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Extreme-value analysis in nano-biological systems: Applications and Implications
Kumiko Hayashi1, Nobumichi Takamatsu1 and Shunki Takaramoto1
1Institute for Solid State Physics, The University of Tokyo, Kashiwano-ha 5-1-5, Kashiwa, Chiba 277-8581,
Japan
*Correspondence to Kumiko Hayashi
hayashi@issp.u-tokyo.ac.jp
ABSTRACT
Extreme value analysis (EVA) is a statistical method that studies the properties of extreme values of datasets,
crucial for fields like engineering, meteorology, finance, insurance, and environmental science. EVA models
extreme events using distributions such as Fréchet, Weibull, or Gumbel, aiding in risk prediction and
management. This review explores EVA's application to nanoscale biosystems. Traditionally, biological
research focuses on average values from repeated experiments. However, EVA offers insights into molecular
mechanisms by examining extreme data points. We introduce EVA's concepts with simulations and review its
use in studying motor protein movements within cells, highlighting the importance of in vivo analysis due to
the complex intracellular environment. We suggest EVA as a tool for extracting motor proteins’ physical
properties in vivo and discuss its potential in other biological systems. While EVA’s use in nanoscale biological
systems is limited, it holds promise for uncovering hidden properties in extreme data, promoting its broader
application in life sciences.
Introduction
Extreme value analysis (EVA) is a branch of statistics that focuses on extreme values. It is the study of the
statistical properties of particularly large or small values within a data set. EVA is widely applied in various
fields where extreme phenomena hold significant importance, such as engineering, meteorology, finance,
insurance, and environmental science (Basnayake et al 2019; de Haan and Ferreira 2006; Dong 2016; Einmahl
and Magnus 2008; Gembris et al. 2002; Kratz 2017; Rootzen 2017; Songchitruksa and Tarko 2006; Tippett et
al. 2016; Tsuduki 2024; Wong and Collins 2020). The central concepts of EVA involve identifying the largest
or smallest values in a dataset, determining whether these extremes follow distributions like the Fréchet
distribution, Weibull distribution, or Gumbel distribution, and analyzing data points that exceed specific
thresholds to estimate their distribution. EVA is utilized in a broad range of fields. For instance, in meteorology,
it is used to predict and assess extreme weather events such as typhoons, floods, and droughts. In finance, it aids
in risk management for extreme price fluctuations, such as market crashes or surges. In engineering, it is applied
to analyze extreme stresses or loads for evaluating the durability and safety design of structures. EVA serves as
a powerful tool for predicting the frequency and impact of extreme events, aiding in risk management and safety
measures.
In this review paper, we discuss the application of EVA to the analysis of experimental data in nanoscale
biosystems. Traditionally, life sciences have elucidated biological phenomena primarily through the average
values of data obtained from repeated experiments. However, by using EVA, it is possible to explore novel
molecular mechanisms in life sciences through the behaviour of extreme values in the data. In the following
sections, we first learn the concepts of EVA through simple simulations using random numbers. Next, we
introduce the application of EVA to the movement of motor proteins within cells, investigated in our recent
research paper (Naoi et al. 2024). The physical properties of motor proteins, such as force and velocity, have
been investigated by in-vitro single-molecule experiments, in which the functions of motor proteins consisting
of minimal complexes were analyzed in glass chambers (Brenner et al. 2020; Elshenawy et al. 2019; Gennerich
et al. 2007; Hirakawa et al. 2000; Mallik et al. 2004; Schnitzer et al. 2000; Toba et al. 2006). However, because
motor proteins function fully in the intracellular environment and are equipped with accessory proteins, the
investigation of motor proteins in vivo is as important as in-vitro single-molecule experiments. We believe that
EVA can be useful as a new informatics tool for extracting the physical properties of motor proteins in the
complex intracellular environment.
Finally, we discuss the potential applications of EVA in nanoscale biosystems, with a specific example of
its application to the analysis of droplet sizes in liquid-liquid phase separation (Takaramoto and Inoue 2024).
While research utilizing EVA in nanoscale biological systems is still limited, it is believed that undiscovered
properties may lie hidden within the extreme values of experimental data.
Simulation using random numbers
In this section, the analytical methods of EVA are reviewed through a simple simulation using uniform
random numbers ranging from 0 to 1 and Gaussian random numbers. In the case of random numbers ranging
from 0 to 1, it is obvious that the maximum value is 1, while Gaussian random numbers do not have a maximum
value. In other words, we examine two random numbers with different extreme properties.
One block of EVA is considered as a data set of elements
!
. In meteorology,
!
is often considered as one
year, which is 365 days. Here we consider the case
!"#$
for the i-th data set of the uniform random numbers,
from which the largest value (
%&'(
)
) is selected. Note that we set
!"#$$$
for the Gaussian random numbers
because of their slow convergence. Then the selection of the block maximum is repeated
*
times to collect the
block maximum data set +
%&'(
)
, (
-"#./.*
where
*
=500, i.e., we repeat the selection 500 times). Using
the block maximum data set +
%&'(
)
,, the return-level plot is investigated by using the ismev and evd packages
in R (R Core Team 2018) (Fig. 1a and 1b). Here, the return level refers to the magnitude of extreme values
expected to occur once within the sample size we are considering, such as 100 years, 1000 people and so on.
For example, when analyzing river water level data for environmental science case, its return level plot can
provide information such as the maximum water level of a flood expected once every 100 years. The distribution
characteristics of the extreme value data, such as the values of location, scale, and shape parameters of the
extreme value distribution (the Fr
0
1chet distribution, Weibull distribution, or Gumbel distribution) can be
estimated from the behavior of the return level plot (Fig. 1c).
The two axes of the return level plot (Fig. 1a and 1b, bottom) represent the return period
23
and return level
43
. For a given probability
5
,
23"6
7
89:;<#65=;
>
?@
, and
43
is defined by the generalized extreme value
distribution as
#65 "A<43=
, where
A
B
43
C
"0DE
F
6
G
#HI
J
436K
L
MN
?@OP
Q
R;;;;;;;;;;;;;;;;;;;;;;;;;;<#=;;;
Note that
43
represents
%
S
&'(
)
, where +
%
S
&'(
)
, is the rearranged data of +
%&'(
)
,, such that
%
S
&'(
@T%
S
&'(
UT
/T%
S
&'(
V
. Roughly,
23
is the sample number. We obtained parameters of the generalized extreme value
distribution
I
,
K
, and
L
in Eq. (1) by using the ismev and evd packages in R(R Core Team 2018). We found
that
I;W $
for the case of uniform random numbers, which have the maximum value (Fig. 1a), and
I;X$
for
the case of Gaussian random numbers, which do not have the maximum value (Fig. 1b). The existence of a
maximum value is related to the value of the shape parameter
I
. Therefore, we particularly focus on the value
of
I
in EVA.
Depending on the value of
I
, it is classified into the Weibull distribution (
I W $
), Gumbel distribution (
I"
$
), or Fr
0
1chet distribution (
I Y $
). In the case of a Weibull distribution, the extreme value
Z[(
is proved to
exist and is estimated using the following equation:
Z[( "K6LOI
. (2)
It has been known that the extreme values of a uniform distribution converge to a Weibull distribution (Fig. 1a),
and those of a Gaussian distribution to Gumbel distribution (Fig. 1b). In our simulations, we obtained the set of
parameters <
K.L.I
=
"<$R\##]$R$$^_`.$R$a\^]$R$$^b_`.6$Rc\$]$R$#d_`=
for uniform random
numbers ranging from 0 to 1. For the set of parameters <
K.L.I
=, the quantity
K6LOI
(Eq. (2)) was calculated
to be 1.00 as the maximum number, as expected.
The properties of the maximum value of the simulation examples shown in Fig. 1 are known from the
beginning. However, the important point is that the extreme value distribution of any data, whose extreme
properties are not known, can be described by Eq. (1), based on the mathematical theorem. In unknown complex
systems like nanoscale biological ones, knowing the functional form (Eq. (1)) for their extreme value
distributions is a significant advantage. In the next section, we consider the application of this strength in EVA
to intracellular motor proteins. The relationships that hold even in the unknown environment such as the
intracellular space are invaluable for measurements.
Applications of EVA to cargo transport by motor proteins.
Motor protein is a general term for proteins that move and function using energy obtained from adenosine
triphosphate (ATP) hydrolysis. Here, we focus on kinesin and dynein among such motor proteins, which are
responsible for cargo transport and form the basis of intracellular material logistics (Fig. 2a). Particularly, we
investigate anterograde transport by kinesin and retrograde transport by dynein in the axons of neurons. In the
logistics of neurons, both anterograde transport, which carries synaptic materials to the terminals, and retrograde
transport, which collects waste materials, are important. Because logistics in long axons is particularly important
for neurons, and disruptions in this process are often associated with neurological diseases (Guedes-
Dias, Holzbaur 2019; Keefe et al. 2023), the application of EVA to axonal transport is considered to be
significant to help elucidate disease mechanisms in the future by increasing in vivo physical quantities we can
estimate.
Cargo motion can be observed using fluorescence microscopy (Hayashi et al. 2018; Naoi et al. 2024). Then
the transport velocity is measured by analyzing the recorded movies obtained from the fluorescence microscopy.
(Note that fluorescence observation is a standard technique in biophysics, and this review will not discuss the
methods of fluorescence observation.) In in vivo cases, the measured velocity values varied affected by the size
of a cargo and the number of motor proteins involved in the cargo transport (Fig. 3a). Especially since cargo
sizes can differ by a factor of 10, due to high viscosity in cells, the average velocity value reflects the cargo size
rather than the performance of the motor proteins, unlike the situations in in vitro single-molecule experiments.
The differences in the mechanical properties of the two different motor proteins, kinesin and dynein, are also
less likely to be reflected in the velocity values, because the influence of cargo size on the velocity values is
larger. Then, the performance of the motor proteins can be considered in the limit of small cargo sizes. Under
this condition, it is believed that the performance of motor proteins can be accurately evaluated using EVA. In
this review, in the following, we introduce the application of EVA to axonal transport in two different species:
C. elegans worms and mice.
C. elegans worms
The measurement method of transport velocity using fluorescence imaging in living C. elegans worms, was
explained the original paper (Naoi et al. 2024). Note that as shown in Fig. 2a, fluorescent proteins were labeled
on the cargo for the fluorescence microscopy. Approximately 2,000 velocity data were collected from about 200
worms for anterograde and retrograde transport. Among ten velocity values (
!"
10), the block maximum value
(
e&'(
)
) was selected. Note that the results did not differ significantly, whether considering one block as one
worm or randomly selecting 10 velocity data among 2,000 velocity data to form one block (Naoi et al. 2024).
(Refer to the original paper (Naoi et al. 2024) for a discussion on how the block data was decided). One thing
to be careful about is the fact that a measurement difficulty in live worms caused the small size of
!
, which is
typically set to of the order of 1000 to obtain a correct extreme value distribution, highlighting a key problem
in the application of EVA to nanoscale biological systems that should be addressed in the future.
Using the block maximum dataset +
e&'(
)
, (
-"#./.*
where
*
=228 for anterograde transport and
;*
=217
for retrograde transport), the return-level plot was calculated by using the ismev and evd packages in R (R Core
Team 2018) (Fig. 2b).
I; W$
for anterograde transport and
I;X$
for retrograde transport. For anterograde
transport, the return level plot shows a convergent behavior as
23
becomes larger, a property specific to a
Weibull distribution (
I; W$
), and the extreme value
Z[(
was estimated to be 4.0±0.4 µm/s using Eq. (2).
Investigating whether this value corresponds to the maximum speed of kinesin in the cell, in other words, the
rate of ATP hydrolysis, is a significant future issue. When
Z[(
is converted to the ATP hydrolysis rate, it is
approximately four times faster compared to in vitro experiments (Schnitzer et al. 2000). While it is known that
velocity values increase in vivo environments compared to in vitro single-molecule experiments, we mention
this as the maximum value that may correspond to the in vivo ATP hydrolysis rate. This possibility needs to be
investigated in the future.
Because the return-level plot of the retrograde transport (Fig. 2b) shows
IX;$
, and
Z[(
cannot be estimated
for retrograde transport by dynein, noting that the maximum exists only for the case
I; W$
. However, it is
physically implausible for there to be no maximum value for retrograde transport. Therefore, there must be a
mechanism by which the measured velocity values did not converge. In the discussion section, we will explore
why dynein appears to exhibit an apparent behavior of Gumbel distribution or Fr
0
1chet distribution.
Mice
To investigate a mammalian case, EVA was also applied to examine the velocities of synaptic cargo transport
by motor proteins in mouse hippocampal neurons (Fig. 2c, left), as originally reported in our previous study
(Hayashi et al 2021). The return-level plot with
I W $
was observed for anterograde transport. Then the
maximum velocity for anterograde transport in mice hippocampal neurons was calculated to be 5.6±1.4 µm/s
using Eq. (2). The maximum velocity was higher than that of the worms. Comparing the maximum velocities
among various species is a future important issue. Return-level plot with
I Y $
was also observed for
retrograde transport (Fig. 2c, right). It is known that dynein exhibits different mechanical properties depending
on the species, but it is expected to have similar mechanical properties in both worms and mice, judging from
the similar behaviors of the return level plots. The mechanical property referred to here is the force-velocity
relationship described in the next discussion section.
Discussion
In our previous paper on the application of EVA to motor proteins (Naoi et al. 2024), the different property
shown in the return level plots (Fig. 2b, c) was attributed to the different force velocity relationship between
kinesin and dynein. The difference in the convexity of force-velocity relationships (Fig. 3b), which have been
clarified based on in-vitro single-molecule studies using optical tweezers, reflects the different mechanisms
underlying the walking behavior of motor proteins on microtubules, although the biological meanings of
convexity are not clearly understood yet. Due to high viscosity in cells, a motor protein experiences a load
proportional to the cargo size transported by it, which is similar to the situation of applying a load by optical
tweezers. Therefore, we predicted that the behavior of velocity values in vivo would be influenced by the force
(load)-velocity relationship. The observed velocity values seem to be particularly related to the properties of the
force-velocity relationship under low load condition areas (green and yellow area depicted in Fig. 3b, right),
because the viscous load dependent on cargo size is estimated to be a few pN, judging from the ratio of the
average velocity (2 µm/s) to the maximum velocity (4 µm/s), and smaller than the load applied by optical
tweezers, which typically reduces the velocity to nearly zero, i.e., the stall forces of motors (about
f
pN
(Schnitzer et al. 2000, Elshenawy et al. 2019)). The property that
IX$
for the retrograde velocity data was
attributed to the fact that the force-velocity relationship was concave. The steep velocity decrease for the case
of the concave force-velocity relationship in the low-load condition (green area in Fig. 3b) caused a major
variation in the larger velocity values, and this behavior tends to generate a major variation in velocity near the
maximum velocity. Then large gaps generated between
egh&.&'(
)
and
egh&.&'(
)i@
for a large
-
in this case. This
gap caused the non-convergence of
e&'(
)
and
IX$
as a result. In the first place, the appearance of small cargos
is a rare event, so in a concave force-velocity relationship, large velocity values are less likely to converge.
We can actually reproduce the behavior observed for anterograde and retrograde transport using a theoretical
model of the force-velocity relationship derived based on the mechanisms underlying ATP hydrolysis by motor
proteins (Sasaki et al. 2018). The force-velocity relationship for the model is represented as follows:
;;;;;;;;;;ejkl[[
<
m
=
"
<
no@ 6noU
=
p;;;;;;;;;;;;;;;;;;;;;;;;;
#
no@ "#
q@H#
r@stuvOwxy;
;;;;;;;;;;;;;;;;;;;;;;;; #
noU "#
rUs?tzvOwxy;;;;;;;;;;;;;;;;;;;;;;;;;;;<^=
See reference
(Sasaki et al. 2018) for the definitions and values of parameters.
The differences in the model
parameters (Eq. (3)) for kinesin and dynein were resulted in the different convexities of the force-velocity
relationship. In Fig. 3c (top) represent the +
egh&
)
, (4000 data) obtained from the simulations using the model
(Eq. (3)). +
egh&.&'(
)
, were chosen from +
egh&
)
, (
!" #$
). Using these +
egh&.&'(
)
, (
*" b$$
), we calculated
the return-level plots for kinesin and dynein (Fig. 3c, bottom). We found that the tendency that
43
(
"egh&.&'(
)
arranged in ascending order) did not converge for a large
23
(Roughly,
23
is the sample number (1
T23T
400))
in the case of dynein, i.e., the gaps created between
egh&.&'(
)
and
egh&.&'(
)i@
for a large
-
. This is because a
large velocity value is likely to be generated in the case of a concave force-velocity relationship, owing to its
steep slope in the load-sensitive regime.
There is one thing that should be mentioned regarding the conclusion obtained from the numerical
experiments depicted in Fig. 3c. We attributed the property
IX$
in the extreme value data for retrograde
transport (specifically the rare occurrence of large velocity values and the divergence observed in the return
level plot at large
23
values) to a concave force-velocity relationship. However, several mechanisms could lead
to the occasional large velocity values in retrograde transport, such as active non-equilibrium fluctuations
originating from the cellular cytoskeleton (Chaubet et al. 2020; Mizuno et al. 2007) and the participation of
multiple molecular motors (Rai et al. 2013). In the future, we wish to further investigate the origin of large
velocity values observed for retrograde transport.
To expand the application of EVA in nanobiology systems, it is crucial to consider applying EVA to systems
entirely different from motor proteins. Inspired by the poster presentation by Takaramoto and Inoue at the
IUPAB2024 Congress (Takaramoto and Inoue 2024) (note that the the collection of abstracts of the IUPAB2024
congress is published as the IUPAB2024 special issue by the journal, Biophysics and Physicobiology), we
became interested in the analysis of droplet sizes formed by liquid-liquid phase separation (LLPS). Within cells,
there are membrane-less organelles formed through LLPS, such as stress granules and nucleoli. These organelles
are formed through local concentration and separation caused by interactions between specific proteins and
RNA, playing critical roles in cellular functions. While membrane-less organelles formed by LLPS are
reversibly assembled and disassembled under normal conditions, in pathological states, this process can break
down, leading to the formation of irreversible aggregates. In neurodegenerative diseases such as ALS and
Parkinson's disease, abnormal irreversible aggregation of proteins has been reported to cause neuronal cell death.
Takaramoto and Inoue investigated the mechanism of liquid condensate formation by the Parkinson's disease-
related protein α-synuclein (αSyn). A previous study (Hoffmann et al. 2021) has shown that when αSyn coexists
with the neuronal protein synapsin, αSyn is incorporated into the droplets formed by synapsin and can undergo
phase separation at lower concentrations than when αSyn forms droplets alone. Takaramoto and Inoue studied
the detailed molecular mechanism of droplet formation by αSyn in the presence of synapsin. Takaramoto and
Inoue confirmed droplet formation in solution of αSyn and synapsin (in vitro experiment) (Fig. 1 of the abstract
in Reference (Takaramoto and Inoue 2024)), and obtained the droplet size distribution. The characteristics of
the synapsin size distribution were similar in their long tails to the distribution of Gaussian random numbers
(Fig. 1b) and the velocity distribution of retrograde transport by dynein (Fig. 2b and 2c), suggesting that it is a
type of extreme value distribution that does not converge to a maximum value. In other words, we predict that
the distribution of the extreme values of the sizes will converge to either the Gumbel or Fréchet type, which
does not have the maximum (private communication). On the other hand, within neurons, a maximum value for
droplet size is expected due to the fixed cell size. It is important to compare both the droplet formation in the in
vitro experiment and cellular environments to investigate its mechanism through EVA. In addition to
experiments, molecular simulations have emerged as powerful tools to elucidate the physicochemical principles
of biomolecular condensates, complementing experimental approaches. At Joseph's keynote session at
IUPAB2024, we learned the potential of LLPS-specific simulations (Mpipi) (Joseph et al. 2021). Pursuing
whether the maximum value of droplet size is related to the mechanism of disease onset and describing life
sciences through extreme values rather than averages is a crucial task for the future.
Declaration
No funding was received for conducting this study.
Conflict of interest
The authors declare no competing interests.
Data availability
Data sharing is not applicable to this article as no new data were created.
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Fig. 1 Simulation using random numbers. The cases for uniform random numbers ranging from 0 to 1 (a) and
Gaussian random numbers (b). Each panel represents the counts, cumulative probability and return level plot
for the block maximum. The shape parameter
I; W$
for the uniform random numbers, and
I;X$
for the
Gaussian random numbers. Red lines represent
A<%&'(=
(Eq. (1)). (c) Examples of extreme value distributions.
Fig. 2 Schematics of synaptic
cargo transport by motor proteins
in the axon of a neuron.
a A cargo is anterogradely
transported by KIF1A (kinesin)
and retrogradely by cytoplasmic
dynein. b , c Return level plot
and histogram of the block
maximum velocity data of
synaptic cargo transport by motor
proteins. Results for anterograde
(red) and retrograde (blue)
transport, in the cases of the motor
neurons of C. elegans worms
(Naoi et al. 2024) (b), and the
hippocampal neurons of mice
(Hayashi et al. 2021) (c).
Fig. 3 Physical parameters of intracellular cargo transport by motor proteins. a The variety in the values of
cargo transport resulted from the number of motor proteins and cargo sizes. b Load dependence of velocity in
convex and concave cases. An existence of a cargo can be a load in the axons of neurons, because of high
viscosity in the cells. c Simulation results for the model (Eq. (3)).
