignore ├── coverletter-orientation.md ├── significance-orientation.md └── README.md /pygbe2-CPC/figs/.DS_Store: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/barbagroup/pygbe-papers/master/pygbe2-CPC/figs/.DS_Store -------------------------------------------------------------------------------- /pygbe2-CPC/figs/Figure1.pdf: -------------------------------------------------------------------------------- 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https://raw.githubusercontent.com/barbagroup/pygbe-papers/master/pygbe2-CPC/figs/molecule_surface_stern.pdf -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | */*.aux 2 | */*.log 3 | */*.out 4 | */*.blg 5 | 6 | pygbe-orientation/CooperBarba-orientationNotes.bib 7 | 8 | pygbe-orientation/CooperBarba-orientation.pdf 9 | 10 | pygbe2-CPC/CooperBarba2014.pdf 11 | 12 | pygbe2-CPC/CooperBarba2014.spl 13 | 14 | *.aux 15 | 16 | *.log 17 | 18 | *.out 19 | 20 | *.gz 21 | 22 | pygbe-orientation/supplementary_material.pdf 23 | 24 | pygbe-orientation/supplementary_material.pdf 25 | -------------------------------------------------------------------------------- /coverletter-orientation.md: -------------------------------------------------------------------------------- 1 | Dear Editor, 2 | 3 | We submit for consideration in the journal our original research article titled 4 | “Probing protein orientation near charged nanosurfaces for simulation-assisted biosensor design,” 5 | intended for the topical category Surfaces; Interfaces; and Materials. 6 | 7 | This study increases our understanding of how nanosurface properties (charge) and preparation conditions (salt levels) affect protein orientation. We also show that our modeling framework can match published results obtained with other methods, both experimental and direct molecular simulation with smaller molecules. 8 | 9 | Favorable orientation of ligand molecules on bioactive surfaces is critical in biosensor sensitivity and performance. But studies so far have not resolved the question of how orientation can be influenced by surface preparation. 10 | 11 | The article presents a physical adsorption study of immunoglobulin G using simulations of electrostatic interactions between the molecule and charged surfaces. The goal is to determine most-probable orientation of molecules on a nanosurface, as a function of the charge and solvent ionic content. Our results find  conditions for high-probability favorable orientation for the antibody iso-type IgG2a, which had not been found to have a preferred orientation in previous studies. 12 | 13 | The research presented in our submission is new and original, and has not been 14 | submitted anywhere else. A preprint is deposited on the arXiv repository with ID 1503.08150v3). 15 | -------------------------------------------------------------------------------- /pygbe2-CPC/conclusion.tex: -------------------------------------------------------------------------------- 1 | %!TEX root = CooperBarba2014.tex 2 | 3 | In this work, we used an implicit-solvent model to study protein-surface interaction. We present for the first time and apply an extension of our open-source \pygbe code to account for the presence of surfaces with imposed potential or charge. The new feature of the code was verified against an analytical solution, which we derived for that purpose. 4 | 5 | To demonstrate the power of this approach in a more realistic setting, we performed tests of protein G B1 D4$^\prime$ near a brick-shaped surface with an imposed charge. The error in energy scaling with the area of boundary elements demonstrates that this extension of \pygbe is capable of resolving the mathematical model correctly. This test was motivated by the biosensing application, where a ligand molecule is adsorbed on a \sam-coated nanoparticle, which can be represented by the brick-shaped surface. 6 | 7 | The addition of a surface with imposed charge or potential in the implicit-solvent model falls naturally in a boundary integral approach. In this case, the region enclosed by the surface is not part of the domain, then, this surface only adds one equation to the linear system, rather than two, which is the case with the molecular solvent-excluded surface. 8 | 9 | We conclude that this implicit-solvent model can offer a valuable approach in protein-surface interaction studies. This tool can be useful for orientation studies of ligand molecules in biosensors, either to find optimal adsorption conditions of salt concentration and surface charge, or to guide the design of better ligand molecules. 10 | -------------------------------------------------------------------------------- /pygbe-orientation/energy.tex: -------------------------------------------------------------------------------- 1 | %!TEX root = CooperBarba-orientation.tex 2 | 3 | We can decompose the total free energy into Coulombic, surface, and solvation energy: 4 | 5 | \begin{equation} 6 | F_\text{Total} = F_\text{Coulomb} + F_\text{surf} + F_\text{solv}. 7 | \end{equation} 8 | 9 | \medskip 10 | 11 | \paragraph*{Coulombic energy---} 12 | 13 | The Coulombic energy arises simply from the Coulomb interactions of all point charges. We compute it by 14 | 15 | \begin{equation} \label{eq:coul_energy} 16 | F_\text{Coulomb} = \frac{1}{2} \sum_j^{N_q}\sum^{N_q}_{\substack{i\\ i\neq j}} q_iq_j\frac{1}{4\pi |\mathbf{r}_i-\mathbf{r}_j|} 17 | \end{equation} 18 | 19 | \paragraph*{Solvation free energy---} 20 | 21 | The solvation energy is the energy contribution of the protein's surroundings: solvent polarization, charged surfaces, and other proteins. We compute it as 22 | 23 | \begin{align} \label{eq:solv_energy} 24 | F_{\text{solv}} &= \frac{1}{2} \int_{\Omega} \rho \,(\phi_{\text{total}} - \phi_{\text{Coulomb}}) \\ 25 | &= \sum_{k=0}^{N_q} q_k (\phi_{\text{total}} - \phi_{\text{Coulomb}})(\mathbf{r}_k), 26 | \end{align} 27 | 28 | \noindent where $\rho$ is the charge distribution, consisting of point charges (which transforms the integral into a sum), and $\phi_\text{reac} = \phi_{\text{total}} - \phi_{\text{Coulomb}}$ is 29 | % 30 | \begin{equation} \label{eq:phi_reac_bem} 31 | \phi_{\text{reac},\mathbf{r}_k} = -K_{L}^{\mathbf{r}_k}(\phi_{1,\Gamma_1}) + V_{L}^{\mathbf{r}_k} \left(\frac{\partial}{\partial \mathbf{n}}\phi_{1,\Gamma_1} \right) 32 | \end{equation} 33 | 34 | \paragraph*{Surface free energy---} 35 | 36 | We use the description of free energy of a surface with prescribed charge (like $\Gamma_2$ in Figure \ref{fig:molecule_surface}) from Chan and co-workers.\cite{ChanMitchell1983,CarnieChan1993} They describe the free energy on a surface as 37 | 38 | \begin{equation} \label{eq:energy_surf} 39 | F_\text{surf} = \frac{1}{2} \int_{\Gamma} G_c \sigma_0^2 d\Gamma, 40 | \end{equation} 41 | 42 | \noindent where $\phi = G_c \sigma_0$. 43 | -------------------------------------------------------------------------------- /significance-orientation.md: -------------------------------------------------------------------------------- 1 | The Journal of Chemical Physics publishes the most significant experimental, computational, and theoretical 2 | results across the full breadth of modern research in chemical physics. A manuscript will be considered for 3 | publication in the Journal only if it meets at least one of the following criteria: 4 | 5 | * novel research that makes a significant advance in improving scientific understanding 6 | in a modern area of chemical physics; 7 | * transformative theoretical methods or advanced experimental techniques for research in chemical physics; 8 | * chemical physics research that takes critical steps toward applications of scientific or technological relevance; 9 | * interdisciplinary scientific advances that generate new avenues of research related to chemical physics; 10 | * initial reports of highly significant research of broad interest to the chemical physics community through rapid Communications. 11 | 12 | Authors are asked to provide a brief explanation below indicating how their manuscripts fulfills one or more of the above criteria for acceptance. 13 | 14 | **Statement of manuscript significance** 15 | 16 | This paper's application of technological relevance is in the field of biosensors, 17 | and the results are instrumental for simulation-assisted biosensor design. 18 | Favorable orientation of ligand molecules on bioactive surfaces is critical 19 | in biosensor sensitivity and performance. But studies so far have not resolved 20 | the question of how orientation can be influenced by surface preparation. 21 | This work considers the effect of electrostatic adsorption, increasing understanding 22 | of how nanosurface properties (charge) and preparation conditions (salt levels) 23 | affect protein orientation. Our results find conditions for high-probability 24 | favorable orientation for the antibody immunoglobulin G iso-type IgG2a, which 25 | was not found to have a preferred orientation in previous studies. The results 26 | also show that local interactions dominate over dipole moment for this protein. 27 | Improving immunoassay sensitivity may thus be assisted by numerical studies using 28 | our method, guiding changes to fabrication protocols or protein engineering of 29 | ligand molecules to obtain more favorable orientations. 30 | -------------------------------------------------------------------------------- /pygbe-orientation/methods.tex: -------------------------------------------------------------------------------- 1 | %!TEX root = CooperBarba-orientation.tex 2 | 3 | \subsection{Discretization and implementation details} 4 | 5 | We solve the system in \eqref{eq:integral_eq} numerically using a boundary element method (\bem). To represent the \ses, we use flat triangular panels where the potential $(\phi)$ and its normal derivative $(\partial \phi /\partial \mathbf{n})$ are constant, and then collocate the discretized equation on the center of each panel. This transforms the integral operators in the matrix equation \eqref{eq:matrix_dphi} into block matrices of size $N_p \times N_p$, where $N_p$ is the number of panels. Each entry of the block matrix is an integral over one panel ($\Gamma_j$), evaluated on the center of panel $\Gamma_i$: 6 | % 7 | \begin{align} \label{eq:layers_element} 8 | K_{L,ij} &= \int_{\Gamma_j} \frac{\partial}{\partial \mathbf{n}} \left[ G_L(\mathbf{r}_{\Gamma_i},\mathbf{r}_{\Gamma_j}) \right]\mathrm{d} \Gamma_j, \nonumber \\ 9 | V_{L,ij} &= \int_{\Gamma_j} G_L(\mathbf{r}_{\Gamma_i},\mathbf{r}_{\Gamma_j}) \mathrm{d} \Gamma_j. 10 | \end{align} 11 | 12 | We classify the integrals in Equation \eqref{eq:layers_element} in three groups, depending on the distance $d$ between the panel and the collocation point. 13 | When the collocation point is inside the panel being integrated, we get a singular integral that we solve with a semi-analytical approach\cite{ZhuHuangSongWhite2001} placing Gauss nodes on the sides of the triangle. 14 | We call near-singular integrals those where $d<2L$ ($L = \sqrt{2\cdot \text{Area}}$). For near-singular integrals, we use a high-order Gauss quadrature rule with 19 or more nodes. 15 | Finally, when the panel and collocation points are further than $2L$ from each other, we only need 1, 3, 4 or 7 Gauss nodes per element to get good accuracy. 16 | 17 | To solve the resulting linear system, we use a general minimal residual method (\gmres). The most time consuming part of the \gmres solver is a matrix-vector multiplication---in principle, an $O(N^2)$ operation---done within every iteration of the solver. But by using a treecode algorithm, we perform this operation in $O(N\log N)$ time.\cite{BarnesHut1986} More details on our implementation of the \bem can be found in our earlier work,\cite{CooperBarba-share154331} and a companion paper.\cite{CooperBarba2015a} 18 | -------------------------------------------------------------------------------- /pygbe-orientation/conclusion.tex: -------------------------------------------------------------------------------- 1 | %!TEX root = CooperBarba-orientation.tex 2 | 3 | Various studies have revealed the importance of protein orientation in immunoassays. One work suggested that highly oriented antibodies could result in 100$\times$ improvement in the affinity of a biosensor.\cite{TajimaTakaiIshihara2011} Thus, a design goal would be to know how to prepare a surface to control protein orientation. Yet, despite much work, control of protein orientation has not been successful. 4 | This study increases our understanding of how nanosurface properties (charge) and preparation conditions (salt levels) affect protein orientation. 5 | We successfully used an implicit-solvent model to study protein orientation near charged surfaces, which in our method can have any geometry. In a companion publication,\cite{CooperBarba2015a} we describe expanding the applicability of our open-source code, \pygbe, to account for the presence of charged surfaces and present grid-convergence studies using an analytical solution and protein \gb. 6 | 7 | Protein \gb behaves like a point dipole near a charged surface, with the dipole-moment vector shifting $\sim$180$^\circ$ when the sign of the surface charge flips. Our results compare well with experimental observations and simulations using combined Monte Carlo and molecular dynamics methods, supporting the use of our approach for probing protein orientation near charged surfaces. 8 | We applied our approach to immunoglobulin G, a biomolecule that is much larger than protein \gb (about $125\times$, by volume) and would be challenging to study via molecular dynamics. 9 | The iso-type \ig 2a was found by previous studies to be hard to control, exhibiting many orientations, but we are able to obtain a preferred orientation that is favorable for biosensing with a positive surface of 0.05C/m$^{2}$ or higher d 37mM of salt in the solvent. We conclude that local electrostatic interactions dominate over the dipole moment, and even this protein can be favorably oriented with the appropriate fabrication protocol. Potentially, protein engineering could be used to obtain ligand molecules that interact with charged surfaces in a desired fashion. 10 | In this application, where ligand molecules undergo little conformational change as they adsorb on the sensor surface, our new implicit-solvent model can offer a valuable approach to assist in biosensor design. In our future work, and in collaboration with experimental researchers, we intend to use this approach to aid the design of better ligand molecules, by looking at the preferred orientations for different ligand mutants. 11 | -------------------------------------------------------------------------------- /pygbe-orientation/model.tex: -------------------------------------------------------------------------------- 1 | %!TEX root = CooperBarba-orientation.tex 2 | 3 | The implicit-solvent model describes a molecular system as a set of continuum dielectric regions, and computes the mean-field potential using electrostatics. 4 | For the case where a protein is dissolved in a solvent, we require two of such regions: inside and outside the protein, interfaced by the solvent-excluded surface (\ses). 5 | The \ses determines the closest a water molecule can get to the protein, and we generate it by rolling a spherical probe of the size of a water molecule around the protein. 6 | The dielectric constant inside the protein is low ($\epsilon= 2\text{ to }4$) and there are point charges placed at the atomic locations. The solvent region has the dielectric constant of water $\epsilon \approx 80$, and we need to account for the presence of salt. 7 | This model results in a system of partial differential equations where the Poisson equation describes the electrostatic potential inside the protein, and the linearized Poisson-Boltzmann equation applies outside the protein. On the \ses, appropriate interface conditions ensure the continuity of the potential and electric displacement. 8 | 9 | 10 | \begin{figure}[h] 11 | \includegraphics[width=0.45\textwidth]{Figure1.pdf} 12 | \caption{Sketch of a molecule interacting with a surface: $\Omega_1$ is the protein, $\Omega_2$ the solvent region, $\Gamma_1$ is the \ses and $\Gamma_2$ a surface with imposed charge.} 13 | \label{fig:molecule_surface} 14 | \end{figure} 15 | 16 | In this work, we use an extension of the implicit-solvent model to consider the effect of charged surfaces. Such is the case of the setup sketched by Figure \ref{fig:molecule_surface}, which is described mathematically by the following equations: 17 | 18 | 19 | \begin{align} \label{eq:pde} 20 | \nabla^2 \phi_1(\mathbf{r}) &= - \sum_k \frac{q_k}{\epsilon_1} \delta(\mathbf{r},\mathbf{r}_k) \ \text{ in solute $(\Omega_1)$,} \nonumber \\ 21 | \nabla^2\phi_2 (\mathbf{r}) &= \kappa^2 \phi_2(\mathbf{r}) \quad \qquad \ \ \ \text{ in solvent $(\Omega_2)$,} \nonumber \\ 22 | \phi_1 &=\phi_2 \nonumber \\ 23 | \epsilon_1 \frac{\partial \phi_1}{\partial \mathbf{n}} &= \epsilon_2 \frac{\partial \phi_2}{\partial \mathbf{n}} \ \qquad \qquad \text{ on interface $\Gamma_1$, and} \nonumber \\ 24 | -\epsilon_2 \frac{\partial \phi_2}{\partial \mathbf{n}} &= \sigma_0 \qquad \qquad \qquad \text{ on surface $\Gamma_2$} 25 | \end{align} 26 | 27 | \noindent where $\phi_i$ is the electrostatic potential in region $\Omega_i$, which has a permittivity $\epsilon_i$, and $\sigma_0$ is a prescribed charge on the surface. The surface $\Gamma_2$ could correspond to a device such as a biosensor. 28 | 29 | -------------------------------------------------------------------------------- /pygbe2-CPC/model.tex: -------------------------------------------------------------------------------- 1 | %!TEX root = CooperBarba2014.tex 2 | 3 | The implicit-solvent model uses continuum electrostatics to describe the mean-field potential in a molecular system. A typical system consists of a protein in a solvent, defining two regions: inside and outside the protein, with an interface marked by the solvent-excluded surface (\ses). The \ses, beyond which a water molecule cannot penetrate into the protein, can be generated by rolling a (virtual) spherical probe of the size of a water molecule around the protein (see Figure \ref{fig:forcefield-ses}). Inside the protein, the domain has low permittivity ($\epsilon= 2\text{ to }4$) and there are point charges located at the positions of the atoms. The solvent region, representing water with salt, has a permittivity of $\epsilon \approx 80$. A system of partial differential equations models this situation, with a Poisson equation governing inside the protein and a linearized Poisson-Boltzmann equation governing in the solvent region. Appropriate interface conditions on the \ses express the continuity of the potential and electric displacement, completing the mathematical formulation. 4 | 5 | \begin{figure}% [h] 6 | \includegraphics[width=0.49\textwidth]{Figure1.pdf} 7 | \caption{Sketch of the process for generating a solvent-excluded surface (\ses): a protein molecule contains a set of atoms that define a radius upon applying a force field and a probe the size of a water molecule is rolled to define the \ses. $\Omega_1$ is the protein region and $\Omega_2$ the solvent region.} 8 | \label{fig:forcefield-ses} 9 | \end{figure} 10 | 11 | This model has been widely applied to investigate interactions between molecules, such as in protein-ligand binding. We are interested here in an extension of the model to consider interactions between proteins and surfaces with an imposed potential or charge. This new setup is sketched in Figure \ref{fig:molecule_surface}, and is described mathematically by the following equations: 12 | 13 | 14 | \begin{align} \label{eq:pde} 15 | \nabla^2 \phi_1(\mathbf{r}) &= - \sum_k \frac{q_k}{\epsilon_1} \delta(\mathbf{r},\mathbf{r}_k) \ \text{ in solute $(\Omega_1)$,} \nonumber \\ 16 | \nabla^2\phi_2 (\mathbf{r}) &= \kappa^2 \phi_2(\mathbf{r}) \quad \qquad \ \ \text{ in solvent $(\Omega_2)$,} \nonumber \\ 17 | \phi_1 &=\phi_2 \qquad \qquad \qquad \text{ on interface $\Gamma_1$,} \nonumber \\ 18 | \epsilon_1 \frac{\partial \phi_1}{\partial \mathbf{n}} &= \epsilon_2 \frac{\partial \phi_2}{\partial \mathbf{n}} \nonumber \\ 19 | \phi_2 = \phi_0 &\text{ or } -\epsilon_2 \frac{\partial \phi_2}{\partial \mathbf{n}} = \sigma_0 \ \ \text{ on surface $\Gamma_2$,} 20 | \end{align} 21 | 22 | \noindent Here, $\phi_i$ is the potential corresponding to the region $\Omega_i$ with permittivity $\epsilon_i$, and $\phi_0$ and $\sigma_0$ are the set potential or charge on the nanosurface. 23 | 24 | \begin{figure} 25 | \includegraphics[width=0.45\textwidth]{Figure2.pdf} 26 | \caption{Sketch of a molecule interacting with a surface: $\Omega_1$ is the protein, $\Omega_2$ the solvent region, $\Gamma_1$ is the \ses and $\Gamma_2$ a nanosurface with imposed charge or potential.} 27 | \label{fig:molecule_surface} 28 | \end{figure} 29 | -------------------------------------------------------------------------------- /pygbe-orientation/prot_orientation.tex: -------------------------------------------------------------------------------- 1 | %!TEX root = CooperBarba-orientation.tex 2 | 3 | We are aiming to investigate the orientation of proteins near self-assembled monolayers (\sam), specifically for biosensing applications. In the framework of the implicit-solvent model, we can represent the \sam\ as a surface charge density, and use Equation \eqref{eq:matrix_dphi} to compute the electrostatic potential. 4 | According to the Boltzmann distribution, the probability of finding the system in micro-state $\lambda$ depends on the total free energy, $F_\text{total}$, as follows: 5 | % 6 | \begin{equation} \label{eq:prob} 7 | P(\lambda) = \frac{\int_{\lambda} \exp \left(-\frac{F_\text{total}}{k_B T} \right) \text{d} \lambda}{\int_{\Lambda} \exp \left(-\frac{F_\text{total}}{k_B T} \right) \text{d} \Lambda}, 8 | \end{equation} 9 | 10 | \noindent where $\Lambda$ is the ensemble of all micro-states, $k_B$ is the Boltzmann constant and $T$ the temperature. To obtain a probability distribution, we assume that electrostatic effects are dominant and use Equation \eqref{eq:prob}, sampling $F_\text{total}$ for different orientations. We define the orientation using the angle between the dipole moment and surface normal vectors as a reference (tilt angle), varying from 0$^\circ$ to 180$^\circ$. For each tilt angle, we rotate the protein about the dipole moment vector by 360$^\circ$ to examine all possible orientations. This process is sketched in Figure \ref{fig:1pgb_orientation}. 11 | 12 | In this case, micro-states are defined by the tilt ($\alpha_{\text{tilt}}$) and rotational ($\alpha_{\text{rot}}$) angles, and we rewrite the integral in the numerator of Equation \eqref{eq:prob} as: 13 | % 14 | \begin{equation} \label{eq:prob_angle} 15 | \int_{\lambda} \exp \left(-\frac{F_\text{total}}{k_B T} \right) \text{d} \lambda = \int \int \exp \left(-\frac{F_\text{total}}{k_B T} \right) \text{d} \alpha_{\text{rot}} \text{d} \alpha_{\text{tilt}}, 16 | \end{equation} 17 | 18 | \noindent where micro-state $\lambda$ is a range of angles $\alpha_{\text{rot}}$ and $\alpha_{\text{tilt}}$. 19 | In biosensors, the ligand is adsorbed on the surface (usually covalently), hence we are interested on the orientation of the molecule very close to the surface, and don't consider configurations away from it. 20 | 21 | 22 | \begin{figure}%[h] 23 | \centering 24 | \includegraphics[width=0.5\textwidth]{Figure2.pdf} 25 | \caption{Setup of the problem for our orientation-sampling studies.} 26 | \label{fig:1pgb_orientation} 27 | \end{figure} 28 | 29 | \subsection{Structure preparation} 30 | 31 | To assess the adequacy of the implicit-solvent model for investigating protein-surface interactions, we studied the orientation of protein \gb mutant near a charged surface, since there are results available in the literature that we could compare to: both experimental observations \cite{BaioWeidnerBaughGambleStaytonCastner2012} and simulations using a combined Monte Carlo and molecular dynamics approach.\cite{LiuLiaoZhou2013} Figure \ref{fig:1pgb} shows the structure of protein \gb (\pdb code {\small 1PGB}), to which we applied mutations {\small E19Q}, {\small D22N}, {\small D46N} and {\small D47N} to obtain the {\small D4$^\prime$} mutant, using the \textsl{SwissPdb Viewer} software.\cite{GuexPeitsch1997} 32 | We then studied the orientation of antibody immunoglobulin G iso-type \ig 2a (\pdb code {\small 1IGT}), a widely used protein in biosensors, whose structure is shown in Figure \ref{fig:1igt}. This is a more interesting case from the point of view of our application, yet it is more difficult to study with molecular simulation due to its size. 33 | In both cases, the vector orientation of the dipole moment (used as reference for the tilt and orientation angles) was obtained using the location of the point charges at the locations of the atoms. 34 | 35 | \begin{figure}%[h] 36 | \centering 37 | \includegraphics[width=0.25\textwidth]{Figure3.pdf} 38 | \caption{Structure of protein \gb (\pdb code {\small 1PGB}).} 39 | \label{fig:1pgb} 40 | \end{figure} 41 | 42 | \begin{figure}%[h] 43 | \centering 44 | \includegraphics[width=0.35\textwidth]{Figure4.pdf} 45 | \caption{Structure of immunoglobulin G (\pdb code {\small 1IGT}).} 46 | \label{fig:1igt} 47 | \end{figure} 48 | 49 | -------------------------------------------------------------------------------- /pygbe2-CPC/introduction.tex: -------------------------------------------------------------------------------- 1 | %!TEX root = CooperBarba2014.tex 2 | 3 | Proteins interacting with solid surfaces appear in many biological processes. Adsorption serves a key function in natural activities, like blood coagulation, and in biotechnologies like tissue engineering, biomedical implants and biosensors. 4 | A full understanding of protein-surface interactions has remained elusive \cite{Gray2004,RabeVerdesSeegel2011}, but adsorption mechanisms are governed by surface energy and often the dominant effect is electrostatics. As a free-energy-driven process, protein-surface interaction is difficult to study experimentally \cite{MijajlovicETal2013}, and thus simulations offer a good alternative. Full atomistic molecular dynamics simulations demand large amounts of computing effort, so we often must resort to other methods. 5 | 6 | Protein electrostatics can be studied via modeling approaches using the Poisson-Boltzmann equation and implicit-solvent representations. These models are popular for computing solvation energies in protein systems \cite{RouxSimonson1999,Bardhan2012}, but few studies have included the effect of surfaces. Lenhoff and co-workers studied surface-protein interactions using continuum models discretized with boundary-element \cite{YoonLenhoff1992,RothLenhoff1993,AsthagiriLenhoff1997} and finite-difference methods \cite{YaoLenhoff2004,YaoLenhoff2005}, in the context of ion-exchange chromatography. They realized that van der Waals effects can be neglected for realistic molecular geometries \cite{RothNealLenhoff1996} and that the model is adequate as long as conformational changes in the protein are slight \cite{YaoLenhoff2004,YaoLenhoff2005}. 7 | 8 | The aim of this work is to develop and assess a computational model to simulate proteins near engineered surfaces of fixed charge, using implicit-solvent electrostatics. 9 | We have added the capability of modeling a protein near a charged surface to our code \pygbe, an open-source code\footnote{\url{https://github.com/barbagroup/pygbe}} that solves the Poisson-Boltzmann equations via an integral formulation, using a fast multipole algorithm and \gpu\ hardware acceleration. Previously, we verified and validated \pygbe in its use to obtain solvation and binding energies, by comparing with analytical solutions of the equations and with results obtained using the well-known \apbs software \cite{CooperBarba-share154331,CooperBardhanBarba2013}. 10 | In the present work, we derived an analytical solution for a spherical molecule interacting with a spherical surface of prescribed charge, and used it to verify the code in its new application and study numerical convergence. 11 | Using the newly extended code, we also studied the interaction between protein \gb and a solid surface of imposed charge, 12 | and conducted a grid-convergence study using this more realistic surface geometry. 13 | 14 | We intend our new modeling tool to be useful in studying the behavior of proteins as they adsorb on surfaces that have been functionalized with self-assembled monolayers (\sam), which are modeled within an implicit-solvent framework as surfaces with prescribed charge. 15 | One application is biosensing, where the target molecules are captured on the sensor via ligand molecules (for which antibodies are a common choice). Favorable orientations of ligand molecules lead to greatly enhanced sensitivity of biosensors \cite{TajimaTakaiIshihara2011,TrillingBeekwilderZuilhof2013}, because binding sites need to be physically accessible to the targets. Studies of protein orientation near charged surfaces might look at how orientation can be influenced by engineering decisions regarding surface preparation, to aid the design of better biosensors. We explore this application in a companion publication that obtains probability of orientations for an antibody near a surface, as function of changing conditions on charge and ionic strength \cite{CooperClementiBarba2015}. 16 | In this paper, we present the details of a new analytical solution for spherical charged surfaces and molecules, grid-convergence studies for the interaction free energy in this case, and grid-convergence studies for protein \gb alone and interacting with a charged surface. The detailed analysis of the model is complemented with a diligent effort for reproducibility and we deposit both input and results data in accessible and permanent archival storage, in addition to the open-source code. 17 | -------------------------------------------------------------------------------- /pygbe2-CPC/discussion.tex: -------------------------------------------------------------------------------- 1 | %!TEX root = CooperBarba2014.tex 2 | 3 | %\subsection{Verification with analytical solution} \label{sec:disc_analytical} 4 | In order to study the interaction of proteins and charged surfaces, we extended \pygbe to account for surfaces with prescribed charge or potential. Unfortunately, there was no analytical solution available in the literature to compare and verify \pygbe's extension. 5 | Section \ref{sec:analytical_solution} derives a closed-form expression for a spherical molecule with a centered charge interacting with a spherical nanosurface with imposed charge or potential. 6 | We used this new analytical solution to conduct a grid-convergence study of the interaction energy (Figure \ref{fig:error_sphere}). The error decays with the area, which is the expected behavior for a boundary element method with constant elements \cite{CooperBardhanBarba2013, CooperBarba-share154331}. 7 | Discretization error is very small for a spherical geometry. To make sure that the errors due to integration, the treecode approximation and the \textsc{gmres} solver were even smaller, we chose all the numerical parameters for high accuracy. This allows us to observe the convergence with respect to the discretization only and extract the order. With more realistic molecular geometries, however, discretization errors will be larger and the requirements can be relaxed in the other numerical parameters of \pygbe, resulting in lower runtimes. 8 | 9 | %\subsection{Protein G B1 D4$^\prime$ near a charged surface} \label{sec:disc_1PGB} 10 | 11 | The results in Figures \ref{fig:convergence_1PGB_isolated} and \ref{fig:convergence_1PGB_sensor} show the applicability of this approach in more realistic situations. The setting in Figure \ref{fig:protein_surface} can model a nanostructure coated with a self-assembled monolayer (\sam) interacting with a protein that will adsorb on that surface. Our application of interest in developing this model is the field of biosensors, where it can assist design through studies of electrostatic adsorption affecting protein orientation near the biosensor. With antibody-based biosensors, orientation determines the accessibility of reaction sites and is critical for sensitivity. In a companion publication \cite{CooperClementiBarba2015}, we present the first studies of protein orientation near charged surfaces using our modeling framework. 12 | 13 | Figures \ref{fig:convergence_1PGB_isolated} and \ref{fig:convergence_1PGB_sensor} show the expected 1/N convergence with a simple protein. Just like in the case of the sphere, the simple geometry of the charged surface forced us to use very fine parameters in order to extract the order of convergence. If we needed to run this computation many times---for example, to sample different protein orientations---we might relax these parameters to obtain shorter computation times. 14 | 15 | The extrapolated values in Table \ref{table:extraPGB} are useful to find more relaxed parameters that still give acceptable results. For example, using a mesh density of 2 elements per square Angstrom on the charged surface and 4 elements per square Angstrom on the protein, and the parameters detailed in Table \ref{table:params3}, we get the results in Table \ref{table:relaxPGB}. These results are less than 2\% away from those in Table \ref{table:extraPGB}, and each run takes less than one minute. Moreover, using the energy values from Table \ref{table:relaxPGB}, the interaction free energy is $-7.61$ [kcal/mol], which is very close to the extrapolated case ($-7.6$ [kcal/mol]). 16 | 17 | \begin{table}[h] 18 | %\centering 19 | %\fontfamily{ppl}\selectfont 20 | \caption{\label{table:params3}Numerical parameters for relaxed runs with protein \gb. } 21 | \begin{tabular}{c c c c c c c} 22 | \hline%\toprule 23 | \multicolumn{3}{l} {\# Gauss points:} & \multicolumn{3}{l}{Treecode:} & \gmres:\\ 24 | \footnotesize{in-element} & \footnotesize{close-by} & \footnotesize{far-away} & $N_{\text{crit}}$ & $P$ & $\theta$ & tol.\\ 25 | \hline%\midrule 26 | 9 per side & 19 & 1 & 300 & 4 & 0.5 & $10^{-5}$\\ 27 | \hline%\bottomrule 28 | \end{tabular} 29 | \end{table} 30 | 31 | \begin{table}[h] 32 | %\centering 33 | %\fontfamily{ppl}\selectfont 34 | \caption{\label{table:relaxPGB}Values of energy for protein \gb using the parameters in Table \ref{table:params3}, and a mesh density of 4 elements per square angstrom in the protein and 2 elements per square angstrom on the charged surface} 35 | \begin{tabular}{c c c} 36 | \hline%\toprule 37 | & \multicolumn{2}{c} {Energy [kcal/mol]} \\ 38 | & Solvation & Surface \\ 39 | \hline%\midrule 40 | Isolated & $-221.56$ & $315.33$ \\ 41 | Interacting & $-225.81$ & $311.96$ \\ 42 | \hline%\bottomrule 43 | \end{tabular} 44 | \end{table} 45 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # pygbe-papers 2 | 3 | ## Submitted 2015 4 | 5 | Read also our [News posting and overview on the group website](http://lorenabarba.com/news/probing-protein-orientation-near-charged-surfaces/) 6 | 7 | **Title 1:** 8 | 9 | *"Probing protein orientation near charged nanosurfaces for simulation-assisted biosensor design"* 10 | 11 | [arXiv:1503.08150v4](http://arxiv.org/abs/1503.08150v4) 12 | 13 | **Abstract:** 14 | 15 | Protein-surface interactions are ubiquitous in biological processes and bioengineering, 16 | yet are not fully understood. In biosensors, a key factor determining the sensitivity and 17 | thus the performance of the device is the orientation of the ligand molecules on the bioactive 18 | device surface. Adsorption studies thus seek to determine how orientation can be influenced by 19 | surface preparation. In this work, protein orientation near charged nanosurfaces is obtained under 20 | electrostatic effects using the Poisson-Boltzmann equation, in an implicit-solvent model. Sampling the 21 | free energy for protein GB1D4' at a range of tilt and rotation angles with respect to the charged surface, 22 | we calculated the probability of the protein orientations and observed a dipolar behavior. This result is 23 | consistent with published experimental studies and combined Monte Carlo and molecular dynamics simulations 24 | using this small protein, validating our method. More relevant to biosensor technology, antibodies such as 25 | immunoglobulin G are still a formidable challenge to molecular simulation, due to their large size. We obtained 26 | the probability distribution of orientations for the iso-type IgG2a at varying surface charge and salt 27 | concentration. This iso-type was not found to have a preferred orientation in previous studies, unlike the 28 | iso-type IgG1 whose larger dipole moment was assumed to make it easier to control. We find that the preferred 29 | orientation of IgG2a can be favorable for biosensing with positive surface charge of $0.05C/m^2$ or higher and 37mM 30 | salt concentration. The results also show that local interactions dominate over dipole moment for this protein. 31 | Improving immunoassay sensitivity may thus be assisted by numerical studies using our method (and open-source code), 32 | guiding changes to fabrication protocols or protein engineering of ligand molecules to obtain more favorable orientations. 33 | 34 | 35 | **Title 2:** 36 | 37 | *"Poisson-Boltzmann model for protein-surface electrostatic interactions and grid-convergence study using the PyGBe code"* 38 | 39 | [arXiv:1506.03745](http://arxiv.org/abs/1506.03745v1) 40 | 41 | *Abstract:* 42 | 43 | Interactions between surfaces and proteins occur in many vital processes and are crucial in biotechnology: the ability to control specific interactions is essential in fields like biomaterials, biomedical implants and biosensors. In the latter case, biosensor sensitivity hinges on ligand proteins adsorbing on bioactive surfaces with a favorable orientation, exposing reaction sites to target molecules. 44 | Protein adsorption, being a free-energy-driven process, is difficult to study experimentally. This paper develops and evaluates a computational model to study electrostatic interactions of proteins and charged nanosurfaces, via the Poisson-Boltzmann equation. 45 | We extended the implicit-solvent model used in the open-source code PyGBe to include surfaces of imposed charge or potential. This code solves the boundary integral formulation of the Poisson-Boltzmann equation, discretized with surface elements. PyGBe has at its core a treecode-accelerated Krylov iterative solver, resulting in O(N log N) scaling, with further acceleration on hardware via multi-threaded execution on GPUs. It computes solvation and surface free energies, providing a framework for studying the effect of electrostatics on adsorption. 46 | We then derived an analytical solution for a spherical charged surface interacting with a spherical molecule, then completed a grid-convergence study to build evidence on the correctness of our approach. The study showed the error decaying with the average area of the boundary elements, i.e., the method is $O(1/N)$, which is consistent with our previous verification studies using PyGBe. 47 | We also studied grid-convergence using a real molecular geometry (protein GB1D4'), in this case using Richardson extrapolation (in the absence of an analytical solution) and confirmed the O(1/N) scaling in this case. 48 | PyGBe is open-source under an MIT license and is hosted under version control at [https://github.com/barbagroup/pygbe](https://github.com/barbagroup/pygbe). In addition, we prepared "reproducibility packages" to supplement this paper, consisting of running and post-processing scripts in Python to allow replication of the grid-convergence studies, all the way to generating the final plots, with a single command. 49 | -------------------------------------------------------------------------------- /pygbe-orientation/bem.tex: -------------------------------------------------------------------------------- 1 | %!TEX root = CooperBarba-orientation.tex 2 | 3 | 4 | We apply Green's second identity to the system of partial-differential equations in \eqref{eq:pde}, and evaluate the resulting equations on $\Gamma_1$ and $\Gamma_2$ to obtain the following system of integral equations: 5 | % 6 | \begin{widetext} 7 | \begin{align} \label{eq:integral_eq} 8 | \frac{\phi_{1,\Gamma_1}}{2}+ K_{L}^{\Gamma_1}(\phi_{1,\Gamma_1}) - V_{L}^{\Gamma_1} \left(\frac{\partial}{\partial \mathbf{n}}\phi_{1,\Gamma_1} \right) = 9 | \frac{1}{\epsilon_1} \sum_{k=0}^{N_q} \frac{q_k}{4\pi|\mathbf{r}_{\Gamma_1} - \mathbf{r}_k|} & \quad \text{on $\Gamma_1$,} \nonumber \\ 10 | \frac{\phi_{1,\Gamma_1}}{2} - K_{Y}^{\Gamma_1}(\phi_{1,\Gamma_1}) + \frac{\epsilon_1}{\epsilon_2} V_{Y}^{\Gamma_1} \left( \frac{\partial}{\partial \mathbf{n}} \phi_{1,\Gamma_1} \right) - 11 | K_{Y}^{\Gamma_1}(\phi_{2,\Gamma_2}) + V_{Y}^{\Gamma_1} \left( -\frac{\sigma_0}{\epsilon_2} \right) = 0& \quad \text{on $\Gamma_1$,} \nonumber \\ 12 | - K_{Y}^{\Gamma_2}(\phi_{1,\Gamma_1}) + \frac{\epsilon_1}{\epsilon_2} V_{Y}^{\Gamma_2} \left( \frac{\partial}{\partial \mathbf{n}} \phi_{1,\Gamma_1} \right) + \frac{\phi_{2,\Gamma_2}}{2} - 13 | K_{Y}^{\Gamma_2}(\phi_{2,\Gamma_2}) + V_{Y}^{\Gamma_2} \left( -\frac{\sigma_0}{\epsilon_2} \right) = 0& \quad \text{on $\Gamma_2$.} 14 | \end{align} 15 | \end{widetext} 16 | 17 | 18 | \noindent The function $\phi_{i,\Gamma_j} = \phi_i(\mathbf{r}_{\Gamma_j})$ is the electrostatic potential at a point that approaches the surface $\Gamma_j$ from the region $\Omega_i$, and 19 | $K$ and $V$ are known as the single- and double-layer potentials, correspondingly: 20 | % 21 | \begin{align} \label{eq:layers} 22 | K_{L/Y}^{\Gamma_k}(\phi_{i,\Gamma_j}) &= \oint_{\Gamma_j} \frac{\partial}{\partial \mathbf{n}} \left[ G_{L/Y}(\mathbf{r}_{\Gamma_k},\mathbf{r}_{\Gamma_j}) \right]\phi_{i,\Gamma_j} \, \mathrm{d} \Gamma, \nonumber \\ 23 | V_{L/Y}^{\Gamma_k} \left( \frac{\partial}{\partial \mathbf{n}} \phi_{i,\Gamma_j} \right) &= \oint_{\Gamma_j} \frac{\partial}{\partial \mathbf{n}} \phi_{i,\Gamma_j} G_{L/Y}(\mathbf{r}_{\Gamma_k},\mathbf{r}_{\Gamma_j}) \, \mathrm{d} \Gamma. 24 | \end{align} 25 | 26 | \noindent In Equation \eqref{eq:layers}, $G_L$ and $G_Y$ are the free-space Green's functions of the Poisson and linearized Poisson-Boltzmann equations, respectively. 27 | The single-layer potential of a distribution $\psi$ on a surface $\Gamma$ evaluated at $\mathbf{r}$, $V^\mathbf{r}(\psi_\Gamma)$, can be interpreted as the potential on $\mathbf{r}$ due to a charge distribution $\psi$ on $\Gamma$. 28 | Similarly, $K^\mathbf{r}(\psi_\Gamma)$ can be seen as the potential induced by a double layer of charges ($\psi$) with opposite sign at $\Gamma$. 29 | 30 | Rearranging terms, we write Equation \eqref{eq:integral_eq} in matrix form, as follows: 31 | % 32 | \begin{align} \label{eq:matrix_dphi} 33 | \left[ 34 | \begin{matrix} % or pmatrix or bmatrix or Bmatrix or ... 35 | \frac{1}{2} + K_{L}^{\Gamma_1} & -V_{L}^{\Gamma_1} & 0 \vspace{0.2cm}\\ 36 | \frac{1}{2} - K_{Y}^{\Gamma_1} & \frac{\epsilon_1}{\epsilon_2} V_{Y}^{\Gamma_1} & -K_{Y}^{\Gamma_1} \vspace{0.2cm} \\ 37 | - K_{Y}^{\Gamma_2} & \frac{\epsilon_1}{\epsilon_2} V_{Y}^{\Gamma_2} & \left(\frac{1}{2} - K_{Y}^{\Gamma_2}\right) \\ 38 | \end{matrix} 39 | \right] \left[ 40 | \begin{matrix} % or pmatrix or bmatrix or Bmatrix or ... 41 | \phi_{1,\Gamma_1} \vspace{0.2cm} \\ 42 | \frac{\partial}{\partial \mathbf{n}} \phi_{1,\Gamma_1} \vspace{0.2cm}\\ 43 | \phi_{2,\Gamma_2}\\ 44 | \end{matrix} 45 | \right] = \nonumber \\ 46 | \left[ 47 | \begin{matrix} % or pmatrix or bmatrix or Bmatrix or ... 48 | \sum_{k=0}^{N_q} \frac{q_k}{4\pi|\mathbf{r}_{\Gamma_1} - \mathbf{r}_k|} \vspace{0.2cm} \\ 49 | V_{Y}^{\Gamma_1} \left( \frac{\sigma_0}{\epsilon_2} \right) \vspace{0.2cm} \\ 50 | V_{Y}^{\Gamma_2} \left( \frac{\sigma_0}{\epsilon_2} \right) 51 | \end{matrix} 52 | \right]. 53 | \end{align} 54 | 55 | The boundary-integral formulation is not limited to represent the protein with a single surface, but can account for solvent-filled cavities inside the protein region and Stern layers.\cite{CooperBardhanBarba2013} In those cases, more than one surface is required to appropriately represent the protein. 56 | Our implementation follows the guidelines from Altman and co-workers\cite{AltmanBardhanWhiteTidor09} to deal with multiple surfaces. 57 | 58 | The boundary-integral formulation of the implicit-solvent model is a popular alternative to compute solvation energies of proteins,\cite{YoonLenhoff1990, Juffer1991a, LuETal2006, BajajETal2011, AltmanBardhanWhiteTidor09, GengKrasny2013, CooperBardhanBarba2013} but the effect of charged surfaces has rarely been considered. The only work that we know of that does include these effects is limited to plane, infinite surfaces.\cite{YoonLenhoff1992} 59 | -------------------------------------------------------------------------------- /pygbe2-CPC/energy.tex: -------------------------------------------------------------------------------- 1 | %!TEX root = CooperBarba2014.tex 2 | 3 | Figure \ref{fig:molecule_surface} shows a system with three types of free energy: Coulombic energy from the point charges, surface energy due to $\Gamma_2$ and solvation energy. The Coulombic energy arises simply from the Coulomb interactions of all point charges. This section describes how we compute the other two components of free energy in the boundary-element framework. 4 | 5 | \medskip 6 | 7 | \paragraph*{Solvation free energy} 8 | 9 | When a protein is in a solvated state, surrounded by water molecules that have become polarized, its free energy differs from its state \emph{in vacuo} by an amount known as the solvation energy. Its free energy again differs in the presence of other structures in the solvent, e.g., other proteins or charged surfaces. In this work we use the term solvation energy to more broadly mean the change in free energy of the protein from its state in a vacuum, to its state in the solvent with any other components or structures. In single-molecule settings, this definition of solvation energy coincides with the energy required to solvate the molecule. 10 | 11 | To calculate the solvation energy, the total minus the Coulomb potential is applied inside the protein, i.e., 12 | 13 | \begin{align} \label{eq:solv_energy} 14 | F_{\text{solv}} &= \frac{1}{2} \int_{\Omega} \rho \,(\phi_{\text{total}} - \phi_{\text{Coulomb}}) \\ 15 | &= \sum_{k=0}^{N_q} q_k (\phi_{\text{total}} - \phi_{\text{Coulomb}})(\mathbf{r}_k), 16 | \end{align} 17 | 18 | \noindent where $\rho$ is the charge distribution, consisting of point charges (which transforms the integral into a sum). 19 | The total minus Coulomb potential includes the reaction potential---representing the response of the solvent by polarization and rearrangement of free ions---and any effects from the immersed surface. 20 | We can also interpret it as the potential generated by the boundary $\Gamma$ of the molecular region $\Omega$. Taking the first expression of Equation \eqref{eq:green_identity} and subtracting out the Coulombic effect yields 21 | % 22 | \begin{equation} \label{eq:phi_reac_bem} 23 | \phi_{\text{reac},\mathbf{r}_k} = -K_{L}^{\mathbf{r}_k}(\phi_{1,\Gamma_1}) + V_{L}^{\mathbf{r}_k} \left(\frac{\partial}{\partial \mathbf{n}}\phi_{1,\Gamma_1} \right) 24 | \end{equation} 25 | 26 | Equation \eqref{eq:solv_energy} requires evaluating $\phi_{\text{reac}}$ for each point-charge location $\mathbf{r}_k$. We obtain this by discretizing Equation \eqref{eq:phi_reac_bem} and using the solution of the linear system in Equation \eqref{eq:matrix_phi} or Equation \eqref{eq:matrix_dphi} as inputs. 27 | 28 | \medskip 29 | \paragraph*{Surface free energy} 30 | 31 | Chan and co-workers \cite{ChanMitchell1983,CarnieChan1993} derived the free energy for a surface with a set charge or potential. They describe the free energy on a surface as 32 | 33 | \begin{align} \label{eq:energy_surf} 34 | F &= \frac{1}{2} \int_{\Gamma} G_c \sigma_0^2 d\Gamma \quad \text{ for set charge, and} \nonumber \\ 35 | F &= -\frac{1}{2} \int_{\Gamma} G_p \phi_0^2 d\Gamma \quad \text{ for set potential,} 36 | \end{align} 37 | 38 | \noindent where $\phi_0$ and $\sigma_0$ are the prescribed potential and surface charge, respectively. The potential is given by $\phi(\sigma, R, \mathbf{x}) = G_c(R, \mathbf{x}) \sigma$ for the first expression and the surface charge by $\sigma(\phi, R, \mathbf{x}) = G_p(R, \mathbf{x}) \phi$ for the second one. This is valid because we are using a linearized Poisson-Boltzmann model. 39 | 40 | Using constant values of $\phi$ and $\frac{\partial \phi}{\partial \mathbf{n}}$ per panel, the discretized version of Equation \eqref{eq:energy_surf} takes the form 41 | 42 | \begin{align} \label{eq:energy_surf_disc} 43 | F &= \frac{1}{2} \sum_{j=1}^{N_p} \phi(\mathbf{r}_j) \sigma_{0j} A_j \text{, and } \nonumber \\ 44 | F &= -\frac{1}{2} \sum_{j=1}^{N_p} \phi_{0j} \sigma(\mathbf{r}_j) A_j. 45 | \end{align} 46 | 47 | \noindent where $A_j$ is the area of panel $j$, and $\sigma = \epsilon \frac{\partial \phi}{\partial \mathbf{n}}$. To obtain the surface free energy, we can plug in the solution of the system in Equation \eqref{eq:matrix_phi} or \eqref{eq:matrix_dphi} to Equation \eqref{eq:energy_surf_disc}. 48 | 49 | \medskip 50 | \paragraph*{Interaction free energy} 51 | When there are two or more bodies in the solvent, they will interact electrostatically. In order to compute the energy of interaction, we need to take the difference between the total energy of the interacting system and the total energy of each isolated component, where the total free energy is given by 52 | % 53 | \begin{equation} \label{eq:total_energy} 54 | F_{\text{total}} = F_{\text{Coulomb}} + F_{\text{surface}} + F_{\text{solv}}. 55 | \end{equation} 56 | 57 | \noindent The interaction free energy is 58 | % 59 | \begin{equation} \label{eq:interaction_energy} 60 | F_{\text{interaction}} = F_{\text{total}}^{\text{assembly}} - \sum_{i=1}^{N_c} F_{\text{total}}^{\text{comp}_i}, 61 | \end{equation} 62 | 63 | \noindent where $N_c$ is the number of components in the system and $F_{\text{total}}^{\text{comp}_i}$ is calculated over the isolated component $i$. -------------------------------------------------------------------------------- /pygbe-orientation/introduction.tex: -------------------------------------------------------------------------------- 1 | %!TEX root = CooperBarba-orientation.tex 2 | 3 | Protein adsorption plays a key role in many biotechnological applications, particularly biomaterials and tissue engineering, biomedical implants and biosensors. 4 | Yet, despite their importance, the specific mechanisms governing protein-surface interactions are not fully understood.\cite{Gray2004,RabeVerdesSeegel2011} 5 | 6 | In the field of biosensors, protein adsorption needs to be engineered to obtain a successful device. 7 | Biosensors detect specific molecules using a nanoscale sensing element, such as a metallic nanoparticle or nanowire covered with a bioactive coating. 8 | The prevailing method to modify a sensor surface is via a self-assembled monolayer (\sam) of a small charged group, with ligand molecules layered on top to achieve the desired function. 9 | Antibodies are a common choice for the ligand molecules, although the newest devices use single-domain or single-chain fragment molecules.\cite{ByunETal2013,TrillingETal2014} 10 | Sensing occurs when a target biomolecule binds to the ligand molecule, changing some physical parameter on the sensor, such as current in nanowires or plasmon resonance frequency in metallic nanoparticles. 11 | 12 | One of the factors crucially affecting biosensor performance is the orientation of ligand molecules.\cite{TajimaTakaiIshihara2011,TrillingBeekwilderZuilhof2013} 13 | These have specific binding sites, which need to be accessible to the target molecule for the biosensor to function well. 14 | Probing protein orientation is thus one key goal of adsorption studies. 15 | The aim of this study is to ascertain how orientation can be influenced by fabrication conditions regarding surface preparation, such as surface charge and ambient salt content. We consider in particular the antibody immunoglobulin G near a solid surface at different charge concentrations and ionic strengths. Using a smaller molecule (protein \gb), we could first confirm agreement of our results with published works reporting experiments\cite{BaioWeidnerBaughGambleStaytonCastner2012} and simulations with a combined Monte Carlo and molecular dynamics method.\cite{LiuLiaoZhou2013} These previous works, among others, also concluded that electrostatic interactions are the dominant effect in the orientation of adsorbed proteins. In the case of immunoglobulin G (\ig), the protein is relevant for biosensor applications, but its large size would make all-atom molecular simulations quite cumbersome and expensive. For this reason, other researchers have studied adsorption of \ig\ using a coarse-grained model that considers each residue as a sphere (united-residue model),\cite{ZhouChenJiang2003} finding that electrostatics dominates the orientation for higher surface charges and that a positive charge can result in the desired ``tail-on'' placement for the \ig 1 iso-type, at low enough salt concentration. 16 | Here, we investigate the preferred orientations for the \ig 2a variant, which other researchers found hard to control. 17 | In addition to obtaining the preferred orientation at different conditions of charge and ionic strength, we also take a detailed look at the probability distribution in the parameter space. 18 | 19 | Our model for protein-surface interactions uses the Poisson-Boltzmann equations in their integral formulation, representing the protein geometry as a dielectric interface in an implicit solvent. We recently verified the model against an analytical solution valid for spherical geometries and studied its numerical convergence in detail.\cite{CooperBarba2015a} 20 | Previous studies on protein-surface interaction using the Poisson-Boltzmann equation showed that such a model is adequate as long as conformational changes in the protein are slight,\cite{YaoLenhoff2004,YaoLenhoff2005} and also that van der Waals effects can be neglected for realistic molecular geometries.\cite{RothNealLenhoff1996} 21 | Conformational changes of the biomolecule can be ignored in this case because binding sites need to remain nearly unmodified during the biosensor fabrication process.\cite{TajimaTakaiIshihara2011} 22 | A continuum framework has been used in the past to study protein orientation,\cite{JufferArgosDevlieg1996} but it included ions explicitly. Other studies have used a coarse-grained model of the molecule, represented as a set of spheres,\cite{ShengTsaoZhouJiang2002,ZhouTsaoShengJiang2004} assigned effective charges at the residue level,\cite{FreedCramer2011,ZhouChenJiang2003} or made approximations to account for pH effects.\cite{BiesheuvelvanderVeenNord2005,HartvigdeWeertOstergaartJorgensenJensen2011} 23 | 24 | The sensor element (functionalized with the \sam) is represented in our model as a charged surface that interacts electrostatically with the biomolecule. A parameter sweep of the protein's rotation and tilt angles with respect to the solid surface provides energy landscapes, where the probability of finding the system in a given micro-state depends on the total free energy. 25 | The continuum approach can thus provide insights to the conditions (surface charge and salt concentration) conducive to a favorable orientation of large proteins, too large for all-atom molecular simulation with today's computing power. It can also represent solid surfaces of any geometry, and we expect that it may in future assist in the design of better ligand-molecule immobilization techniques for high-sensitivity biosensors. 26 | 27 | -------------------------------------------------------------------------------- /pygbe-orientation/supplementary_material.tex: -------------------------------------------------------------------------------- 1 | %% ****** Start of file aiptemplate.tex ****** % 2 | %% 3 | %% This file is part of the files in the distribution of AIP substyles for REVTeX4. 4 | %% Version 4.1 of 9 October 2009. 5 | %% 6 | %!TEX encoding = UTF-8 Unicode 7 | % This is a template for producing documents for use with 8 | % the REVTEX 4.1 document class and the AIP substyles. 9 | % 10 | % Copy this file to another name and then work on that file. 11 | % That way, you always have this original template file to use. 12 | 13 | \documentclass[aip,notitlepage,reprint]{revtex4-1} 14 | %\documentclass[reprint]{revtex4-1} 15 | 16 | \usepackage{amsfonts} 17 | \usepackage{amsmath} 18 | \usepackage{amssymb} 19 | \usepackage{booktabs} 20 | \usepackage{caption} 21 | \usepackage{color} 22 | \usepackage{comment} 23 | \usepackage{graphicx} 24 | \usepackage{hyperref} 25 | %\usepackage[utf8]{inputenc} % allows using accents directly in text, like ÔøΩ 26 | \usepackage{subfig} 27 | \usepackage{xspace} 28 | 29 | 30 | \captionsetup{justification=raggedright, 31 | singlelinecheck=false 32 | } 33 | 34 | \newcommand{\pygbe}{\texttt{PyGBe}\xspace} 35 | \newcommand{\gb}{{\small G\,B1\,D4$^\prime$}\xspace} 36 | \newcommand{\ig}{{\small IgG}} 37 | \newcommand{\pdb}{{\small PDB}\xspace} 38 | \newcommand{\gmres}{\textsc{gmres}\xspace} 39 | \newcommand{\bem}{\textsc{bem}\xspace} 40 | \newcommand{\ses}{\textsc{ses}\xspace} 41 | \newcommand{\sam}{\textsc{sam}} 42 | \newcommand{\gpu}{\textsc{gpu}} 43 | \newcommand{\cpu}{\textsc{cpu}} 44 | \newcommand{\apbs}{\textsc{apbs}\xspace} 45 | \newcommand{\nvidia}{\textsc{nvidia}\xspace} 46 | \newcommand{\msms}{\texttt{\textsc{msms}}\xspace} 47 | \newcommand{\amber}{\texttt{\textsc{amber}}\xspace} 48 | \newcommand{\ccby}{\textsc{cc-by}\xspace} 49 | 50 | \graphicspath{{figs/}} % PATH to figure files-- change to ./ for submission 51 | 52 | 53 | 54 | \begin{document} 55 | 56 | % Use the \preprint command to place your local institutional report number 57 | % on the title page in preprint mode. 58 | % Multiple \preprint commands are allowed. 59 | %\preprint{} 60 | 61 | \title{Probing protein orientation near charged nanosurfaces for simulation-assisted biosensor design} 62 | 63 | 64 | % Explanatory text should go in the []'s, 65 | % actual e-mail address or url should go in the {}'s for \email and \homepage. 66 | % \affiliation command applies to all authors since the last \affiliation command. 67 | % The \affiliation command should follow the other information. 68 | 69 | \author{Christopher D. Cooper} 70 | \email[]{cdcooper@bu.edu,christopher.cooper@usm.cl} 71 | %\thanks{} 72 | \affiliation{Department of Mechanical Engineering, Boston University, Boston, MA.} 73 | \affiliation{Mechanical Engineering, Universidad T\'ecnica Federico Santa Mar\'ia, Valpara\'iso, Chile.} 74 | % Note: 75 | %At FINAL paper stage, move USM to \affiliation instead instead of \altaffiliation 76 | %\affiliation{Department of Mechanical Engineering, Universidad T\'ecnica Federico Santa Mar\'ia, Valpara\'iso, Chile} 77 | 78 | \author{Natalia C. Clementi} 79 | \email[]{ncclementi@gwu.edu} 80 | \affiliation{Department of Mechanical \& Aerospace Engineering, The George Washington University, Washington, DC.} 81 | 82 | \author{Lorena A. Barba} 83 | \email[]{labarba@gwu.edu} 84 | \homepage[]{http://lorenabarba.com/} 85 | \affiliation{Department of Mechanical \& Aerospace Engineering, The George Washington University, Washington, DC.} 86 | 87 | % Collaboration name, if desired (requires use of superscriptaddress option in \documentclass). 88 | % \noaffiliation is required (may also be used with the \author command). 89 | %\collaboration{} 90 | %\noaffiliation 91 | 92 | %\date{\today} 93 | 94 | \maketitle %\maketitle must follow title, authors, abstract and \pacs 95 | 96 | \begin{center} 97 | \textbf{Supplementary Material} 98 | \end{center} 99 | For the main result in the paper, looking at the orientation of \ig 2a near a surface with charge $\sigma=0.1$C/m$^2$ immersed in a solvent with 37mM of salt, we show here the contribution of each energy component to the probability distribution. 100 | We studied the contribution of the solvation and surface energies to the orientation probability distribution to show that the orientation mechanism is guided by the solvation energy in this case. 101 | Fig.~\ref{fig:supp} shows the probability distribution computed in three ways: using the total energy (Fig.~\ref{fig:full}), neglecting the surface energy (Fig.~\ref{fig:solv}), and neglecting the solvation energy (Fig.~\ref{fig:surf}). 102 | Both the angle combination of the preferred orientation and the probability magnitude are similar in Fig.~\ref{fig:solv} and Fig.~\ref{fig:full}, indicating that the influence of neglecting the surface energy is small. 103 | Hence, the orientation of \ig 2a under these conditions is dominated by the solvation energy. 104 | 105 | 106 | \begin{figure} 107 | \centering 108 | \subfloat[Probability distribution with the total energy.]{\includegraphics[width=0.43\textwidth]{figs/supp_1.pdf} \label{fig:full}}\\ 109 | \subfloat[Probability distribution neglecting the surface energy.]{\includegraphics[width=0.43\textwidth]{figs/supp_2.pdf} \label{fig:solv}}\\ 110 | \subfloat[Probability distribution neglecting the solvation energy.]{\includegraphics[width=0.40\textwidth]{figs/supp_3.pdf} \label{fig:surf}} 111 | \caption{} 112 | \label{fig:supp} 113 | \end{figure} 114 | 115 | \clearpage 116 | 117 | \end{document} 118 | -------------------------------------------------------------------------------- /pygbe2-CPC/methods.tex: -------------------------------------------------------------------------------- 1 | 2 | %!TEX root = CooperBarba2014.tex 3 | 4 | \subsection{Discretization} 5 | 6 | To numerically solve the system in \eqref{eq:matrix_dphi}, we discretize the boundaries into flat triangular panels and assume that $\phi$ and $\frac{\partial \phi}{\partial \mathbf{n}}$ are constant within those panels. The discretized form of the integral operators is as follows: 7 | % 8 | \begin{align} \label{eq:layers_disc} 9 | &K_{L,\text{disc}}^{\mathbf{r}_i}\left(\phi(\mathbf{r}_{\Gamma})\right) = \sum_{j=1}^{N_p}\phi(\mathbf{r}_{\Gamma_j})\int_{\Gamma_j} \frac{\partial}{\partial \mathbf{n}} \left[ G_L(\mathbf{r}_{i},\mathbf{r}_{\Gamma_j}) \right]\mathrm{d} \Gamma_j, \nonumber \\ 10 | &V_{L,\text{disc}}^{\mathbf{r}_i} \left( \frac{\partial}{\partial \mathbf{n}} \phi(\mathbf{r}_{\Gamma}) \right) = \sum_{j=1}^{N_p} \frac{\partial}{\partial \mathbf{n}} \phi(\mathbf{r}_{\Gamma_j}) \int_{\Gamma_j} G_L(\mathbf{r}_{i},\mathbf{r}_{\Gamma_j}) \mathrm{d} \Gamma_j, 11 | \end{align} 12 | 13 | \noindent where $N_p$ is the number of discretization elements on $\Gamma$, and $\phi(\mathbf{r}_{\Gamma_j})$ and $\frac{\partial}{\partial \mathbf{n}} \phi(\mathbf{r}_{\Gamma_j})$ are the constant values of $\phi$ and $\frac{\partial \phi}{\partial \mathbf{n}}$ on panel $\Gamma_j$ (we are somewhat abusing the nomenclature here by reusing the symbol $\Gamma$, which previously referred to the complete surface). By collocating $\mathbf{r}_i$ on the center of each panel, we get a linear system of equations that looks just like \eqref{eq:matrix_phi} or \eqref{eq:matrix_dphi}, but the coefficient matrix is formed by sub-matrices of size $N_p \times N_p$ rather than integral operators. Each element of a sub-matrix is an integral over one panel $\Gamma_j$, with $\mathbf{r}_i$ located at the center of the collocation panel $\Gamma_i$, as follows: 14 | 15 | \begin{align} \label{eq:layers_element} 16 | K_{L,ij} &= \int_{\Gamma_j} \frac{\partial}{\partial \mathbf{n}} \left[ G_L(\mathbf{r}_{\Gamma_i},\mathbf{r}_{\Gamma_j}) \right]\mathrm{d} \Gamma_j, \nonumber \\ 17 | V_{L,ij} &= \int_{\Gamma_j} G_L(\mathbf{r}_{\Gamma_i},\mathbf{r}_{\Gamma_j}) \mathrm{d} \Gamma_j. 18 | \end{align} 19 | 20 | The terms on the right-hand side and the unknown vectors in the discretized form of Equation \eqref{eq:matrix_phi} are sub-vectors of size $N_p$. In this case, each element is the evaluation on the collocation panel $\Gamma_i$, written as 21 | % 22 | \begin{align} \label{eq:vector_disc} 23 | \phi_{1,\Gamma_1} &= \phi_1(\mathbf{r}_i), \nonumber \\ 24 | \frac{\partial}{\partial \mathbf{n}}\phi_{1,\Gamma_1} &= \frac{\partial}{\partial \mathbf{n}}\phi_1(\mathbf{r}_i), \nonumber \\ 25 | \sum_{k=0}^{N_q} \frac{q}{4\pi|\mathbf{r}_{\Gamma_1} - \mathbf{r}_k|} &= \sum_{k=0}^{N_q} \frac{q}{4\pi|\mathbf{r}_i - \mathbf{r}_k|}, 26 | \end{align} 27 | 28 | \noindent where $\mathbf{r}_i$ is located at the center of panel $\Gamma_i$. We avoid any complications related to non-smooth boundaries in \bem by using centered collocation. 29 | 30 | 31 | In our numerical solution, integrals are calculated in three possible ways, depending on how close the panel is to the collocation point. When the collocation point is inside the element being integrated, we use a semi-analytical technique, with Gauss points placed along the edges of the element \cite{HessSmith1967,ZhuHuangSongWhite2001}. If the integrated element is closer than $2L$ from the collocation point ---where $L = \sqrt{2\cdot A_j}$ for $A_j$ the area of panel $j$--- we use a fine Gauss quadrature rule, with 19 or more points per element. Beyond a distance of $2L$, elements have only 1, 3, 4 or 7 Gauss points, depending on the case. 32 | 33 | 34 | \subsection{Treecode-accelerated boundary element method} 35 | 36 | Most modern implementations of the boundary element method (\bem) use Krylov methods to solve the linear system, usually a general minimal residual method (\gmres), which is agnostic to the structure of the matrix. In practice, Krylov solvers for \bem require $O(n \cdot N_p^2)$ operations to obtain the unknown vector, where $n$ is the number of iterations to get a desired residual, and is much smaller than $N_p$. The $O(N^2)$ scaling is given by a matrix-vector product (with a dense matrix) done in every iteration; this is the most time-consuming part of the algorithm, and makes \bem prohibitive for more than a few thousand discretization elements. 37 | 38 | But when we inspect the approximation of the integrals in \eqref{eq:layers_element} with Gauss quadrature rules, we see that the matrix-vector product has the form of an $N$-body problem, similar to gravitational potential calculations in planetary systems. In this case, the Gauss quadrature points act analogously to planets (sources of mass) and the collocation points are analogous to the locations where the gravitational potential is computed (target points). There are several ways to accelerate this kind of computations, for example fast-multipole methods \cite{GreengardRokhlin1987}, treecodes \cite{BarnesHut1986}, and fast-Fourier-transform methods \cite{PhillipsWhite1997}. 39 | In our numerical solution (developed as the open-source code \pygbe), we accelerate the $N$-body calculation with a treecode \cite{BarnesHut1986,LiJohnstonKrasny2009}, making this part of the algorithm scale as $O(N\log N)$ rather than $O(N^2)$. 40 | 41 | The treecode algorithm groups the sources and targets in a tree-structured set of boxes and approximates interactions between far-away boxes using a series expansion---a Taylor series, in our case. This allows for controllable accuracy that depends on the number of terms used in the expansion and the multipole-acceptance criterion that defines the threshold where the distance between source and target is far enough to approximate the interactions with expansions. The interactions of targets with close-by sources and expansion centers are computationally intensive calculations, which \pygbe offloads to the \gpu. Details of our implementation of the treecode in \pygbe can be found in our previous work \cite{CooperBarba-share154331}. 42 | -------------------------------------------------------------------------------- /pygbe-orientation/CooperBarba-orientation.tex: -------------------------------------------------------------------------------- 1 | %% ****** Start of file aiptemplate.tex ****** % 2 | %% 3 | %% This file is part of the files in the distribution of AIP substyles for REVTeX4. 4 | %% Version 4.1 of 9 October 2009. 5 | %% 6 | %!TEX encoding = UTF-8 Unicode 7 | % This is a template for producing documents for use with 8 | % the REVTEX 4.1 document class and the AIP substyles. 9 | % 10 | % Copy this file to another name and then work on that file. 11 | % That way, you always have this original template file to use. 12 | 13 | %\documentclass[aip,graphicx]{revtex4-1} 14 | \documentclass[aip,reprint]{revtex4-1} 15 | 16 | \usepackage{amsfonts} 17 | \usepackage{amsmath} 18 | \usepackage{amssymb} 19 | \usepackage{booktabs} 20 | \usepackage{caption} 21 | \usepackage{color} 22 | \usepackage{comment} 23 | \usepackage{graphicx} 24 | \usepackage{hyperref} 25 | %\usepackage[utf8]{inputenc} % allows using accents directly in text, like ÔøΩ 26 | \usepackage{subfig} 27 | \usepackage{xspace} 28 | 29 | 30 | \captionsetup{justification=raggedright, 31 | singlelinecheck=false 32 | } 33 | 34 | \newcommand{\pygbe}{\texttt{PyGBe}\xspace} 35 | \newcommand{\gb}{{\small G\,B1\,D4$^\prime$}\xspace} 36 | \newcommand{\ig}{{\small IgG}} 37 | \newcommand{\pdb}{{\small PDB}\xspace} 38 | \newcommand{\gmres}{\textsc{gmres}\xspace} 39 | \newcommand{\bem}{\textsc{bem}\xspace} 40 | \newcommand{\ses}{\textsc{ses}\xspace} 41 | \newcommand{\sam}{\textsc{sam}} 42 | \newcommand{\gpu}{\textsc{gpu}} 43 | \newcommand{\cpu}{\textsc{cpu}} 44 | \newcommand{\apbs}{\textsc{apbs}\xspace} 45 | \newcommand{\nvidia}{\textsc{nvidia}\xspace} 46 | \newcommand{\msms}{\texttt{\textsc{msms}}\xspace} 47 | \newcommand{\amber}{\texttt{\textsc{amber}}\xspace} 48 | \newcommand{\ccby}{\textsc{cc-by}\xspace} 49 | 50 | \graphicspath{{figs/}} % PATH to figure files-- change to ./ for submission 51 | 52 | 53 | 54 | \begin{document} 55 | 56 | % Use the \preprint command to place your local institutional report number 57 | % on the title page in preprint mode. 58 | % Multiple \preprint commands are allowed. 59 | %\preprint{} 60 | 61 | \title{Probing protein orientation near charged nanosurfaces for simulation-assisted biosensor design} 62 | 63 | 64 | % Explanatory text should go in the []'s, 65 | % actual e-mail address or url should go in the {}'s for \email and \homepage. 66 | % \affiliation command applies to all authors since the last \affiliation command. 67 | % The \affiliation command should follow the other information. 68 | 69 | \author{Christopher D. Cooper} 70 | \email[]{cdcooper@bu.edu,christopher.cooper@usm.cl} 71 | %\thanks{} 72 | \affiliation{Mechanical Engineering, Boston University, Boston, MA} 73 | \affiliation{Mechanical Engineering, Universidad T\'ecnica Federico Santa Mar\'ia, Valpara\'iso, Chile} 74 | 75 | \author{Natalia C. Clementi} 76 | \email[]{ncclementi@gwu.edu} 77 | \affiliation{Mechanical \& Aerospace Engineering, The George Washington University, Washington, DC.} 78 | 79 | \author{Lorena A. Barba} 80 | \email[]{labarba@gwu.edu} 81 | \homepage[]{http://lorenabarba.com/} 82 | \affiliation{Mechanical \& Aerospace Engineering, The George Washington University, Washington, DC.} 83 | 84 | % Collaboration name, if desired (requires use of superscriptaddress option in \documentclass). 85 | % \noaffiliation is required (may also be used with the \author command). 86 | %\collaboration{} 87 | %\noaffiliation 88 | 89 | \date{\today} 90 | 91 | \begin{abstract} 92 | 93 | Protein-surface interactions are ubiquitous in biological processes and bioengineering, yet are not fully understood. 94 | In biosensors, a key factor determining the sensitivity and thus the performance of the device is the orientation of the ligand molecules on the bioactive device surface. Adsorption studies thus seek to determine how orientation can be influenced by surface preparation, varying surface charge and ambient salt concentration. 95 | In this work, protein orientation near charged nanosurfaces is obtained under electrostatic effects using the Poisson-Boltzmann equation, in an implicit-solvent model. 96 | Sampling the free energy for protein \gb at a range of tilt and rotation angles with respect to the charged surface, we calculated the probability of the protein orientations and observed a dipolar behavior. This result is consistent with published experimental studies and combined Monte Carlo and molecular dynamics simulations using this small protein, validating our method. 97 | More relevant to biosensor technology, antibodies such as immunoglobulin G are still a formidable challenge to molecular simulation, due to their large size. 98 | With the Poisson-Boltzmann model, we obtained the probability distribution of orientations for the iso-type {\small IgG2a} at varying surface charge and salt concentration. 99 | This iso-type was not found to have a preferred orientation in previous studies, unlike the iso-type {\small IgG1} whose larger dipole moment was assumed to make it easier to control. 100 | Our results show that the preferred orientation of {\small IgG2a} can be favorable for biosensing with positive charge on the surface of 0.05C/m$^{2}$ or higher and 37mM salt concentration. 101 | The results also show that local interactions dominate over dipole moment for this protein. 102 | Improving immunoassay sensitivity may thus be assisted by numerical studies using our method (and open-source code), guiding changes to fabrication protocols or protein engineering of ligand molecules to obtain more favorable orientations. 103 | 104 | \end{abstract} 105 | 106 | \pacs{}% insert suggested PACS numbers in braces on next line 107 | 108 | \maketitle %\maketitle must follow title, authors, abstract and \pacs 109 | 110 | % Body of paper. 111 | 112 | \section{Introduction}\label{sec:intro} 113 | \input{introduction} 114 | 115 | \section{Implicit-solvent model for proteins near charged surfaces} \label{sec:implicit_solvent} 116 | \input{model} 117 | 118 | \paragraph*{Boundary integral formulation---} \label{sec:bie} 119 | \input{bem} 120 | 121 | 122 | %============= 123 | \section{Methods}\label{sec:methods} 124 | \input{methods} %discretization and treecode 125 | 126 | \subsection{Energy calculation} \label{sec:energy} 127 | \input{energy} 128 | 129 | \subsection{Orientation sampling of a protein near a charged surface} \label{sec:prot_orientation} 130 | \input{prot_orientation} 131 | 132 | 133 | %============= 134 | 135 | \section{Results} \label{sec:results} 136 | \input{results} 137 | 138 | \subsection{Reproducibility and data management} 139 | We have a consistent reproducibility practice that includes releasing code and data associated with a publication. The \pygbe code was released at the time of submitting our previous publication,\cite{CooperBardhanBarba2013} under an MIT open-source license, and we maintain a version-control repository. As with our previous paper, we also release with this work all of the data needed to run the numerical experiments reported here, including running scripts and post-processing code in Python for producing the figures. To support our open-science goals, we prepared such a \emph{``reproducibility package''} for each of the results presented in Figures \ref{fig:1PGB_probability}, \ref{fig:1IGT_convergence}, and the probability plots in Figures \ref{fig:1IGT_negcharge} and \ref{fig:1IGT_poscharge}. The included running scripts invoke the \pygbe code with the correct input data and meshes (also included), and post-process the results to give the final figure, all with just one command. Please see the respective captions for a reference to the reproducibility packages, hosted on the \textbf{figshare} repository. 140 | 141 | \section{Discussion} \label{sec:discussion} 142 | \input{discussion} 143 | 144 | \section{Conclusion} 145 | \input{conclusion} 146 | 147 | 148 | 149 | 150 | \begin{acknowledgments} 151 | This work was supported by ONR via grant \#N00014-11-1-0356 of the Applied Computational Analysis Program. LAB also acknowledges support from NSF CAREER award OCI-1149784 and from NVIDIA, Inc.\ via the CUDA Fellows Program. 152 | We are grateful for many helpful conversations with members of the Materials and Sensors Branch of the Naval Research Laboratory, especially Dr. Jeff M. Byers and Dr. Marc Raphael. 153 | \end{acknowledgments} 154 | 155 | % Create the reference section using BibTeX: 156 | \bibliography{compbio,bem,scicomp,fastmethods,scbib,biosensors} 157 | 158 | \end{document} 159 | -------------------------------------------------------------------------------- /pygbe2-CPC/CooperBarba2014.tex: -------------------------------------------------------------------------------- 1 | %% Use the option review to obtain double line spacing 2 | %% \documentclass[preprint,review,12pt]{elsarticle} 3 | %% Use the options 1p,twocolumn; 3p; 3p,twocolumn; 5p; or 5p,twocolumn 4 | %% for a journal layout: 5 | %% \documentclass[final,1p,times]{elsarticle} 6 | %% \documentclass[final,1p,times,twocolumn]{elsarticle} 7 | %% \documentclass[final,3p,times]{elsarticle} 8 | %%\documentclass[final,3p,times,twocolumn]{elsarticle} 9 | %% \documentclass[final,5p,times]{elsarticle} 10 | \documentclass[final,5p,times,twocolumn]{elsarticle} 11 | 12 | \usepackage{amsfonts} 13 | \usepackage{amsmath} 14 | \usepackage{amssymb} 15 | \usepackage{booktabs} 16 | \usepackage{caption} 17 | \usepackage{color} 18 | \usepackage{comment} 19 | \usepackage{graphicx} 20 | \usepackage{hyperref} 21 | \usepackage[utf8]{inputenc} % allows using accents directly in text, like ÔøΩ 22 | \usepackage{subfig} 23 | \usepackage{xspace} 24 | 25 | 26 | \captionsetup{justification=raggedright, 27 | singlelinecheck=false 28 | } 29 | 30 | \newcommand{\pygbe}{\texttt{PyGBe}\xspace} 31 | \newcommand{\gb}{{\small G\,B1\,D4$^\prime$}\xspace} 32 | \newcommand{\gmres}{\textsc{gmres}\xspace} 33 | \newcommand{\bem}{\textsc{bem}\xspace} 34 | \newcommand{\ses}{\textsc{ses}\xspace} 35 | \newcommand{\sam}{\textsc{sam}} 36 | \newcommand{\gpu}{\textsc{gpu}} 37 | \newcommand{\cpu}{\textsc{cpu}} 38 | \newcommand{\apbs}{\textsc{apbs}\xspace} 39 | \newcommand{\nvidia}{\textsc{nvidia}\xspace} 40 | \newcommand{\msms}{\texttt{\textsc{msms}}\xspace} 41 | \newcommand{\amber}{\texttt{\textsc{amber}}\xspace} 42 | \newcommand{\ccby}{\textsc{cc-by}\xspace} 43 | 44 | \graphicspath{{figs/}} % PATH to figure files-- change to ./ for submission 45 | 46 | %% The lineno packages adds line numbers. Start line numbering with 47 | %% \begin{linenumbers}, end it with \end{linenumbers}. Or switch it on 48 | %% for the whole article with \linenumbers after \end{frontmatter}. 49 | %% \usepackage{lineno} 50 | 51 | %% natbib.sty is loaded by default. However, natbib options can be 52 | %% provided with \biboptions{...} command. Following options are 53 | %% valid: 54 | 55 | %% round - round parentheses are used (default) 56 | %% square - square brackets are used [option] 57 | %% curly - curly braces are used {option} 58 | %% angle - angle brackets are used