Evolving LSystems to Capture Protein Structure Native Conformations

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Evolving L-Systems to Capture Protein
StructureNative Conformations

• Gabi Escuela1, Gabriela Ochoa2 and Natalio
  Krasnogor3
• 1,2 Department of Computer Science, Universidad
  Simon Bolivar, Caracas, Venezuela
• 1gabiescuela_at_netuno.net.ve, 2gabro_at_ldc.usb.ve
• 3 School of Computer Science and I.T., University
  of Nottingham
• Natalio.Krasnogor_at_nottingham.ac.uk

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Content

• Proteins
• Protein Structure Prediction (PSP)
• The HP model
• EA approaches to PSP current encoding
• L-Systems
• Why a grammatical encoding?
• Methods and Results
• Discussion and Future Work

3D structure of myoglobin, showing coloured alpha
helices.
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Proteins

• Linear chains of 30-400 units from 20 different
  amino acids
• Fold into a unique functional structure native
  state or tertiary structure

Show repeated substructures alpha helices and
beta sheets
1A8M 3-D Structure
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Protein Structure Prediction (PSP)

• Goal Determining the 3D structure of proteins
  from their amino acid sequences
• Strategy find an amino acid chain's state of
  minimum energy
• Solution will have practical consequences in
  medicine, drug development and agriculture

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The 2D HP Model

• Hydrophobic effect is the main force governing
  folding
• q ?H, P, each letter of q has to be put in
  vertex of a given lattice L (at each point turn
  90º Left or Right, or continue ahead)
• Scoring function adds -1 for each contact
  between two Hs adjacent in the lattice that are
  not consecutive in q

2 Amino acids types hydrophobic (H) and polar or
hydrophilic (P)
HPHPPHHPHPPHPHHPPHPH
Square Lattice
9 H-H bonds Score -9

• Objective Find the organization (embedding) of q
  in L of minimum score (maximum contacts)

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EA approaches to PSP Current (Direct) Encoding

• EAs and other stochastic methods global
  optimization of a suitable energy function
• Encoding Cartesian Coordinates, Distance
  Geometries, Internal Coordinates
• Absolute structure encoded as a string of
  symbols. For example In the 2D Square
• s Up, Down, Left, Right
• Relative each move is interpreted in terms of
  the previous one
• s Forward, TurnLeft, TurnRight

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Protein HPHPPHHPHPPHPHHPPHPH L 20
Absolute Encoding
RDDLULDLDLUURULURRD L 19
R
D
D
L
First position is fixed
Relative Encoding
RFRRLLRLRRFRLLRRFR L 18
R
R
F
R
First and second position are fixed
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L-Systems (Lindenmayer, 1968)

• A model of morphogenesis, based on formal
  grammars
• Rewriting Define complex objects by replacing
  parts of a simple object using a set of
  productions.

• Symbols F, f, , -, ,
• Axiom (S)
• Production (replacement) rules

r1
S F
r2
f
F
F
start
Ff
1
2
3
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Why a Grammatical Encoding?

• Specifies how to construct the phenotype
• Can achieve greater scalability through
  self-similar and hierarchical structure
• Proteins exhibit high degree of regularity, and
  repeated motifs
• Current encoding may not be suitable for
  crossover and building block transfer between
  individuals

Protein Structure
3D L-System
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Method

• Prove of principle Can a folded protein be
  captured (encoded) by an L-system?
• How to find that L-system An EA used to evolve
  an L-system that capture a folded protein
  (inverse problem)

Output L-system L that once derived, will
produce the target string RFRRLLRLRRFRLLRRFR
Input Folded structure in Relative
Coordinates RFRRLLRLRRFRLLRRFR
EA
Axiom 01F Rules 0RFR1, 12L2, 2R0L
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Proposed Grammatical Encoding

• D0L-system (deterministic and context free)
• Alphabet ??t ? ?nt
• ?tF,L,R terminal symbols (relative coord.)
  ?nt0,1,2,...,m-1 non-terminal symbols
  (rewriting rules), m max. number of rules
• Axiom a ? ?
• Rewriting rules i wi , where i ? ?nt and wi ?
  ?

axiom R2 rules 0R03F 1R01L
2F310 3LRL3
Example
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Evolutionary Algorithm

• Generational with rank based selection
• Randomly generated initial population
• Prefixed maximum number of rules
• Axiom and Rules randomly generated strings of
  prefixed maximum length
• Genetic operators
• Uniform-like (homologous) recombination (rate
  1.0) complete production rules are interchanged
• Per symbol mutation in both axioms and rules
  (deletion (30), insertion (10),
  modification(60))

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Derivation, and Fitness Function

• Derivation from genotype (axiom and rules) to
  phenotype (folded structure)
• Post-processing non-terminal symbols pruning
• Fitness calculation number of matches between
  the target string and the solution Min. 0, Max
  length of the desired folding.

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Results (1)
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Results (2)
Evolutionary progression towards the target
structure
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Discussion

• The proposed EA discovered L-systems that capture
  a target folding under the HP model in 2D
  lattices
• We are not solving the PSP yet, but ..
• We are proposing a novel and potentially useful,
  generative encoding for evolutionary approaches
  to PSP

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Future work

• Incorporate problem knowledge about secondary
  structures

Beta Turn
Beta Sheet
Alpha Helix

• Explore longer chains and 3D lattices