On Stable States in a Topologically Driven
Protein Folding Model
Zheng Dai1 , David Becerra1 , and Jérôme Waldispühl1
School of Computer Science, McGill University, Montreal, Canada,
[email protected], [email protected]
Abstract. Theoretical models of protein folding often make simplifying
assumptions that allow analysis, yielding interesting theoretical results.
In this paper, we study models where folding dynamics is primarily driven
by local topological features in an iterative manner. We illustrate the
merit of the proposed approach through its ability to simulate realistic
protein folding processes even when the sequence content information is
reduced to just hydrophobic and polar. We then analyze our models and
show that under our simple assumptions certain structures are inherently
unstable, and that determining whether structures can be stable is an
N P-hard problem. Interestingly, we find that when our model has only
two amino acids, the problem becomes solvable in polynomial time.
Keywords: Protein folding, inverse folding, complexity, minimalist model,
HP model, o↵-lattice
2
1
Introduction
The development and analysis of theoretical models simulating the folding of
proteins is an e↵ective approach to study a phenomenon that cannot be directly
observed. Despite more than 40 years of intensive research, our understanding
of the rules shaping the protein folding landscape remains incomplete, mostly
because of the sheer size of the systems and inherent computational complexity
of the underlying problems.
Computer simulations of atomistic models are arguably among the most popular approaches to study biomolecular systems [Karplus and McCammon, 2002].
While those methods are constantly developed and enhanced, they are facing both technological limitations and complexity factors that restrain their
progress and analysis. Particularly, these computational approaches are hampered by the inherent complexity involved in modelling the physical systems
that (i) limits applications of this technology to the simulation of small molecules
[Lane et al., 2013], and more importantly (ii) prevents any formal analysis of the
properties of the folding landscape generated by these models.
By contrast, minimalist models o↵er a di↵erent perspective to study protein
folding mechanisms [Kolinski and Skolnick, 2004,Head-Gordon and Brown, 2003].
The simplification of amino acids properties, reduction of the complexity of interactions, and discretization of the energy landscape allows the conception of
mathematically simple models whose behaviour can be analytically calculated.
Such insights provides important clues on fundamental principles governing the
folding of proteins.
One of the earliest example is the Zimm-Bragg model which has been designed to analyze helix-coil transitions [Zimm and Bragg, 1959]. More sophisticated frameworks such as the hydrophobic zipper model [Dill et al., 1993] have
been subsequently developed to study cooperativity in protein folding. Among
them, the hydrophobic/polar (HP) lattice model, which limits amino acids interaction to hydrophobic contacts and restricts the conformational space to a lattice
[Dill, 1985,Istrail and Lam, 2009], probably remains the most popular one.
HP models have been used to show that the protein folding problem is N Phard on the 3D cubic lattice [Berger and Leighton, 1998] and even on the 2D
square lattice [Crescenzi et al., 1998]. Interestingly, HP lattice models have also
been used to show that the complexity of the inverse folding problem (i.e. deciding if it exists a sequence that folds into a pre-determined target structure) can
be di↵erent than the one of the folding problem [Yue and Dill, 1992]. Indeed,
while the inverse folding problem is polynomial in 2D lattices, the inverse folding problem turns out to be N P-complete on 3D lattice [Berman et al., 2007].
Variants of the HP lattice models allowing the creation of disulfide bridges between cysteines have also been introduced, and efficient algorithms to solve the
inverse folding problem have been designed for 2D [Khodabakhshi et al., 2009b]
and 3D lattices [Khodabakhshi et al., 2009a].
However, the simplicity of the HP models limits the impact of the theoretical results derived from it on our understanding of the biological principles
driving the folding of proteins. First, lattices strongly constrain the accessible
3
conformational landcape [Guyeux et al., 2014]. O↵-lattice HP models were introduced to circumvent this limitation [Istrail and Lam, 2009], but little is known
about the computational complexity of the folding and inverse folding problems
[Pelta and Carrascal, 2007]. Next, the restriction to the HP alphabet does not
enable us to capture the impact of the diversity of amino acids properties. Finally, the dynamics of the folding is not considered either. Simple protein folding
models with better expressivity but whose behaviour can still be characterized
are thus needed to fill this gap and bring us closer to the biological reality.
It is generally accepted that the folding of proteins is an iterative process, in
which the protein moves from one configuration to the next until a stable configuration (e.g. minimum energy structure) is reached [Bryngelson et al., 1995]. Although minimalist models and variants of the HP model have been introduced to
simulate aspects of this dynamical processes [Hockenmaier et al., 2007,Dill et al., 1993],
they did not lead to complexity results.
Given the impracticality of sampling the complete conformational space of
proteins, a popular approach consists of defining instead a set of rules that explicitly determine how an amino acid must move at each moment in time, taking
into account the positions of neighboring amino acids [Krasnogor et al., 2005].
These methods have mainly used cellular automata and Lindenmayer systems
to define the rules and dynamics of the folding process. The complexity of the
proposed models, given by the enormous number of transition rules, hampered
the success, applicability, and convergence of these methods [Reyes et al., 2014].
In this paper, we lay the groundwork for a system on which we can construct
models that simulate the protein folding process. Underlying our system is the
hypothesis that protein folding is driven primarily through perturbations resulting from local graph topology. Remarkably, in such systems the concept of stable
structures emerge naturally as a fixed point in the conformational landscape.
We have applied the proposed system to describe a computationally tractable
set of models based on the (HP) alphabet and assess their performance on real
proteins. Our results demonstrate that although our models remain sufficiently
simple to permit a formal analysis of their combinatorial property, they can also
achieve satisfying predictive capabilities. This aspect is important to guarantee
the biological relevance of our complexity results.
We then further investigate some of the theoretical implications of our system. Specifically we demonstrate that determining whether specific structures
can be stable is an N P-hard problem. To our knowledge it is the first result
to suggest that the difficulty of the inverse folding problem scales with the size
of the protein as opposed to the energy model in an o↵-lattice model. Interestingly, we also find that if we restrict our alphabet to two amino acids (e.g.
HP model) the stable structure decision problem becomes solvable in polynomial
time. These results are suggesting the practical difficulty to identify “designable”
protein structures.
4
2
2.1
Methodology
The Role of Local Topology in Folding Dynamics
A protein is a chain of amino acids held together through peptide bonds. This
chain usually folds into a particular conformation, which are held together with
interactions between non-adjacent amino acids. Protein structures can therefore
be easily described by a graph, where each node represents an amino acid, and
each edge represents an interaction. This representation of protein structures provides us a convenient framework to study the folding process [Vendruscolo et al., 1999].
In order to accurately represent the dynamic process used to transform an
unfolded polypeptide into a stable fold, we need an additional information. Since
peptide bonds are more stable than most other interactions, we need a way of
di↵erentiating between edges that represent peptide bonds than those that represent other interactions. Here, we will refer to edges representing peptide bonds
as permanent, and we will refer to the other edges as transient. For convenience,
we will also say each node has a permanent edge pointing to itself.
In this paper, our key assumption about folding dynamics is that protein
folding is driven primarily though perturbations resulting from local topology.
We argue that this principle models an important aspect of the dynamics of
protein folding. Here, we also assume that folding occurs in discrete time steps,
and that folding is deterministic.
At a given time step, for any pair of amino acids, it may exist an edge between
them. If the edge is permanent, then the edge will exist in the next time step,
and for all future time steps. However, if the edge is transient, or if there does
not exist an edge between them. Then the existence of the edge between them
in the next time step depends on neighbourhood of that pair, and the identities
of the amino acids.
The neighbourhood of a contact between two amino acid is easy to describe
in the context of contact maps, which are equivalent to adjacency matrices since
we are using here graph terminology. The order of the rows and columns of the
matrix corresponds to the primary sequence of the protein. An entry of this
matrix is equal to 1 if an edge exists (i.e. a contact exists) and 0 otherwise. Each
pair (i, j) of amino acids is an entry in the adjacency matrix. In this paper, we
define the neighbourhood N (i, j) of that pair (i, j) to be the 3 by 3 sub-matrix
with that entry in the middle (i.e. the set of contacts N (i, j) = {(i 1, j 1), (i
1, j), (i 1, j + 1), (i, j 1), (i, j), (i, j + 1), (i + 1, j 1), (i + 1, j + 1), (i+, j + 1)}).
For any pair that lies on the edge of the adjacency matrix, we will consider the
undefined entries in the neighbourhood to be 0.
Using this definition, we can see how local features can propagate through the
protein. In figure 1 we provide a toy model that illustrates how the model works,
and how features may arise. Further motivating our definition of a neighbourhood
is the strong parallels it has to models of statistical potential that also makes
use of local structures.
We can now define a transition function that takes as input a pair of amino
acids (i, j) with its neighbourhood, and output the status of the same edge (i, j)
5
Fig. 1: A toy model with two amino acids ”A” and ”B”. In here the transition
function only cares about two cells in the matrix, (1,1) and (1,2), and it converges
to a cycle of length 2 on the sequence ”AAAABBBB”.
at the next time step. More precisely, the transition function outputs a 1 or a 0,
where 1 means an edge should exist in the next time step, and 0 means an edge
should not exist in the next time step.
Finally, we can define an overall step function, which takes a graph as an
input, and computes the transition function on each pair of vertices that does
not have a permanent edge. Importantly, the original graph is left unchanged
throughout this computation. The individual results of the transition functions
are instead cached in a new graph, which replaces the original one at the end
of the computation of the overall step function. This new graph contains all
permanent edges the original graph had, and has a transient edge for each pair
of vertices that fulfills the following 3 requirements: (i) the pair does not have a
permanent edge between them in the original graph; (ii) the transition function
6
outputs a 1 for the pair; and (iii) if an edge did not exist between them in the
previous graph, then the degree of neither vertices can exceed a constant we will
call the interaction limit. The reasoning for the last restriction is because there is
no implicit constraint in our formulation of protein folding that stops a protein
from transitioning into states that are physically infeasible.
Algorithm 1 illustrates the process described above. Noticeably, we initialize
the contact map on line 2, and refer to an arbitrary convergence criterion on line
13. These steps are left undefined here in this generic algorithm. However, in our
experiments we define the initial state as the matrix consisting of 0 everywhere
except for the main diagonal and sub-diagonal, which can be interpreted as the
graph containing only permanent edges. We also use a convergence criterion that
halts the process after a specified number of steps.
Algorithm 1 Protein Folding Process
1: Take an amino acid sequence S of length n as input
2: Initialize an n ⇥ n matrix A to an initial state
3: loop
4:
Initialize a blank n ⇥ n matrix B
5:
for each cell (i, j) in B where j  i do
6:
if | i j | 1 then
7:
Set Bi,j to 1, since it is a permanent edge
8:
else
9:
Construct a 3 ⇥ 3 matrix U from cell Ai,j , and the 8 cells surrounding it
10:
If any cells land outside the matrix, pretend its value is 0
11:
Feed U , Si , and Sj to the transition function and set Bi,j to its output
12:
Set A to B
13:
if some convergence criteria is met then
14:
return A
2.2
HP-X8 Models
Our description above requires parametrization of the transition function before
it can be used to iterate on graphs to simulate folding dynamics. We first need
to set the number of di↵erent amino acids the transition function is capable
of recognizing, and what the transition function outputs for every unique set of
inputs. We will refer to such parametrization of a transition function as a model.
In order to verify the validity of our overall approach, we need to find models
that have the ability to simulate biological phenomena. Searching for such a
model in the full space of possible models is obviously intractable, so instead
we introduce a subset of the model space which is both tractable and contains
models that are complex enough to capture biological phenomena. First, we
reduce the amino acid alphabet from 20 letters down to 2 (Hydrophobic and
Polar), and limit the number of interactions to 8. Next, in the transition function,
we will only consider two types of amino acid inputs: either both amino acids are
hydrophobic, or not. In particular, it means that for any matrix M, the transition
7
function will give the same output on inputs (M,P,P) and (M,H,P), where H and
P are the hydrophobic and polar amino acids respectively.
Next, for the matrix input, the transition function for a pair of amino acids
(i, j) will only consider the entries at the coordinates (i 1, j 1), (i 1, i +
1), (i, j), (i + 1, i 1), and(i + 1, j + 1), which together form an X shape. We also
consider two matrices to be equivalent if they are identical after a series of 90
degree rotations, which means the transition function ignores the relative linear
positions of the amino acids on the chain. Under these hypotheses, our model
can be defined using the 12 types of matrix inputs shown in figure 2.
Finally, we will say that (H,H) interactions are stronger than all other interactions. It means that if the transition function outputs 0 for a matrix and an
(H,H) input, it must output 0 for any other amino acid inputs.
We call this set of models HP-X8 models. Due to these restrictions, there
are only a total of 531441 models with di↵erent parameters. This reduction in
complexity allows brute force enumeration and testing of these models; however,
there is a significant sacrifice in the amount of information that is lost. Although
it is unlikely for HP-X8 models to be able to make detailed predictions of protein
dynamics, we find that some of these models do capture interesting phenomena,
and their ability to do so despite the loss of information are indicative of the
expressivity and biological relevance that our overall approach has.
Fig. 2: All unique input matrices of HP-X8 models.
3
3.1
Results and Discussion
Brute Force Enumeration
We tested all HP-X8 models on random sequences, and we considered those
that did not halt within 5 time steps to be viable, since heuristically speaking
5 time steps would not be enough to generate interesting structures. For our
experiments, we always assumed that folding starts with no transient edges.
This filtering process left us with 55404 viable models.
In order to test the viable models, we extracted a diverse set of 95 proteins
from the BetaSheet916 dataset [Cheng and Baldi, 2005], with sizes ranging from
8
41 to 120 residues, for which the native 3D structures are available with a resolution better than 2.5Å. The extracted proteins belong to all of the main protein
families (such as ↵/ , ↵ + and all- ) as defined by SCOP. The sequence identity between any pair of sequences in the data set is between 15 20%, and only
proteins containing less than 6 strands were extracted. BetaSheet916 is a benchmark that has been routinely adopted set for -residue pairs, -strand pairs,
and -sheet prediction methods. Additionally, the proteins in BetaSheet916 have
played a central role in protein folding studies by being the system of choice in
a vast body of experimental and theoretical studies. The table for reduction of
the 20 amino acids to the HP alphabet is provided in figure 3.
Fig. 3: Translation table for the 20 amino acid alphabet to the HP alphabet.
For each protein we ran each model for 100 time steps and computed the
harmonic mean of the precision and recall (F1 Score). For the purposes of scoring
how well a model performs, we define an interaction as existing if a pair of amino
acids are closer than 10nm apart, and we say we have a true positive if a predicted
interaction lies within 2 amino acids of the real contact, since those parameters
appear produce the best results. Due to the abundance of short range contacts,
models that generate large amounts of those indiscriminantly usually end up
with the best F1 score. However, it does appear to be a decent heuristic, as some
models that do score reasonable well do display interesting features. Therefore,
we took the 50 models that scored best with the F1 score and ran each model
for 200 time steps, this time applying a complementary scoring function:
X q p
p
p
F1 ⇥ (
cd d 4 + c4 + c3 )
(1)
d25,6,7...
Where cd is the number of contacts detected that are d amino acids apart,
and F1 is the F1 score. This function takes into account that it is better to have a
diverse range of contacts at all distances by using a sum of concave functions, and
includes a range dependent factor so that long range contacts are more heavily
favoured than close range ones. We also performed the whole process for a control
set, which consisted of the same proteins, but we shu✏ed the sequence of the
amino acids before starting the process. Figure 4 contains some examples of the
resulting contact maps, and a comparison to the controls. (A comprehensive list
of the results can be found in the supplementary material).
Figure 4 illustrates an interesting observation. We discover that for many
sequences there exists an HP-X8 model that takes a protein from a state with
9
no transient edges to one whose adjacency matrix reasonably resembles the real
protein contact map. These results support our hypothesis by showing that local perturbations in topology is indeed capable of generating protein folds that
look realistic. By comparing our results to our controls, we observe that without
sequence information, HP-X8 models are less capable of carrying a protein into
a state that resembles the real protein contact map, which rules out the possibility that our results are a consequence of the robustness of the HP-X8 models.
We have included the data pertaining to this comparison in the supplementary
materials.
Fig. 4: On the top row are optimal HP-X8 models iterated on 1B13, 1SOY, and
1NR2 for 190, 197, and 155 time steps respectively from left to right. Below
them are the results of the same experiment on the same proteins, where the
sequence was randomized. The lower triangle is the contact map extracted from
the PDB, while the upper triangle is the output of the model.
3.2
Consensus Model
In the previous section, we have provided evidence that our approach has merit.
Nonetheless, if our models are reflective of reality, it is reasonable to suspect
that all the models we have captured in the last experiment are noisy approximations of a single model. Therefore, we would like to assess the degree to which
the models we have captured approximate reality. To this end we measure the
agreements between the transition functions of the captured models.
We allow each protein to place a vote for what should happen for each input
for the transition function, and we have recorded the results of the vote in
10
figure 5a. Under the null hypothesis that each protein selects its best model
uniformly at random, we observe statistically significant (p-value under 0.05)
agreement for all inputs except 3.
Similarly, we also checked model agreement for our control set in the same
way. Figure 5b shows our results. The overall agreement levels seem to be approximately the same as our result set.
Unsurprisingly, these results suggest that the HP-X8 models are far too simplistic to fully capture a general underlying model. This is quite plausible, since
models based on the HP alphabet lose a large amount of information when transitioning from a 20 letter alphabet to a 2 letter alphabet. Despite this, it is
clear that our framework captures a signal, since there are statistically significant agreements in most of the rules. We hypothesize that we are capturing the
necessary rules that are required for a transition function to generate biological structures. Our control experiment supports this notion, since the aggregate
function from the control experiment is very similar to the aggregate function
from our main experiment with similar levels of agreement.
(a) The agreements between the best models of each protein.
(b) The agreements between the best models of each protein for the control experiment.
Fig. 5: Consensus results
11
3.3
Stable States
We are now presenting the main theoretical results of this study. Importantly,
in the remainder of this manuscript we will consider the full generalized models
instead of the sub-set of HP-X8 models we have previously used to demonstrate
the biological relevance of this system.
Our study aims to characterize the difficulty for a sequence to have a stable
conformation (i.e. state) in this framework. Interestingly, our models enable us
to naturally define the concept of a stable state. Let a state st be a graph (or its
adjacency matrix) representing a protein structure at a given time step t. Using
our models, a state gives all the information required to compute the next state
t + 1. Thus, we can decide if a folding is complete if the next state st+1 given by
the step function is identical to the current state st . The state st is this called a
stable state (or stable structure).
Intuitively, the stability of a state is dependent on the sequence. This observation leads naturally to several interesting questions such as: given a transition
function, are there certain states that are inherently unstable, and if yes, is it
possible to find them? Such questions are closely related to the problem of protein design, since if a state is inherently unstable it becomes also undesignable.
To answer these questions, let us consider a problem which we will call the
stable state characterization problem.
Definition 1. Given a graph with n vertices representing a state and a set of k
amino acids along with a transition function, determine if there exists a sequence
of n amino acids taken from the set such that the state is a stable state.
Theorem 1. Stable state characterization is N P-hard.
Proof. Stable state characterization is polynomially reducible from vertex colouring, which is a classical N P-complete problem.
It suffices to introduce amino acids which correspond to colours, and make
edges unstable when two amino acids representing the same colour interacts
with each other, and make edges stable when the amino acids are di↵erent. As a
construction detail, we make sure that edges do not spontaneously form. A string
of amino acids that stabilizes the state must then encode a graph colouring. Note
that the interaction limit does not come into play here, since that only restricts
the formation of new edges, and not whether an edge could break.
As a construction detail, we need to interlace the original graph with noninteracting amino acids. We can take the adjacency matrix representing the
original problem, and interlace blank rows and columns between each row and
column of the original adjacency matrix. The reason this is required is because
adjacent amino acids are connected by default, which would limit the graph
topologies that could be encoded if we do not interlace.
If a sequence that stabilizes the state exists, then we can read the colouring o↵
the parts of the sequence that correspond to the original matrix. The colouring
would be valid, since otherwise one of the edges would be unstable. Conversely,
12
if a valid colouring exists, then we can make a sequence of amino acids corresponding to that colouring and interlace that sequence with any amino acid,
since the corresponding rows and columns are blank and therefore stable. This
will provide a sequence that would stabilize the new adjacency matrix, since the
rows and columns in the original adjacency matrix would be stabilized by the
colouring by our construction.
Thus, a solution to the stable state characterization problem exists if and only
if a solution to the original vertex colouring problem exists. If the original graph
was also represented as an adjacency matrix, then the new matrix can be created
in linear time. Thus, there is a polynomical reduction from vertex colouring to
stable state characterization, and therefore stable state characterization is N Phard.
t
u
Corollary 1. Stable state characterization is N P-complete.
Proof. This follows immediately from the above theorem, since a sequence of
amino acids that allow the state to be stable would be a suitable certificate. The
evaluation given such a sequence takes time O(n2 ), where n is the length of the
sequence.
t
u
Thus, in the general setting it is impossible to provide an efficient method of
deciding whether a particular structure can be stable or not, unless P = N P.
remarkably, we note that this also proves that there can be states that are
inherently unstable.
Since a vertex colouring problem can encode 3-SAT at 3 colours, stable state
characterization can be N P-complete if the transition function uses 3 amino
acids or more. But perhaps the problem only becomes N P-complete when the
states are unrealistic, for instance when a single amino acid has an unrealistic
amount of interactions. The following theorem shows that that is not the case.
Theorem 2. Vertex colouring with k colours where the maximum degree is 2k
1 is N P-complete, given that k > 2.
Proof. We can reduce to this problem from arbitrary k-colouring. Note that if a
vertex has more than 2k 1 incoming edges, we can construct a device, where
two unconnected nodes are both connected to the same k 1 other nodes. This
would then require the two nodes to be the same colour. We can keep doing
this to create a chain, where each link must be the same colour. This way, each
link only requires one more incoming edge, thus the maximum degree of this
graph is 2(k 1) + 1 = 2k 1. Since each node originally can have at most as
many incoming edges as there are nodes, this modification represents at most a
polynomial increase in the number of nodes, thus the reduction is polynomial.
t
u
From the way our proof of N P-completeness is constructed, it follows from
the theorem that realistic instances of the stable state characterization problem
do not necessarily simplify the problem.
13
3.4
Implications for the Inverse Folding Problem
The results presented in the last section has implications for the inverse folding
problem, where given a structure one has to produce a sequence that can stabilize
the structure. If the inverse folding problem can be solved efficiently, then we
can create an oracle algorithm to solve stable state characterization efficiently,
where we run inverse folding on our sequence, and output whether a sequence
was found or not. Thus it follows that there exists no efficient algorithm for
inverse folding, unless P = N P.
However, this conclusion must be completed. In our model, it is possible that
a stable conformation for a protein is represented by a small cycle of states rather
than by a single stable state. Since the size of the cycle roughly corresponds to
the stability of the conformation, an analog of the stable state characterization
in this case is to find a sequence for which the cycle the state is a part of is
minimized. This is also N P-hard, since if the optimal cycle contains only 1
state, a correct optimization algorithm will find a sequence that allows this, and
will in turn solve the stable state characterization problem.
In our reduction of the stable state characterization problem, we showed that
for a specific model with fixed parameters, the inverse folding problem is N Pcomplete. To our knowledge, this is the first time evidence has been presented
which shows that the difficulty of the inverse folding problem scales with the
size of the length of the protein, as opposed to the size of the parameters of the
model in o↵-lattice models.
3.5
Complexity of Stable State Characterization with 2 Amino
Acids
Our proof above show that stable state characterization becomes N P-complete
at 3 amino acids. We will now show that for any transition function that only
recognizes 2 amino acids, the problem lies in P. In particular, this shows that
for the HP-X8 models, stable state characterization can be done efficiently, and
therefore the inverse folding problem can also be solved efficiently.
Theorem 3. Stable state characterization with 2 unique amino acids is in P.
Proof. For simplicity, we will show this by reducing the problem to 2-SAT, which
can be solved in polynomial time, instead of providing an actual algorithm.
Let us call the 2 amino acids T and F . We will also assign each amino acid a
boolean value, such that T is true and F is f alse. Let S be a sequence consisting
of T and F , and let A be the adjacency matrix representing the state we want to
stabilize. Each cell, based on its surroundings, can be constrained in a way that
can be expressed as a 2-CNF, where a and b are the interacting amino acids.
14
The
The
The
The
The
The
The
The
cell
cell
cell
cell
cell
cell
cell
cell
is
is
is
is
is
is
is
is
stable only if the interaction is between T and T
stable only if the interaction is between F and F
stable only if the interaction is between T and F
unstable only if the interaction is between T and T
unstable only if the interaction is between F and F
unstable only if the interaction is between T and F
always stable
always unstable
(a _ a) ^ (b _ b)
(¬a _ ¬a) ^ (¬b _ ¬b)
(a _ b) ^ (¬a _ ¬b)
(¬a _ ¬b)
(a _ b)
(a _ ¬b) ^ (¬a _ b)
(a _ ¬a)
(a _ a) ^ (¬a _ ¬a)
We join all this with conjunctions to produce the 2-CNF, which can be satisfied if and only if there exists a sequence of T and F which stabilize the state.
This reduction produces a 2-CNF that is no larger than a constant factor of the
size of the adjacency matrix. Thus, this reduction can be done in polynomial
time, and since 2-SAT can be solved in polynomial time, this problem can be
solved in polynomial time.
Therefore, stable state characterization with 2 unique amino acids is in P.
t
u
3.6
Expressivity of Transition Functions
It is easy to solve stable state characterization if we restrict the number of amino
acids we use to 2. So is there any reason to use any more? The following result
shows that systems with more amino acids are more expressive in terms of the
stable states that it can allow, implying that there is value in having a system
that recognizes more amino acids.
Theorem 4. Suppose we are given an adjacency matrix which requires n unique
amino acids to ensure its stability. Then supposing that no interaction of an
amino acid with itself is completely stable, we can construct an adjacency matrix
which must require k amino acids to stabilize, where k > n.
Proof. We construct a new adjacency matrix, where we start o↵ by taking the
original adjacency matrix and adding an extra row to the top and bottom, and
an extra column to the left and right. We then take this matrix and copy it
diagonally n times.
If this results in a adjacency matrix which requires n + 1 unique amino acids
to ensure stability, we are done. Otherwise, we required at least n unique amino
acids to ensure stability of this new adjacency matrix, since each submatrix that
resembles the original matrix requires at least n unique amino acids to ensure
stability. Furthermore, in any stabilizing sequence, it is possible to pick a set
of n amino acids such that they are all distinct, and there are at least 2 amino
acids between any two selected, since each submatrix requres all n amino acids
to stabilize.
Let S be the set of (sequence ⇥ transitionf unction) tuples that ensure stability of the matrix with only n unique amino acids in the sequence. Assume
that each transition function makes use of at most n amino acids. This is inconsequential, since if only n unique amino acids are used, then any transition
15
function using a larger set of amino acids can be truncated to only those used.
Therefore, S is clearly finite.
For each element s present in S, add 3 rows to the bottom of the adjacency
matrix. Then pick out n unique amino acids in the sequence such that no 2
amino acids are spaced closer than 2 amino acids apart. For each amino acid,
find the cell at which that amino acid interacts with the one on the 2nd row
that was added, then arrange that cell such that if the identity of the new amino
acid was itself, it would be unstable according to the transition function. This
is always possible by our assumption.
Once we have finished doing this, it ensures that for that particular sequence
and transition function, there exists no amino acid that stabilizes that contact
matrix. If we do it for all (sequence ⇥ ruleset) tuples, then there exists no
ruleset in which some sequence can stabilize the adjacency matrix with less than
n elements.
Thus more than n unique amino acids are required to stabilize this contact
matrix. The new number is clearly finite, since if we have a sequence where each
element is unique, then trivially there is a transition function which stabilizes
the contact matrix.
t
u
Here, we place the restriction that no interaction of an amino acid with itself
is completely stable because if that is the case, every state can be stable if we
take a sequence consisting of just that amino acid. Since such system is not a
very expressive, we exclude it from our consideration. Also, we note that since
for any matrix we know we need at least 1 amino acid to have stability, the
induction can begin.
Although this result has some limitations. Indeed, we have not proven that
with each amino acid added we can improve expressivity. However, we have
shown that if we keep adding amino acids we will eventually improve. We have
also not considered the role of interaction limits, but we can always set it arbitrarily high so that the theorem works.
4
Conclusion
We have presented in this work a system which lays the groundwork for an energy
independent paradigm of protein folding. As we have demonstrated, even with
relatively simple parametrization our system is capable of producing surprisingly
reasonable predictions. An analysis of the specific parameters that produce good
models suggests that on some level, processes occurring in reality are being
replicated by our system.
We have also provided some preliminary analysis into the general properties of the models of this type. We have shown that inverse folding, under this
paradigm, is intractable, and thus may require simplifications and approximations. We also showed that certain kinds of simplifications, while they make
problems more tractable, also reduce the robustness of the models.
Nonetheless, many questions are still open, in particular we did not investigate stable cycles of states, which intuitively should be more frequent than
16
stable states as the end product of a folding process. Although we have established a connection between stable states and stable cycles through the problem
of minimizing the length of a stable cycle, a thorough investigation has yet to
be done.
We believe that the work presented here provides a strong basis for future
work in theoretical protein folding. We hope that future work will further elucidate the features of this paradigm, and use it to create novel and interesting
models of protein folding.
17
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