Editing Lattice protein

'''Lattice proteins''' are highly simplified models of protein-like heteropolymer chains on lattice conformational space which are used to investigate [[protein folding]].<ref name=":2">{{Cite journal | vauthors = Lau KF, Dill KA |year=1989 |title=A lattice statistical mechanics model of the conformational and sequence spaces of proteins |journal=Macromolecules |volume=22 |issue=10 |pages=3986–97 |doi=10.1021/ma00200a030 |bibcode=1989MaMol..22.3986L }}</ref> Simplification in lattice proteins is twofold: each whole residue ([[amino acid]]) is modeled as a single "bead" or "point" of a finite set of types (usually only two), and each residue is restricted to be placed on vertices of a (usually cubic) [[lattice (group)|lattice]].<ref name=":2" /> To guarantee the connectivity of the protein chain, adjacent residues on the backbone must be placed on adjacent vertices of the lattice.<ref name=":1" /> Steric constraints are expressed by imposing that no more than one residue can be placed on the same lattice vertex.<ref name=":1" />

Because proteins are such [[Macromolecule|large molecules]], there are severe computational limits on the simulated timescales of their behaviour when modeled in all-atom detail. The [[millisecond]] regime for all-atom simulations was not reached until 2010,<ref>{{cite journal | vauthors = Voelz VA, Bowman GR, Beauchamp K, Pande VS | title = Molecular simulation of ab initio protein folding for a millisecond folder NTL9(1-39) | journal = Journal of the American Chemical Society | volume = 132 | issue = 5 | pages = 1526–8 | date = February 2010 | pmid = 20070076 | pmc = 2835335 | doi = 10.1021/ja9090353 }}</ref> and it is still not possible to fold all real proteins on a computer. Simplification significantly reduces the computational effort in handling the model, although even in this simplified scenario the protein folding problem is [[NP-completeness|NP-complete]].<ref>{{cite journal | vauthors = Berger B, Leighton T | title = Protein folding in the hydrophobic-hydrophilic (HP) model is NP-complete | journal = Journal of Computational Biology | volume = 5 | issue = 1 | pages = 27–40 | year = 1998 | pmid = 9541869 | doi = 10.1089/cmb.1998.5.27 }}</ref>

== Overview ==
Different versions of lattice proteins may adopt different types of lattice (typically square and triangular ones), in two or three dimensions, but it has been shown that generic lattices can be used and handled via a uniform approach.<ref name=":1">{{cite journal | vauthors = Bechini A | title = On the characterization and software implementation of general protein lattice models | journal = PLOS ONE | volume = 8 | issue = 3 | pages = e59504 | year = 2013 | pmid = 23555684 | pmc = 3612044 | doi = 10.1371/journal.pone.0059504 | bibcode = 2013PLoSO...859504B | doi-access = free }}</ref>

Lattice proteins are made to resemble real proteins by introducing an ''[[energy function]]'', a set of conditions which specify the [[interaction energy]] between beads occupying adjacent lattice sites.<ref name=":0" /> The energy function mimics the interactions between amino acids in real proteins, which include [[steric hindrance|steric]], [[hydrophobic effect|hydrophobic]] and [[hydrogen bonding]] effects.<ref name=":1" /> The beads are divided into types, and the energy function specifies the interactions depending on the bead type, just as different types of amino acids interact differently.<ref name=":0" /> One of the most popular lattice models, the hydrophobic-polar model ([[HP model]]),<ref>{{cite journal | vauthors = Dill KA | title = Theory for the folding and stability of globular proteins | journal = Biochemistry | volume = 24 | issue = 6 | pages = 1501–9 | date = March 1985 | pmid = 3986190 | doi = 10.1021/bi00327a032 }}</ref> features just two bead types—[[hydrophobic]] (H) and [[dipole|polar]] (P)—and mimics the [[hydrophobic effect]] by specifying a favorable interaction between H beads.<ref name=":0" />

For any sequence in any particular structure, an energy can be rapidly calculated from the energy function. For the simple HP model, this is an enumeration of all the contacts between H residues that are adjacent in the structure but not in the chain.<ref>{{Cite book | vauthors = Su SC, Lin CJ, Ting CK  | date = December 2010 | pages = 51–56 | publisher = IEEE | doi = 10.1109/BIBMW.2010.5703772 | isbn = 978-1-4244-8303-7 | chapter = An efficient hybrid of hill-climbing and genetic algorithm for 2D triangular protein structure prediction | title = 2010 IEEE International Conference on Bioinformatics and Biomedicine Workshops (BIBMW) | s2cid = 44932436 }}</ref> Most researchers consider a lattice protein sequence ''protein-like'' only if it possesses a single structure with an energetic state lower than in any other structure, although there are exceptions that consider ensembles of possible folded states.<ref>{{cite journal |last1=Bertram |first1=Jason |last2=Masel |first2=Joanna |title=Evolution Rapidly Optimizes Stability and Aggregation in Lattice Proteins Despite Pervasive Landscape Valleys and Mazes |journal=Genetics |date=April 2020 |volume=214 |issue=4 |pages=1047–1057 |doi=10.1534/genetics.120.302815|pmid=32107278 |pmc=7153934 |doi-access=free }}</ref> This is the energetic ground state, or [[native state]]. The relative positions of the beads in the native state constitute the lattice protein's [[Protein tertiary structure|tertiary structure]]{{Citation needed|date=December 2018}}. Lattice proteins do not have genuine [[Protein secondary structure|secondary structure]]; however, some researchers have claimed that they can be extrapolated onto real protein structures which do include secondary structure, by appealing to the same law by which the [[phase diagram]]s of different substances can be scaled onto one another (the [[theorem of corresponding states]]).<ref>{{cite journal | vauthors = Onuchic JN, Wolynes PG, Luthey-Schulten Z, Socci ND | title = Toward an outline of the topography of a realistic protein-folding funnel | journal = Proceedings of the National Academy of Sciences of the United States of America | volume = 92 | issue = 8 | pages = 3626–30 | date = April 1995 | pmid = 7724609 | pmc = 42220 | doi = 10.1073/pnas.92.8.3626 | bibcode = 1995PNAS...92.3626O | doi-access = free }}</ref>

By varying the energy function and the bead sequence of the chain (the [[Protein primary structure|primary structure]]), effects on the native state structure and the [[chemical kinetics|kinetics]] of folding can be explored, and this may provide insights into the folding of real proteins.<ref>{{cite journal | vauthors = Moreno-Hernández S, Levitt M | title = Comparative modeling and protein-like features of hydrophobic-polar models on a two-dimensional lattice | journal = Proteins | volume = 80 | issue = 6 | pages = 1683–93 | date = June 2012 | pmid = 22411636 | pmc = 3348970 | doi = 10.1002/prot.24067 }}</ref> Some of the examples include study of folding processes in lattice proteins that have been discussed to resemble the two-phase folding kinetics in proteins. Lattice protein was shown to have quickly collapsed into compact state and followed by slow subsequent structure rearrangement into native state.<ref>{{Cite journal|last1=Socci|first1=Nicholas D.|last2=Onuchic|first2=José Nelson | name-list-style = vanc |date=1994-07-15|title=Folding kinetics of proteinlike heteropolymers|journal=The Journal of Chemical Physics|volume=101|issue=2|pages=1519–1528|doi=10.1063/1.467775|issn=0021-9606|arxiv=cond-mat/9404001|bibcode=1994JChPh.101.1519S |s2cid=10672674}}</ref> Attempts to resolve [[Levinthal's paradox|Levinthal paradox]] in protein folding are another efforts made in the field. As an example, study conducted by Fiebig and Dill examined searching method involving constraints in forming residue contacts in lattice protein to provide insights to the question of how a protein finds its native structure without global exhaustive searching.<ref>{{Cite journal|last1=Fiebig|first1=Klaus M.|last2=Dill|first2=Ken A. | name-list-style = vanc |date=1993-02-15|title=Protein core assembly processes |journal=The Journal of Chemical Physics|volume=98|issue=4|pages=3475–3487|doi=10.1063/1.464068 |bibcode=1993JChPh..98.3475F }}</ref> Lattice protein models have also been used to investigate the [[energy landscape]]s of proteins, i.e. the variation of their internal [[Thermodynamic free energy|free energy]] as a function of conformation.{{Citation needed|date=December 2018}}

== Lattices ==
A [[Lattice (discrete subgroup)|lattice]] is a set of orderly points that are connected by "edges".<ref name=":1" /> These points are called vertices and are connected to a certain number other vertices in the lattice by edges. The number of vertices each individual vertex is connected to is called the [[coordination number]] of the lattice, and it can be scaled up or down by changing the shape or [[dimension]] (2-dimensional to 3-dimensional, for example) of the lattice.<ref name=":1" /> This number is important in shaping the characteristics of the lattice protein because it controls the number of other [[Amino acid|residues]] allowed to be adjacent to a given residue.<ref name=":1" /> It has been shown that for most proteins the coordination number of the lattice used should fall between 3 and 20, although most commonly used lattices have coordination numbers at the lower end of this range.<ref name=":1" />

Lattice shape is an important factor in the accuracy of lattice protein models. Changing lattice shape can dramatically alter the shape of the energetically favorable conformations.<ref name=":1" /> It can also add unrealistic constraints to the protein structure such as in the case of the [[Parity (mathematics)|parity]] problem where in square and cubic lattices residues of the same parity (odd or even numbered) cannot make hydrophobic contact.<ref name=":0" /> It has also been reported that triangular lattices yield more accurate structures than other lattice shapes when compared to [[X-ray crystallography|crystallographic]] data.<ref name=":1" /> To combat the parity problem, several researchers have suggested using triangular lattices when possible, as well as a square matrix with diagonals for theoretical applications where the square matrix may be more appropriate.<ref name=":0" /> Hexagonal lattices were introduced to alleviate sharp turns of adjacent residues in triangular lattices.<ref>{{cite journal | vauthors = Jiang M, Zhu B | title = Protein folding on the hexagonal lattice in the HP model | journal = Journal of Bioinformatics and Computational Biology | volume = 3 | issue = 1 | pages = 19–34 | date = February 2005 | pmid = 15751110 | doi = 10.1142/S0219720005000850 }}</ref> Hexagonal lattices with diagonals have also been suggested as a way to combat the parity problem.<ref name=":1" />

== Hydrophobic-polar model ==
[[File:Thermodynamically Stable Lattice Protein.jpg|thumb|A schematic of a thermodynamically stable conformation of a generic polypeptide. Note the high number of hydrophobic contacts. amino acid residues are represented as dots along the white line. Hydrophobic residues are in green while polar residues are in blue. See also [https://math.hawaii.edu/~bjoern/?page=labbyfold&string=111111000011000001111000111111&moves=wwwwdssdwwdsssaasdddwwwwdssss&xs=40&ys=40&sheets=1 this example in LabbyFold]]]
[[File:Thermodynamically Unstable Lattice Protein.jpg|thumb|A schematic of a thermodynamically unstable conformation of a generic polypeptide. Note the lower number of hydrophobic contacts than above. Hydrophobic residues are in green and polar residues are in blue. See also [https://math.hawaii.edu/~bjoern/?page=labbyfold&string=111111000011000001111000111111&moves=wwwwdssssdwwwwdssssdwwwwdssss&xs=40&ys=40&sheets=1 this example in LabbyFold]]]
The [[Hydrophobic-polar protein folding model|hydrophobic-polar]] [[protein]] model is the original lattice protein model. It was first proposed by Dill et al. in 1985 as a way to overcome the significant cost and difficulty of predicting protein structure, using only the [[Hydrophobe|hydrophobicity]] of the [[amino acid]]s in the protein to predict the protein structure.<ref name=":0">{{cite journal | vauthors = Dubey SP, Kini NG, Balaji S, Kumar MS | title = A Review of Protein Structure Prediction Using Lattice Model | journal = Critical Reviews in Biomedical Engineering | volume = 46 | issue = 2 | pages = 147–162 | date = 2018 | pmid = 30055531 | doi = 10.1615/critrevbiomedeng.2018026093 }}</ref> It is considered to be the paradigmatic lattice protein model.<ref name=":1" /> The method was able to quickly give an estimate of protein structure by representing proteins as "short chains on a 2D square lattice" and has since become known as the hydrophobic-polar model. It breaks the protein folding problem into three separate problems: modeling the protein conformation, defining the energetic properties of the amino acids as they interact with one another to find said conformation, and developing an efficient algorithm for the prediction of these conformations. It is done by classifying amino acids in the protein as either hydrophobic or polar and assuming that the protein is being [[Protein folding|folded]] in an [[Aqueous solution|aqueous]] environment. The lattice statistical model seeks to recreate protein folding by minimizing the [[Gibbs free energy|free energy]] of the contacts between hydrophobic amino acids. Hydrophobic amino acid residues are predicted to group around each other, while hydrophilic residues interact with the surrounding water.<ref name=":0" />

Different lattice types and algorithms were used to study protein folding with HP model. Efforts were made to obtain higher approximation ratios using [[approximation algorithm]]s in 2 dimensional and 3 dimensional, square and triangular lattices. Alternative to approximation algorithms, some [[genetic algorithm]]s were also exploited with square, triangular, and face-centered-cubic lattices.<ref>{{cite journal | vauthors = Shaw D, Shohidull Islam AS, Sohel Rahman M, Hasan M | title = Protein folding in HP model on hexagonal lattices with diagonals | journal = BMC Bioinformatics | volume = 15 Suppl 2 | issue = 2 | pages = S7 | date = 2014-01-24 | pmid = 24564789 | pmc = 4016602 | doi = 10.1186/1471-2105-15-S2-S7 | doi-access = free }}</ref>

== Problems and alternative models ==
The simplicity of the hydrophobic-polar model has caused it to have several problems that people have attempted to correct with alternative lattice protein models.<ref name=":0" /> Chief among these problems is the issue of [[Degeneracy (mathematics)|degeneracy]], which is when there is more than one minimum energy [[Protein structure|conformation]] for the modeled protein, leading to uncertainty about which conformation is the native one. Attempts to address this include the HPNX model which classifies amino acids as hydrophobic (H), [[Ion|positive]] (P), negative (N), or neutral (X) according to the [[Ion|charge]] of the amino acid,<ref>{{cite journal | vauthors = Backofen R, Will S, Bornberg-Bauer E | title = Application of constraint programming techniques for structure prediction of lattice proteins with extended alphabets | journal = Bioinformatics | volume = 15 | issue = 3 | pages = 234–42 | date = March 1999 | pmid = 10222411 | doi = 10.1093/bioinformatics/15.3.234 | doi-access = free }}</ref> adding additional parameters to reduce the number of [[Gibbs free energy|low energy]] conformations and allowing for more realistic protein simulations.<ref name=":0" /> Another model is the Crippen model which uses protein characteristics taken from [[X-ray crystallography|crystal structures]] to inform the choice of native conformation.<ref>{{cite journal | vauthors = Crippen GM | title = Prediction of protein folding from amino acid sequence over discrete conformation spaces | journal = Biochemistry | volume = 30 | issue = 17 | pages = 4232–7 | date = April 1991 | pmid = 2021616 | doi = 10.1021/bi00231a018 }}</ref>

Another issue with lattice models is that they generally don't take into account the space taken up by amino acid [[side chain]]s, instead considering only the [[Amino acid|α-carbon]].<ref name=":1" /> The side chain model addresses this by adding a side chain to the vertex adjacent to the α-carbon.<ref>{{cite journal | vauthors = Dill KA, Bromberg S, Yue K, Fiebig KM, Yee DP, Thomas PD, Chan HS | title = Principles of protein folding--a perspective from simple exact models | journal = Protein Science | volume = 4 | issue = 4 | pages = 561–602 | date = April 1995 | pmid = 7613459 | pmc = 2143098 | doi = 10.1002/pro.5560040401 }}</ref>{{clear}}

== References ==
{{reflist}}

[[Category:Protein structure]]
[[Category:NP-complete problems]]