A robust class of stable proteins
in the 2D HPC model
Alireza Khodabakhshi, Jan Manuch, Arash
Rafiey, Arvind Gupta.
Simon Fraser University, Canada
Presentation Summary
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Background
Hydrophobic-Polar (HP) model
Inverse Protein Folding (IPF) in HP model
Constructible structures
Wave structures
Stability of wave structures in 2D HPC model
Open problems
Proteins
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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.
Protein Folding
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Protein folds as to minimize the free energy of the system.
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It is not known how a protein can choose the minimum energy fold among
all possible folds, cf. Dill, Bromberg,Yue, Fiebig, Yee, Thomas, Chan (1995).
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Many forces act on the protein which contribute to changes in free
energy including:
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hydrophobic interactions.
hydrogen bonding,
van der Waals interactions,
intrinsic propensities,
ion pairing,
disulphide bonds,
Hydrophobic-Polar (HP) Model
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Hydrophobic interactions are the most significant forces driving the
minimum energy conformation (fold), cf. Dill (1990)
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Amino acids are of two types: hydrophobic or polar depending on
their affinity to water.
Hydrophobic-Polar (HP) Model
In HP model (Chan and Dill, 1985) proteins are:
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represented as sequences over {0, 1} ( or {H, P}).
embedded on a (2D or 3D) lattice with each amino acid occupying exactly one
square and neighboring amino acids occupy neighboring squares.
2D square lattice
3D cubic lattice
Hydrophobic-Polar (HP) Model
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The fold of a protein is a self-avoiding walk and its energy is the
negative of the number of HH bonds (contacts).
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An HH bond occurs if we have adjacent H amino acids in the lattice
which are not consecutive in the protein sequence.
protein: p = 01100110100001110100110
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A fold with minimum
energy is called a
native fold.
The energy of this fold is -4.
This is not the minimum energy.
The energy of this fold is -8.
This is the minimum energy.
Hydrophobic-Polar (HP) Model
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There might be several native folds of a protein sequence.
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A protein is stable if it has a unique native fold.
The total number of native folds of this protein is 82.
Folding in HP model is still hard!
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To find a native fold for a given HP sequence is NP-hard for
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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
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3/8 for 3D square lattice, cf. Hart, Istrail (1995)
6/11 and 3/5 for 2D and 3D triangular lattices, cf. Agarwala, Batzoglou,
Danˇcík, Decatur, Farach, Hannenhalli, Skiena (1997)
Inverse Protein Folding or Protein Design
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In many applications such as drug design or nanotechnology, we
are interested in the complement problem to protein folding.
F
IPF
S
IPF under HP model
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For a given target fold G, find a sequence S that:
1.
2.
has the minimum energy when folded into G (positive design).
has the minimum possible degeneracy (negative design).
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Degeneracy-d(S)-of a sequence is the number of its native folds.
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The complexity of IPF is unknown but it is conjectured to be NPhard.
Canonical and Grand canonical models
Two alternative formulations have been proposed:
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Canonical model:
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For a given target conformation G and a real value λ<1 find a
sequence S with at most [λn] H monomers that achieves the
minimum energy.
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This problem is polynomial for 2D square lattice, and it is NP-hard for
3D cubic lattice (Berman et al., 2004).
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Grand canonical model:
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Limits the number of H monomers by penalizing the solvent
accessible H monomers.
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IPF under GC model is in P (Kleingberg 1999)
Structure approximating IPF
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Canonical and GC methods do not guarantee neither positive nor
negative criteria.
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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.
Inverse Protein Folding
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Constructible structures was introduced to approximate any given
shape in 2D square lattice Gupta, M., Stacho, JCB (2005).
Constructible structures are formed by a system of tiles
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(a) a starting tile:
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and (b)
a regular tile
Saturated folds
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Constructible structures have a special
property: every H amino acid has
2 HH bonds.
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If a protein is from the set 0{0,1}*0, two is the maximum
number of bonds possible for each H monomer.
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A fold in which every H amino acid has two bonds is called
saturated.
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Observation 1: If there exists a saturated fold of a protein p then
the fold is native, and any native fold of p is saturated.
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Constructible structures satisfy positive design criteria.
Linear constructible structures
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In linear structure S every tile except for the first and the last tile is
attached to two tiles.
Bend
Hydrophobic-Polar (HP) Model
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Note that in a saturated fold hydrophobic “1” amino acids form
cycles (called 1-cycles) where all edges are bonds.
A 1-cycle of size 4 is called a core.
Stability of constructible structures
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Conjecture. All constructible structures are stable.
It was proved for arbitrary long structures forming I- and L-shapes.
L1 (one bend)
L0 (no bend)
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We verified by computer that all constructible structures with up to 9
tiles are stable (over 50,000 structures).
Toward solving the conjecture
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Our goal is to solve the conjecture in a special case which is general
enough to be still able to approximate (although more coarsely) any
given shape.
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In this work we
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augment the HP model by distinguishing cysteine and
non-cysteine hydrophobic monomers: HPC model
introduce a special class of linear constructible structures — wave
structures
design a program for automatic case analysis (2DHPSolver)
finally, we demonstrate that these tools is sufficient to show that all wave
structures in the HPC model
HPC model
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Three types of amino acids:
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depicts a polar “0” amino acid
depicts a non-cysteine hydrophobic “1” amino acid
depicts a cysteine “2” amino acid
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Hydrophobic contacts (any two neighboring
hydrophobic amino acids):
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In addition, cysteine monomers
form SS-bridges (pairs) such that
each cysteine can be involved in
at most one SS-bridge:
3 contacts and 1 SS-bridge
HPC model—native and saturated folds
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Energy model:
Econtact·(# of contacts)+ ESS-bridge·(# of SS-bridges)
for some negative constants Econtact and ESS-bridge
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Native fold is the one which minimizes energy
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Saturated fold:
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each hydrophobic monomer has 2 contacts
every cysteine monomers is paired in SS-bridge
Observation. Any saturated fold is native.
Wave structures
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Subclass of linear structures with the following properties:
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every second core we have to “bend” (change directions).
do not contain the following forbidden motifs:
Wave structures
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Wave structures can be built from super-tiles.
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Theorem. The protein sequences of wave structures are stable
under the HPC model.
Stability of wave structures
Proof techniques:
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Using induction on the boundaries to prove some key property π for any
saturated fold of the designed proteins.
For instance π can be: every H monomer belongs to a core
Assume that π holds for every H monomer to the right of the boundary,
show that π holds for any H monomer on the boundary as well.
We show that if π does not
hold for an H monomer on
the boundary then a property
of the design protein is
violated.
Toward solving the conjecture
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The induction step requires the analysis of many cases.
We have developed a program called 2DHPSolver that helps in
analyzing the many cases.
2DHPSolver is a semi-automatic tool used for proving the induction
step by case analysis.
Cores
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A core c is called monochromatic if all its H-vertices are either
cysteines or non-cysteines.
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Let c1 and c2 be two cores in a saturated fold F. We say c1 and c2
are correctly aligned if they are connected in one of the forms in
Figure 6.
Configuration with correctly cores.
Proof
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We prove the following two lemmas using 2DHPSolver. The
computer generated proofs are available on our website:
http://www.sfu.ca/~ahadjkho/2dhpsolver
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Lemma1: Let F be an arbitrary saturated fold of a wave protein. Every
H-vertex in F belongs to a monochromatic core.
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Lemma2: Let c1 and c2 be two adjacent monochromatic cores in F.
Then either c1 and c2 are aligned correctly or there is a third core c3
such that c1, c2 and c3 form the configuration in Figure below.
The only configuration
of misaligned cores
Proof
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Next we show that no fold with misaligned cores is possible.
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Assume that such a fold exist. Based on lemma2 there are three
cores c1 , c2 and c3 misaligned cores while all other cores are
aligned correctly.
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Consider a graph G where each core ci represents a vertex vi in G
and vi and vj are connected if ci and cj are connected.
c1
c2
v2
v1
v3
c3
Proof
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G cannot have a cycle other than (v1, v2, v3, v1), otherwise we get a
closed sequence not containing the whole sequence.
c1
c2
v2
v1
v3
c3
Proof
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G cannot have a cycle other than (v1, v2, v3, v1), otherwise we get a
closed sequence not containing the whole sequence.
c1
c2
v2
v1
v3
c3
Proof
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G cannot have a cycle other than (v1, v2, v3, v1), otherwise we get a
closed sequence not containing the whole sequence.
Therefore G has at lease three vertices of degree 1. These vertices
correspond to terminating cores in F, creating separated occurrences
of the subsequence t = (001)3.
c1
c2
v2
v1
v3
c3
t=001001001
Proof
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G cannot have a cycle other than (v1, v2, v3, v1), otherwise we get a
closed sequence not containing the whole sequence.
Therefore G has at lease three vertices of degree 1. These vertices
correspond to terminating cores in F, creating separated occurrences
of the subsequence t = (001)3.
Hence, in total F contains at least 5 occurrences of t while there must
be only two occurrences of t in F.
c1
c2
t
v2
v1
v3
c3 t t
t
t=001001001
Proof
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With similar arguments we show that the tiles are connected as a
linear chain (F is a linear constructible structure)
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Finally, it is not hard to show that the cores in F must connect as in
original fold.
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Therefore, wave proteins are stable.
Open problems
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Prove the conjecture for less restricted subclass of constructible
structures.
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Extend the HPC model for the 3D lattices and come up with
powerful and stable 3D designs.
Thank you!