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Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model J an Ma nuch Arvind Gupta Mehdi Karimi Alireza Hadj Khodabakhshi Arash Rafiey Simon Fraser University, Canada J an Ma Design of

SLIDE 1

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model

J´ an Maˇ nuch Arvind Gupta Mehdi Karimi Alireza Hadj Khodabakhshi Arash Rafiey Simon Fraser University, Canada

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 1/17

SLIDE 2

Proteins

Protein is a polymer constructed from a linear sequence (chain) of amino acids.

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 2/17

SLIDE 3

Proteins

Protein is a polymer constructed from a linear sequence (chain) of amino acids. When placed into a solvent it will fold into a unique 3D spatial structure with minimal energy. The structure (shape) determines the function of the protein.

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 2/17

SLIDE 4

Protein Folding

Many forces act on the protein which contribute to changes in free energy including: hydrogen bonding, van der Waals interactions, intrinsic propensities, ion pairing, disulphide bonds, hydrophobic interactions.

f0; 1g 1

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 3/17

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Protein Folding

Many forces act on the protein which contribute to changes in free energy including: hydrophobic interactions — most significant, cf. Dill (1990)

f0; 1g 1

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 3/17

SLIDE 6

Protein Folding

Many forces act on the protein which contribute to changes in free energy including: hydrophobic interactions — most significant, cf. Dill (1990) Amino acids are of two types: hydrophobic or polar depending

n their affinity to water.

Hence, we can model proteins as sequences over

f0; 1g,

where

1 represents a hydrophobic amino acid, and 0 represents a hydrophilic (polar) amino acid.

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 3/17

SLIDE 7

Hydrophobic-Polar Model

introduced by Dill (1985) the protein sequence is embedded on a lattice (e.g., 2D: square, 3D: cubic) with each amino acid occupying exactly

ne node and consecutive amino acids occupy

neighboring nodes

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 4/17

SLIDE 8

Hydrophobic-Polar Model

introduced by Dill (1985) the protein sequence is embedded on a lattice (e.g., 2D: square, 3D: cubic) with each amino acid occupying exactly

ne node and consecutive amino acids occupy

neighboring nodes Example in 2D square lattice: protein:

p = 011001101000011 101 00 11

– depicts a polar “

0” amino acid

– depicts a hydrophobic “

1” amino acid

– depicts a peptide bond between neighboring amino acids in the sequence

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 4/17

SLIDE 9

Hydrophobic-Polar Model

introduced by Dill (1985) the protein sequence is embedded on a lattice (e.g., 2D: square, 3D: cubic) with each amino acid occupying exactly

ne node and consecutive amino acids occupy

neighboring nodes free energy of the fold — in the HP model only hydrophobic interactions (contacts) are considered Example in 2D square lattice:

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 4/17

SLIDE 10

Hydrophobic-Polar Model

the protein sequence is embedded on a lattice (e.g., 2D: square, 3D: cubic) with each amino acid occupying exactly

ne node and consecutive amino acids occupy

neighboring nodes free energy of the fold — in the HP model only hydrophobic interactions (contacts) are considered a fold with minimal free energy corresponds to a fold with the largest number of (hydrophobic) contacts Example in 2D square lattice: 8 contacts (maximum possible) — native fold

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 4/17

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Folding in HP model is still hard!

To find a native fold is NP-hard for 2D square lattice, cf. Crescenzi, Goldman, Papadimitriou, Piccolboni, Yannakakis (1998) 3D square lattice, cf. Berger, Leighton (1998)

3=8 6=11 3=5

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 5/17

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Folding in HP model is still hard!

To find a native fold is NP-hard for 2D square lattice, cf. Crescenzi, Goldman, Papadimitriou, Piccolboni, Yannakakis (1998) 3D square lattice, cf. Berger, Leighton (1998) there are linear time approximations with approximating factor

3=8 for 3D square lattice, cf. Hart, Istrail (1995) 6=11 and 3=5 for 2D and 3D triangular lattices, cg.

Agarwala, Batzoglou, Danˇ cík, Decatur, Farach, Hannenhalli, Skiena (1997)

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 5/17

SLIDE 13

Inverse protein folding

In many applications such as drug design, nanotechnology, we are interested in the complement problem to protein folding: For a target fold/shape, find a protein sequence whose native fold is the target. target fold

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 6/17

SLIDE 14

Inverse protein folding

In many applications such as drug design, nanotechnology, we are interested in the complement problem to protein folding: For a target fold/shape, find a protein sequence whose native fold is the target. A variation of this problem is hard in 3D square lattice: NP-complete for 3D square lattices, cf. Hart (1997) “The problem”: given constant

E and K, find a sequence

with at most

K hydrophobic (“1”) amino acids and score at

least

this problem is polynomial for 2D square lattice, cf. Berman et al. (2004)

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 6/17

SLIDE 15

Inverse protein folding

In many applications such as drug design, nanotechnology, we are interested in the complement problem to protein folding: For a target fold/shape, find a protein sequence whose native fold is the target. We are interested in a more natural formulation of this problem: For a given shape find a protein with a native fold approximating the shape.

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 6/17

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Designs in 2D square lattice

Gupta, M., Stacho, JCB (2005):

Theorem. (Constructible structures)

Connecting many basic building blocks: we can approximate any given shape:

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 7/17

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Saturated folds

Why it works? The protein string does not start or end with

Thus, in 2D square lattice, every

1 can have at most two

hydrophobic contacts. Since our designs have the maximal number of contacts (2), i.e., they must be native: We call such folds “saturated” folds.

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 8/17

SLIDE 18

Stability

Example in 2D square lattice: protein:

p = 011001101000011 10 10 01 10

has 82 native folds:

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 9/17

SLIDE 19

Stable designs in 2D square lattice

In article “Prototeins” (1998) in American Scientist, Brian Hayes posed an open problem: Is it possible to design stable sequences of all pos- sible length in HP model for 2D square lattice?

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 10/17

SLIDE 20

Stable designs in 2D square lattice

In article “Prototeins” (1998) in American Scientist, Brian Hayes posed an open problem: Is it possible to design stable sequences of all pos- sible length in HP model for 2D square lattice? Aichholzer et al. (2001/2006): designed stable cyclic sequences for all even length; and stable sequences for all length divisible by 4 in 2D square lattice. Also showed examples of sequences with exponentially many folds.

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 10/17

SLIDE 21

Stable designs in 2D square lattice

In article “Prototeins” (1998) in American Scientist, Brian Hayes posed an open problem: Is it possible to design stable sequences of all pos- sible length in HP model for 2D square lattice? Aichholzer et al. (2001/2006): designed stable cyclic sequences for all even length; and stable sequences for all length divisible by 4 in 2D square lattice. Also showed examples of sequences with exponentially many folds. Li, Zhang, Chen (2005): designed infinite family of stable sequence in 2D triangular lattice.

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 10/17

SLIDE 22

Stability of constructible structures

Gupta, M., Stacho, JCB (2005): We conjecture that all constructible structures are stable. We proved it for arbitrary long structures forming

I- and L-shapes.

We verified by computer that all constructible structures with up to 9 tiles are stable.

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 11/17

SLIDE 23

Hexagonal prism lattice

Degree of the lattice is 5. Assuming no

1 at the end of the string, each 1 can have at

most three contacts. A fold in which each

1 has exactly three contacts, will be

called saturated.

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 12/17

SLIDE 24

Basic building block — a tube

Formed by 6 “plates”

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 13/17

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Basic building block — a tube

Formed by 6 “plates” such that hydrophobic cores of plates come together

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 13/17

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Basic building block — a tube

Formed by 6 “plates” such that hydrophobic cores of plates come together and connecting the plates with “small loops” at the top and bottom

a tube of height 10

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 13/17

SLIDE 27

Basic building block — a tube

Formed by 6 “plates” such that hydrophobic cores of plates come together and connecting the plates with “small loops” at the top and bottom

a tube of height 12 coiled coil structure in protein 1AIK

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 13/17

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Connecting tubes together

by overlapping a top loop of one tube with a bottom loop of the other tube: disadvantage: only one tube can be attached at the top of any tube

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 14/17

SLIDE 29

Tubular structures

Linear sequence of interconnected tubes. Since their folds are sat- urated, they are also na- tive folds of correspond- ing proteins.

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 15/17

SLIDE 30

Stability of tubular structures

We conjecture all of them are stable. We have formal proof for tubular structures with 1 or 2 tubes.

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 16/17

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Stability of tubular structures

We conjecture all of them are stable. We have formal proof for tubular structures with 1 or 2 tubes. Idea of the proof: Properties of the protein: no

111, no 1101 and only two

ccurrences of

101.

Possible neighborhoods of

1-node x:

x x x x x x x

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 16/17

SLIDE 32

Stability of tubular structures

We conjecture all of them are stable. We have formal proof for tubular structures with 1 or 2 tubes. Idea of the proof: Properties of the protein: no

111, no 1101 and only two

ccurrences of

101.

Possible neighborhoods of

1-node x:

x x

x x x x

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 16/17

SLIDE 33

Stability of tubular structures

We conjecture all of them are stable. We have formal proof for tubular structures with 1 or 2 tubes. Idea of the proof: Properties of the protein: no

111, no 1101 and only two

ccurrences of

101.

Possible neighborhoods of

1-node x:

x x

x x x

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 16/17

SLIDE 34

Stability of tubular structures

We conjecture all of them are stable. We have formal proof for tubular structures with 1 or 2 tubes. Idea of the proof: Properties of the protein: no

111, no 1101 and only two

ccurrences of

101.

Possible neighborhoods of

1-node x:

x x

x x x

Show that any connected component of

1 in one layer

is either completely copied to the layer above (bellow)

r completely replaced with

0 in this layer.

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 16/17

SLIDE 35

Stability of tubular structures

We conjecture all of them are stable. We have formal proof for tubular structures with 1 or 2 tubes. Idea of the proof: Properties of the protein: no

111, no 1101 and only two

ccurrences of

101.

Possible neighborhoods of

1-node x:

x x

x x x

Show that any connected component of

1 in one layer

is either completely copied to the layer above (bellow)

r completely replaced with

0 in this layer.

Show that any connected component of

1 in one layer

is hexagon.

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 16/17

SLIDE 36

Future work

Generalize the design to a broader class of structures.

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 17/17

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Future work

Generalize the design to a broader class of structures. Verify the design in real world (or on a folding software): first,

0 and 1 has to be replaces with 20 amino acids;

for that one could either pick only 2 amino acids: the most polar and the most hydrophobic;

r use all 20 of them in such a way that the triples of

amino acids in designed sequence allow for bond angle which is used in the design for this triple.

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 17/17

SLIDE 38

Future work

Generalize the design to a broader class of structures. Verify the design in real world (or on a folding software): first,

0 and 1 has to be replaces with 20 amino acids;

for that one could either pick only 2 amino acids: the most polar and the most hydrophobic;

r use all 20 of them in such a way that the triples of

amino acids in designed sequence allow for bond angle which is used in the design for this triple. Design protein sequences which fold to a similar (desired) structure when folded to several lattices (independence of design from specific properties of the lattice used).

Design of artificial tubular protein structures in 3D hexagonal prism lattice under HP model * * * J´ an Maˇ nuch * * * BIOCOMP 2007, Las Vegas – p. 17/17