[This article was first published on r on Everyday Is A School Day, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)Want to share your content on R-bloggers? click here if you have a blog, or here if you don't. Back to basics! Learning non-polar amino acids, what zwitterions actually are, and dipping into the applied math — Rodrigues rotation and Lennard-Jones potential. Slowly building toward optimal phi/psi!Motivations We’ve explored quite a bit lately in molecular dynamic simulation and then protein-protein docking as well the last time. There is still so much to learn. I’ve decided to go back to basics, revisiting our old friends amino acids and try to understand the natural properties behind each one and see if that will make more sense in the future when we’re exploring more. While making notes for myself of all the amino acids, I’ll also try to understand some of the basic math behind the structures. Are you ready !? Lol, I’m not, but let’s go anyway! Objectives: Amino AcidsNon-polarRodriguez Rotation FormulaLennard-Jone Potential EnegeryBond StretchBond AngleProper dihedralNon-bondedCalculating LJOpportunities For ImprovementLessons LearntAmino Acids Amino acids are the building blocks of proteins, each sharing a common backbone: a central α-carbon bonded to an amino group (–NH₂), a carboxyl group (–COOH), a hydrogen atom, and a variable side chain (R group) that defines each amino acid’s identity and chemistry.Non-polar amino acids Non-polar amino acids have hydrophobic side chains — they avoid water and tend to cluster in the interior of folded proteins, forming the hydrophobic core that drives protein stability. Understanding each one’s shape and bulk is directly relevant to how they pack, how they constrain backbone flexibility, and how substitutions affect enzyme active sites.library(tibble)library(kableExtra)aa_nonpolar dplyr::select(aa:mw_da) |> kbl()aaaa3namefunctional_groupsmiles_sidechaincharge_ph7mw_daGGlyGlycineH (none)[H]Neutral75.03AAlaAlanineMethylCNeutral89.09VValValineIsopropylCC(C)Neutral117.15LLeuLeucineIsobutylCCC(C)CNeutral131.17IIleIsoleucinesec-ButylCCC(C)Neutral131.17PProProlinePyrrolidine ringC1CCNC1Neutral115.13FPhePhenylalanineBenzylCc1ccccc1Neutral165.19WTrpTryptophanIndolylmethylCc1c[nH]c2ccccc12Neutral204.23MMetMethionineThioetherCCSCNeutral149.20aa_nonpolar |> dplyr::select(aa,aa3,md_note,main_function) |> kbl()aaaa3md_notemain_functionGGlyMinimal VDW radius; unrestricted phi/psi; near-zero excluded volumeConformational flexibility; tight turns; active site geometryAAlaLow steric perturbation; high alpha-helix propensity in force fieldsHelix former; hydrophobic core; alanine-scanning mutagenesisVValBeta-branching restricts psi; favors extended beta-sheet; large gamma-carbonsBeta-sheet core; hydrophobic packing; sickle-cell HbS Glu6ValLLeuFlexible chi2; common rotamers at -65/-65 and -65/175; high hydrophobic SASAHydrophobic core; leucine zippers; most abundant non-polar in proteomesIIleBeta-branching + gamma-branch; most restricted chi1/chi2; large buried SASAHydrophobic core; beta-barrel interiors; transmembrane helicesPProFixed phi ~-60; no backbone NH donor; cis/trans isomerism at Xaa-Pro bondHelix breaker; beta-turns; collagen Gly-Pro-X repeatsFPheRigid aromatic ring; pi-pi stacking and cation-pi in MD energy decompositionHydrophobic core; aromatic clusters; ligand binding pocketsWTrpIndole NH can H-bond; amphipathic at membrane interface; strong 280nm absorbanceMembrane anchoring; fluorescence probe; ligand binding; rarest standard AAMMetFlexible sulfur geometry; oxidizable to sulfoxide in long MD runs; check reactive FFTranslation initiation; hydrophobic core; redox sensingClaude generated most of the above information. We’ll add onto the md_note section as we encounter certain things during our MD sims.What’s Zwitterion? A zwitterion is a molecule that has both positive and negative charges but is overall electrically neutral. In amino acids, the amino group (–NH₂) can accept a proton to become positively charged (–NH₃⁺), while the carboxyl group (–COOH) can lose a proton to become negatively charged (–COO⁻). At physiological pH (~7.4), most amino acids exist as zwitterions, with the amino group protonated and the carboxyl group deprotonated. This dual charge allows amino acids to interact with both polar and non-polar environments, contributing to their solubility in water and their ability to form various interactions in proteins.What Does Non-polar Actually Mean? It is worth clarifying what “non-polar” actually refers exclusively to the side chain (R group) — specifically that it consists largely of carbon and hydrogen bonds with no net dipole and no ionizable groups, making it hydrophobic and largely indifferent to water. It says nothing about the backbone, which is the same for all amino acids and always carries polar bonds (C=O, N–H). In fact, as mentioned above, all amino acids including non-polar ones exist as zwitterions at physiological pH — a property that comes entirely from the backbone, not the side chain.Note to self: All amino acids’ backbones are zwitterions; the R-side chain determines polarity and hydrophobicity. Also, net charge neutral == overall charges equals zero, does not mean the molecule is non-polar.Rodriguez Rotation Formula Rodrigues’ rotation formula is a method for rotating a 3D vector in space around a specified axis by a given angle. The formula is expressed as:\(v_{rotation} = v.\cos(\theta) + \sin(\theta)(k \times v) + (1 - \cos(\theta))(k(k \cdot v))\)where v is the original vector, k is the unit vector along the axis of rotation, and θ is the angle of rotation in radians.This formula apparently is very popular in computer graphics and robotics, but I can see how it can be useful in molecular dynamics as well when we want to rotate a molecule or a part of it around an axis. Especially when we want to estimate the least energy conformation of a molecule. The direction application of this formula in amino acid sequence would be in rearranging the atoms based on phi and psi which are the angles of rotations around the N-Cα and Cα-C bonds of the amino acid backbone, respectively. By applying Rodrigues’ rotation formula, we can calculate the new positions of the atoms in the amino acid after rotating them by the specified angles, allowing us to explore different conformations of the molecule. How I remember which angle is which is Nancy Phi (sounds like some detective show and also N->C) and C C Psi (All with S sound, also Carbon to carbon). We’ll leave the hand calculation until next time, but let’s learn how to rotate a coordinate based on an axis with Rodriguez!Below I’ll write the code first, then explain. Please feel free to use your mouse to hover over the plotly object and check out the coordinates.library(plotly)library(pracma)#### Let's start simplex1