Finding Protein Folding Funnels inRandom Networks
Macoto Kikuchi (菊池誠)Cybermedia Center, Osaka Univ. (大阪大學)

DDAP9 (17 Dec. 2016)

Outline1

2
3

Introduction: Rareness, Funnel picture,
Network representation
Model and Method
Results
Probability distribution of the ideal funnels
Network size dependence of the probability andS.D.

4

Summary and Discussions

Introduction: rarenessProteins are not at all typical random polypeptide
If we make a random polypeptide, its
low-energy state will become glassy.
Many diﬀerent conformations have low energy
close to each other.

A protein folds into its specific nativeconformation spontaneously under thephysiological conditions.
Anfinsen’s dogma: The native conformation isdetermined by the amino acid sequence.

Such a special property of proteins have beendeveloped through Darwinian evolution.
Proteins are good examples of the fact that theevolution can create very rare states of matter.

Introduction: funnel picture
Funnel picture of the energy landscape has been
accepted widely after 1990s as a mechanism behindthe protein folding
Minimum frustration principe (Bryngelson andWolynes)
Consistency principle (Go)funnel picture
The number of conformations decreases
monotonously as energy lowers and approaches tothe native state.

Introduction: motivation of this study
We consider the following question.
Main question
How rare are the funnel-like energy landscapes?
To answer this question, at least partly, weintroduce a simple and abstract model based on therandom energy model on random networks, which
expresses the energy landscape of the proteins.

Introduction: network discription
Network model has widely been used so far tounderstand the protein folding dynamics.
Markov state model has been used to describe
the relationship between many conformations
obtained during the MD simulations.
Hori et al. (PNAS 2009) tried to determineinterconnection between all the conformationsof some proteins including the conformations
that do not appear in MD. And compare theobtained network with that of random
polypeptide.

Modelconformation network
1
Give a random network, which represent theconnections between conformations.
Node: metastable ensenble of conformations.
Edge: possible transitions between the nodes
We consider that the network structure isdetremined by the native conformations
1 to 1 correspondence between the native
conformation and the network structure

We assume any random network corresponds tosome native state.

random order model
1
Each node is assigned an integer randomly.
Integer represents the energy of the node (RandomEnergy Model)
We need only the order of the nodes according to theenegy.

We consider that the arrangement of the numbers
is determined by the amino acid sequence

We assume all the arrangements of the numbers arepossible.

Construction of the model: 1
Make a simple
random graph
N: number ofnodes
L: average
number of edges
connected toeach node2
Select one of thenodes having thelargest number ofedges as U
(unfolded state)
Select one of thefarthest nodes
from U as F
(native foldedstate)

3

The network is
rejected if it isseparated when U
is deleted,4
Assign integers
from 1 to N − 2
randomly toremaining nodes.
U and F are
fixed to 0 andN − 1,
respectively

5Draw arrows fromthe edge ofsmaller number tothat of larger
number, if two
node areconnected.
The arrows
represent thedirections oftransitions.
We assume onlytheenergy-loweringtransitionsIdeal Funnel
We introduce a concept of ideal funnel
definition
Starting from U node, if all the energy-lowering pathlead to F node, we call the network is ideal funnel
The energy-lowering transitions between nodes
eventually leads to the native state without being
trapped by misfolded state.

non-ideal funnel(misfolded states 7 and8 exist)

ideal funnel

Question again
In the language of proteins
Given one netive state, how rare are the amino acidsequence that the energy landscape becomes theideal funnel among all the possible sequences.
In terms of the model
Given a random graph, estimate the appearance
probalibity of the ideal funnel among all the possible
arrangement of integers.

Method
N = 8 ∼ 27
L = 3, 4, 5 (We will present the result only for
L = 3)
Generate 1000 random graphs for each (N, L)
Estimate the appearance probability of ideal
funnel among all the possible arrangement ofnumbers for each graph.

Exact enumeration for N ≤ 14
Multicanonical Monte Carlo method for
N > 14This method with Wang-Landau weight learning is
powerful for counting the number of rare states ofcombinatorial problems (application to counting
the number of the magic squares is: A. Kitajima
and MK, PLOS One 2015) .

Results

PDF of ideal funnels (N = 16, L = 3)

PDF of ideal funnels (N = 22, L = 3)

PDF of ideal funnels (N = 27, L = 3). The solidline represents the Log-Normal distribution

We expect that the appearance probability of theideal funnels approaches the Log-Normal
distribution for large N
Implication of Log-Normal
Very small number of networks have a large
probabilitySince the diﬀerent arrangement of the numbercorresponds to the diﬀerent sequence, suchnetworks are rubost against mutation.
Possible explanation for the fact that there all only
relatively small number of folds for known proteins.

0-2<log10P>-4-6-8-10-12-14
10

15

20
N-2

25

Ndependence of log P (bars: 3×S.D.)30

P decreases exponentially with N (as expected)
Number of robust networks decreases more
slowly than typical networks
Evolutionally favorable

Summary and Discussion
We estimated the rareness of the folding
funnels using the random-energy model on therandom network.
PDF of appearance probability of the ideal
funnel is close to Log-Normal type.
There are very small number of the native
conformations that are robust against mutations.

Typecal networks decrease exponentially withN
The rubust networks decrease also exponentiallybut more slowly.

Remark

The model is very simple, abstract, and rather
arbitrary. So is far from reality. We, however, still
consider that we can get some insight on rareness ofthe foldable proteins and its implication toevolutions from this study.