Finding Protein Folding Funnels inRandom Networks
Macoto Kikuchi
kikuchi@cmc.osaka-u.ac.jp
Cybermedia Center, Osaka Univ.

CCP2017 (11 Jul. 2017)
paper in preparation

Outline
123
45Introduction: Rareness, Fold, Funnel picture,
Variation of the funnel structure,Networkrepresentation
Question
Model
MethodResults
Probability distribution of the ideal funnels
Network size dependence of the probability andS.D.

6Summary and Discussions

Introduction: rarenessA protein folds into its specific nativeconformation spontaneously under thephysiological conditions.
Anfinsen’s dogma: The native conformation is thethermodynamically stable one determined by theamino acid sequence.
Proteins 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.

ex. The native conformation of lysozyme

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 beenaccepted widely after 1990s as a mechanism behindthe protein folding
Consistency principle (Go)Minimum frustration principle (Bryngelson andWolynes)
The energy landscape is determined by the nativestructure.

Energyfunnel picture
The number ofconformations
decreases
monotonously as
energy lowers andapproaches the nativestate.

It is well known that the GO-like model, in which
the interaction between the residues are determinedby the native structure, can nicely reproduce theexperimentally observed folding processes,
GO-like model is the simplest model thatrealizes the funnel-like energy landscape.
The reason is not yet so clear. But anyway thefunnel picture well describes the protein folding.

Introduction: folds
Number of ”folds” is very small compared to thenumber of proteins.
Fold: skelton of the native structures.
ex. classification by SCOP reported the number offolds is 1195 in the year 2009.
http://scop.mrc-lmb.cam.ac.uk
Althought the definition of the ”fold” is still
ambiguous, it is clear that many proteins have thesame ”fold”.
Why do the folds so scarce?

Introduction: variation of the funnelstructure
Free-energy landscape can have a variety even
within the framework of the GO-like model.
Folding pathways are restricted by the nativestructure, but not uniquely determined.
ex. All the experimentally observed variety of thefolding processes of lysozyme family can be
reproduced by the extended GO-like model, in which
the relative interaction strength of two domains aretuned as a parameter.
Kanzaki and MK, Chem. Phys. Lett. 427

Variety of the free-energy landscape at the folding
temperature for lysozyme by the extended GO-likemodel
Kanzaki and MK, Chem. Phys. Lett. 427(2006) 414.

Three points to be considered

Proteins are rare states of polypeptedes.
Folds are very scarece.
Free-energy landscape and folding pathway can havea variety within the funnel picture.

Motivation of the 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.

Structural network
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 nativeconformation 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 simplerandom 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 and
N − 1,
respectively

5Draw arrows fromthe node ofsmaller number tothat of larger
number, if two
node areconnected (DAG).
The arrowsrepresent thedirections oftransitions.
We assume only
theenergy-lowering

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
Rare event sampling
We estimate the appearance probability of ideal
funnel among all the possible arrangement ofnumbers using the Multicanonical Ensemble Monte
Carlo method with parameters determined by
Wang-Landau procedure.
The smallest number of the dead-end node that canbe reached from the unfolded state is used as energyof the arrangement for Multicanonical method.

We estimated the appearance probability of large
magic squares. (PlosONE 10(5) e0125062)

Multicanonical ensemble method with theWang-Landau learning is highly suitable for
estimating the appearance probability of very rare
conformations
Since the total number of conformations is known
for the present mode, we can estimate ”absolute”
appearance probability of the ideal funnels.

Computaion DetailN = 8 ∼ 27

Exact enumeration for N ≤ 14. Multicanonical for
N > 14.

L=3
Generate 1000 random graphs for each N
Estimate the appearance probability of ideal
funnel for each graph.

ResultsPDF 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
probability
Since the diﬀerent arrangement of the number
corresponds 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

25

N-2

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.
The rubust networks decrease also exponentiallybut more slowly.

Multicanonical and Wang-Landau method issuitable for estimating the probability of rare
conformations.

Remark
The model is very simple, abstract, and ratherarbitrary.
We did not consider special fearures of strucruralgraphs, such as small-worlk, scale-free, hubstructure etc.
The present study, however, will serve as a startpoint to consier rareness of the foldable proteins
and its implication to evolutions from this study.

Study of similar direction is on the way.
Gene reguration network, Genetic code etc.

Rareness will be an important keyword inconsidering life related phenomena in the fieldof Biophysics.

The paper is in preparation.
Contact kikuchi@cmc.osaka-u.ac.jp if you’reinterested in the rareness of life-related
phenomena.