RESEA
RCH
ARTICL
E
Interactions
between
calmodulin
and
neurogranin
govern
the
dynamics
of
CaMKII
as
a leaky
integrator
Mariam
Ordyan
ID
1,2
, Tom
Bartol
2
, Mary
Kennedy
ID
3
, Padmini
Rangamani
ID
4
*
,
Terrence
Sejnowski
ID
1,2
*
1
Institute
for
Neural
Computation,
Universit
y of Californi
a San
Diego,
La
Jolla,
California,
United
States
of
America,
2
Computat
ional
Neurobi
ology
Laborat
ory,
Salk
Institute
for
Biological
Sciences,
La
Jolla,
California
,
United
States
of America,
3
The
Division
of Biology
and
Biological
Engineeri
ng,
California
Institute
of
Technolo
gy,
Pasadena,
Californi
a, United
States
of America,
4
Department
of Mechanical
and
Aerospac
e
Engineeri
ng,
Universit
y of California
San
Diego,
La
Jolla,
California,
United
States
of America
*
prangam
ani@ucsd.
edu
(PR),
terry@salk.
edu
(TS)
Abstract
Calmodulin-depe
ndent
kinase
II (CaMKII)
has
long
been
known
to play
an
important
role
in
learning
and
memory
as
well
as
long
term
potentiation
(LTP).
More
recently
it has
been
sug-
gested
that
it might
be
involved
in the
time
averaging
of synaptic
signals,
which
can
then
lead
to the
high
precision
of information
stored
at a single
synapse.
However,
the
role
of the
scaffolding
molecule,
neurogranin
(Ng),
in governing
the
dynamics
of CaMKII
is not
yet
fully
understood.
In this
work,
we
adopt
a rule-based
modeling
approach
through
the
Monte
Carlo
method
to study
the
effect
of
Ca
2+
signals
on
the
dynamics
of CaMKII
phosphorylation
in the
postsynaptic
density
(PSD).
Calcium
surges
are
observed
in synaptic
spines
during
an
EPSP
and
back-propagating
action
potential
due
to the
opening
of NMDA
receptors
and
voltage
dependent
calcium
channels.
Using
agent-based
models,
we
computationally
inves-
tigate
the
dynamics
of phosphorylation
of CaMKII
monomers
and
dodecameric
holoen-
zymes.
The
scaffolding
molecule,
Ng,
when
present
in significant
concentration,
limits
the
availability
of free
calmodulin
(CaM),
the
protein
which
activates
CaMKII
in the
presence
of
calcium.
We
show
that
Ng
plays
an
important
modulatory
role
in CaMKII
phosphorylation
fol-
lowing
a surge
of high
calcium
concentration.
We
find
a non-intuitive
dependence
of this
effect
on
CaM
concentration
that
results
from
the
different
affinities
of CaM
for
CaMKII
depending
on
the
number
of calcium
ions
bound
to the
former.
It has
been
shown
previously
that
in the
absence
of phosphatase,
CaMKII
monomers
integrate
over
Ca
2+
signals
of cer-
tain
frequencies
through
autophosphor
ylation
(Pepke
et al,
Plos
Comp.
Bio.,
2010).
We
also
study
the
effect
of multiple
calcium
spikes
on
CaMKII
holoenzyme
autophosphorylatio
n, and
show
that
in the
presence
of phosphatase,
CaMKII
behaves
as
a leaky
integrator
of calcium
signals,
a result
that
has
been
recently
observed
in vivo
.
Our
models
predict
that
the
param-
eters
of this
leaky
integrator
are
finely
tuned
through
the
interactions
of Ng,
CaM,
CaMKII,
and
PP1,
providing
a mechanism
to precisely
control
the
sensitivity
of synapses
to calcium
signals.
Author
Summary
not
valid
for
PLOS
ONE
submissions.
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OPEN
ACCESS
Citation:
Ordyan
M, Bartol
T, Kennedy
M,
Rangaman
i P, Sejnowski
T (2020)
Interaction
s
between
calmodulin
and neurogran
in govern
the
dynamics
of CaMKII
as a leaky
integrator
. PLoS
Comput
Biol 16(7):
e1008015.
https://do
i.org/
10.1371/
journal.pcbi.10
08015
Editor:
Joanna
J
ę
drzejew
ska-Szmek,
Instytut
Biologii
Doswiadc
zalnej
im M Nenckiego
Polskie
j
Akademii
Nauk,
POLAND
Received:
December
10, 2019
Accepted:
June
4, 2020
Published:
July 17, 2020
Peer Review
History:
PLOS
recognize
s the
benefits
of transpar
ency
in the peer review
process;
therefore,
we enable
the publication
of
all of the content
of peer review
and author
response
s alongside
final,
published
articles.
The
editorial
history
of this article
is available
here:
https://doi.o
rg/10.1371/jo
urnal.pcbi
.1008015
Copyright:
©
2020
Ordyan
et al. This is an open
access
article
distributed
under
the terms
of the
Creative
Commons
Attribution
License,
which
permits
unrestricte
d use, distribu
tion, and
reproduction
in any medium,
provided
the original
author
and source
are credited.
Data
Availabilit
y Statement:
All bngl files are
available
on GitHub
at the following
reposito
ry:
Author
summary
Neurons
communicate
with
each
other
through
synapses.
The
strength
of
a particular
synapse
is effectively
the
level
of
sensitivity
of
the
postsynaptic
neuron
in
response
to
fir-
ing
of
the
presynaptic
neuron.
The
process
of
changing
synaptic
strength
is dubbed
synap-
tic
plasticity,
a foundational
aspect
of
learning
and
memory.
In
this
work,
we
create
a
computational
model
of
a calcium
signaling
pathway
that
sets
off
a chain
reaction
in
CaM-
KII
phosphorylation,
eventually
leading
to
synaptic
plasticity.
Computational
modeling
provides
a unique
way
to
tease
apart
and
understand
the
non-intuitive
results
of
interac-
tions
between
the
molecules
involved.
Our
model
successfully
predicts
the
experimentally
observed
activation
dynamics
of
this
crucially
important
enzyme
which
is necessary
for
learning.
These
dynamics,
along
with
other
pathways,
regulate
the
size
of
the
synapse,
which
is known
to
be
highly
correlated
with
synaptic
strength.
In
this
work,
we
reveal
quantitative
characteristics
of
CaMKII
activation
for
various
stimuli,
leading
to
important
insights
regarding
the
potential
role
of
Neurogranin,
a scaffolding
protein
in
this
pathway.
Introduction
Information
is stored
in
the
brain
through
synaptic
plasticity.
It has
been
reported
that
synap-
tic
strength
is highly
correlated
with
the
size
of
the
spine
head,
and
the
precision
of
informa-
tion
stored
at
a single
synapse
is quite
high
despite
the
stochastic
variability
of
synaptic
activation
[1–3].
Structural
changes
to
the
postsynaptic
spine
that
can
lead
to
spine
enlarge-
ment,
and
thus
structural
plasticity
are
triggered
by
Ca
2+
signaling
[4,
5].
Time
averaging
of
these
calcium
signals
has
been
suggested
as
a plausible
mechanism
for
achieving
the
high
pre-
cision
of
information
processing
observed
in
spines.
Furthermore,
phosphorylation
of
cal-
cium/calmodulin-dependent
protein
kinase
II (CaMKII)
has
been
postulated
as
the
most
probable
pathway
satisfying
the
long
time
scales
predicted
for
averaging
[1].
CaMKII
is an
autophosphorylating
kinase
[6,
7]
[8–10];
in
postsynaptic
densities
(PSD),
CaMKII
has
been
shown
to
play
an
important
role
in
learning
and
memory
[11].
Specifically,
mice
with
a mutation
in
a subtype
of
CaMKII
exhibit
deficiencies
in
long-term
potentiation
(LTP)
and
spatial
learning
[10,
12,
13].
Moreover,
CaMKII
expression
regulates
the
rate
of
dendritic
arborization
and
the
branch
dynamics
[14,
15],
highlighting
its
importance
in
struc-
tural
plasticity.
Additionally,
CaMKII
has
been
shown
to
bind
actin,
the
major
cytoskeletal
protein
in
dendritic
spines
[16–18],
further
emphasizing
its
role
in
structural
plasticity.
Activa-
tion
of
CaMKII
is exquisitely
regulated
at
multiple
levels
as
summarized
below.
•
CaMKII
activation
by
calmodulin
(CaM)
:
CaMKII
is activated
by
calmodulin
(CaM)
[6],
which
is a protein
with
4
Ca
2+
binding
sites:
2 on
its
C-terminal
and
2 on
N-terminal
(Fig
1A)
[19,
20].
CaM
binds
Ca
2+
cooperatively
and
is able
to
activate
CaMKII
more
potently
if
it has
more
Ca
2+
ions
bound
to
it [21].
•
Neurogranin
(Ng)-CaM
interactions
:
In
the
absence
of
Ca
2+
, CaM
is bound
to
scaffolding
protein
neurogranin
(Ng),
which
dramatically
reduces
the
affinity
of
CaM
for
Ca
2+
(Fig
1B).
On
the
other
hand,
Ca
2+
decreases
binding
affinity
of
CaM
for
Ng
[22].
Thus,
CaM
activa-
tion
and
therefore
CaMKII
activation
depend
on
the
competitive
effects
of
Ca
2+
and
Ng.
•
The
role
of
structure
in
CaMKII
activation
:
Further
complexity
for
CaMKII
activation
is
built
into
the
structure
of
the
molecule
itself.
CaMKII
is a dodecamer
arranged
in
2 stacked
hexomeric
rings
[6,
23].
Monomers
of
CaMKII
truncated
to
remove
the
association
domain
PLOS COMP
UTATIONAL
BIOLOGY
Interactio
ns
between
calmoduli
n and
neurogran
in govern
the
dynamics
of CaMKI
I
PLOS
Computationa
l Biology
| https:/
/doi.org/10.13
71/journal.p
cbi.1008015
July
17,
2020
2 / 29
https://github
.com/marordy
an/CaMKII_we
ll_
mixed/.
Funding:
PR,MO,TB
,TS: FA9550-18-
1-0051
US Air
Force
https://
www.airforce.c
om/, TB,TS:
P41GM103
712 National
Institute
of Health
https://
www.nih.go
v/ MK,TB,TS:
NS44306MK
National
Institute
of Health
https://ww
w.nih.gov/
MK,TB,T
S:
DA030749
National
Institute
of Health
https://ww
w.
nih.gov/
The funders
had no role in study
design,
data collection
and analysis
, decision
to publish,
or
preparation
of the manuscript.
Competing
interests
:
The authors
have declared
that no competing
interests
exist.
(mCaMKII)
contain
a kinase
domain,
CaM-binding
domain,
and
phosphorylation
sites
T286
(287),
and
T305(306)
[9,
24–26].
When
the
CaM-binding
domain
is unoccupied,
and
the
T286(287)
site
is unphosphorylated,
the
monomer
is in
a conformation
such
that
the
kinase
is
inactive
(Fig
1C,
top).
When
CaM
is bound
to
a CaMKII
monomer,
the
latter
undergoes
a
conformational
change,
such
that
the
kinase
is now
active.
An
active
CaMKII
monomer
can
phosphorylate
other
Ca
2+
/
CaM
-bound
CaMKII
monomers
resulting
in
CaMKII
autopho-
sphorylation.
A CaMKII
monomer
can
only
be
phosphorylated
if the
calmodulin
bound
to
it has
at
least
one
calcium
ion,
and
once
phosphorylated
[27],
the
monomer
remains
in
the
active
state
even
after
unbinding
from
CaM
(Fig
1C
middle).
The
same
is true
for
full
length
monomers
that
are
bound
within
the
holoenzyme.
In
this
manner,
a brief
Ca
2+
influx
initiates
a prolonged
CaMKII
activation
[7,
9, 26,
27,
27–29].
Within
the
holoenzyme
each
of
the
monomers
can
phosphorylate
neighbors,
as
long
as
the
former
is active
and
the
latter
has
CaM
bound
to
it (Fig
1D)
[8,
27].
The
phosphorylation
rate
of
the
CaMKII
monomers
depends
on
how
many
Ca
2+
ions
are
bound
to
the
CaM
bound
to
that
substrate
[30].
Fig
1.
Schematic
represe
ntation
of
the
interaction
s between
calmodulin
(CaM),
calcium,
CaMKII,
and
neurogranin
(Ng).
(A)
Calmodulin
has
4 calcium
binding
sites;
it binds
Ca
2+
cooperatively
and
its
ability
to
activate
CaMKII
depends
on
how
many
of
these
sites
are
occupied.
(B)
Neurograni
n (Ng)
is a scaffoldin
g molecule;
upon
binding
calmodulin
it
dramatical
ly
reduces
the
latter’s
affinity
for
calcium.
In
this
work,
we
assume
that
CaM
cannot
bind
calcium
and
Ng
simultane
ously.
(C)
The
default
conformatio
n of
CaMKII
monome
rs
is inactive
(top);
they
can
be
activated
by
CaM
bindin
g. Once
bound
to
CaM,
the
monomers
can
be
phosphoryl
ated
by
another
active
CaMKII
protein.
In
this
case,
the
CaMKII
monome
r will
remain
active
even
after
losing
CaM.
Protein
phosph
atase
1 (PP1)
dephosphoryl
ates
CaMKII
(bottom
). (D)
CaMKII
holoenzyme
is a dodecamer
, which
consists
of
2 stacked
hexomeric
rings
of
CaMKII
monome
rs.
Within
the
ring,
a given
CaMKII
monomer
can
be
phosphoryla
ted
by
its
neighbor
provid
ed
that
they
are
both
in
the
active
conformatio
n.
This
is commo
nly
referred
to
as
autophospho
rylation.
https://d
oi.org/10.1371
/journal.pcbi.10
08015.g001
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UTATIONAL
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Interactio
ns
between
calmoduli
n and
neurogran
in govern
the
dynamics
of CaMKI
I
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•
Phosphatases
are
important
:
Various
types
of
protein
phosphatases
(PP1,
PP2A,
PP2B,
PP2C)
can
dephosphorylate
CaMKII
in
the
brain
and,
if CaM
is not
bound
to
the
latter,
bring
it back
to
its
inactive
state
[25,
31].
In
the
PSD,
however,
protein
phosphatase
1 (PP1)
has
been
shown
to
be
mainly
responsible
for
CaMKII
dephosphorylation
[31,
32]
(Fig
1C
bottom).
Many
computational
models
of
CaMKII
dynamics
have
been
developed
in
the
literature
to
probe
the
different
interactions
in
this
cascade
at
varying
levels
of
detail
[11,
33–40].
A large
majority
of
these
models
focused
on
the
bistability
behavior
of
CaMKII
[11,
33,
35,
37,
38,
40],
which
resulted
from
the
nonlinear
rate
functions
used
in
the
model.
However,
experiments
suggest
that
in
long-term
potentiation,
CaMKII
is only
transiently
activated
and
does
not
exhibit
bistable
behavior
[41,
42].
Furthermore,
as
noted
recently
[43],
the
behavior
of
CaMKII
is complicated
by
its
multimeric
nature
and
by
the
intersubunit
interaction
necessary
for
phos-
phorylation
of
T286,
which
is affected
by
the
presence
of
Ng
and
protein
phosphatases.
There-
fore,
a more
complete
computational
model
of
CaMKII
dynamics
needs
to
account
for
both
the
behavior
of
the
monomers
and
the
dynamics
of
CaMKII
holoenzyme
in
the
presence
of
Ng
and
PP1.
Here
we
sought
to
examine
how
the
competition
between
Ca
2+
-mediated
activation
of
CaM
and
Ng
scaffolding
of
CaM
affects
the
response
of
CaMKII
to
calcium
signals.
To
do
so,
we
developed
two
computational
models—Model
1 that
accounts
for
CaMKII
monomer
activation
and
Model
2 that
investigates
holoenzyme
kinetics—in
an
agent-based
framework.
Model
1 is
built
on
a previously
published
model
of
activation
of
CaMKII
monomers
by
Ca
2+
/
CaM
[30]
and
now
includes
the
role
of
the
scaffolding
molecule
Ng
and
the
protein
phosphatase
PP1.
Using
these
models
we
investigated
the
dynamics
of
monomeric
and
dodecameric
CaMKII
phosphorylation
as
a function
of
the
dynamics
of
Ca
2+
-influx
and
of
interactions
with
Ng.
An
important
distinction
between
our
model
and
those
presented
in
[34]
and
[44]
is that
we
did
not
explicitly
construct
our
model
to
replicate
desired/observed
CaMKII
activation
dynamics.
Rather,
our
model
hinges
solely
on
rate
constants
for
interactions
between
molecules
based
on
biochemical
experiments
and
presented
in
Tables
1 and
2. Our
results
show
that
under
condi-
tions
of
our
model
CaMKII
behaves
as
a leaky
integrator
and,
more
importantly,
that
the
scaf-
fold
molecule,
Ng,
tunes
the
behavior
of
this
leaky
integrator.
Methods
We
constructed
the
models
at
different
scales
to
characterize
CaMKII
phosphorylation
at
increasing
levels
of
complexity.
First,
we
added
the
scaffolding
molecule
Ng
to
the
model
from
Pepke
et al
[30],
to
investigate
the
effect
of
Ng
on
CaMKII
phosphorylation
dynamics.
Second,
we
added
PP1
to
this
CaMKII
monomers
model
to
simulate
the
phosphorylation-dephospho
r-
ylation
cycle
and
characterize
the
effects
of
Ng
on
this
system.
Finally,
we
built
a model
of
a
dodecameric
holoenzyme
and
looked
at
the
response
of
the
holoenzyme-phospha
tase
system
to
calcium
signals
and
how
it is affected
by
Ng.
An
important
assumption
of
all
the
models
developed
in
this
study
is that
only
one
of
the
phosphorylation
sites
(T286/7)
of
CaMKII
is
considered
throughout.
The
phosphorylation
of
T305/6
site
is a slower
reaction
that
is known
to
inhibit
CaM
binding
to
T286/7-unphosphory
lated
CaMKII
subunit
[114],
and
is omitted
from
our
simulations.
Model
description
Model
of
CaMKII
monomers.
We
begin
with
the
model
of
CaMKII
monomers
(mCaM-
KII)
described
by
Pepke
et
al.
[30].
This
model
includes
Ca
2+
binding
rates
to
CaM,
CaM
PLOS COMP
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4 / 29
binding
rates
to
mCaMKII,
and
phosphorylation
rates
of
active
CaMKII
monomers
by
one
another,
all
depending
on
how
many
Ca
2+
ions
are
bound
to
the
CaM
molecules
involved.
We
incorporate
Ng
binding
to
CaM
with
the
rate
constants
from
[55],
and
assume
that
the
binding
of
CaM
to
Ca
2+
and
Ng
is mutually
exclusive
(Table
1).
In
addition
to
the
reactions
in
[30],
we
included
CaM
unbinding
and
binding
to
phosphorylated
CaMKII,
albeit
with
slower
kinetics
[52].
This
reaction
is important
for
the
timescales
of
our
interest
(on
the
order
of
minutes)
and
we
adapt
these
reaction
rates
from
[52].
Finally,
CaMKII
dephosphorylation
by
PP1
is modeled
as
Michaelis-Menten
kinetics,
with
the
rate
constants
from
[56]
(Table
1).
Table
1.
Reaction
rates
for
the
model.
Descriptio
n
Parameter
Value
Reference
Paramet
er
Value
Reference
Ca
2+
binding
to
CaM
k
1
C
on
4
μ
M
-1
s
-1
[21,
30,
45,
46]
k
1
C
off
40.24
s
-1
[47–49]
k
2
C
on
10
μ
M
-1
s
-1
[21,
30,
45,
46]
k
2
C
off
9.3
s
-1
[49]
k
1
N
on
100
μ
M
-1
s
-1
[21,
30,
45,
46]
k
1
N
off
2660
s
-1
[47,
49–51]
k
2
N
on
150
μ
M
-1
s
-1
[21,
30,
45,
46]
k
2
N
off
990
s
-1
[47–51]
CaM
binding
to
unphosph
orylated
CaMKII
k
CaM
0
on
3.8
�
10
−
3
M
-1
s -1
[30,
52]
k
CaM
0
off
6.56
s
-1
[30,
52]
k
CaM
1
C
on
59
�
10
−
3
μ
M
-1
s
-1
[30,
52]
k
CaM
1
C
off
6.72
s
-1
[30,
52]
k
CaM
2
C
on
0.92
μ
M
-1
s
-1
[30,
52]
k
CaM
2
C
off
6.35
s
-1
[30,
52]
k
CaM
1
C
1
N
on
0.33
μ
M
-1
s
-1
[30,
52]
k
CaM
1
C
1
N
off
5.68
s
-1
[30,
52]
k
CaM
2
C
1
N
on
5.2
μ
M
-1
s
-1
[30,
52]
k
CaM
2
C
1
N
off
5.25
s
-1
[30,
52]
k
CaM
1
N
on
22
�
10
−
3
μ
M
-1
s
-1
[30,
52]
k
CaM
1
N
off
5.75
s
-1
[30,
52]
k
CaM
2
N
on
0.1
μ
M
-1
s
-1
[30,
52]
k
CaM
2
N
off
1.68
s
-1
[30,
52]
k
CaM
1
C
2
N
on
1.9
μ
M
-1
s
-1
[30,
52]
k
CaM
1
C
2
N
off
2.09
s
-1
[30]
k
CaM
4
on
30
μ
M
-1
s
-1
[30,
52]
k
CaM
4
off
1.95
s
-1
[30,
52]
Ca
2+
binding
to
CaM-Ca
MKII
k
K
1
C
on
44
μ
M
-1
s
-1
[30,
49]
k
K
1
C
off
29.04
s
-1
[30,
49]
k
K
2
C
on
44
μ
M
-1
s
-1
[30,
49]
k
K
2
C
off
2.42
s
-1
[30,
49]
k
K
1
N
on
75
μ
M
-1
s
-1
[30,
49]
k
K
1
N
off
301.5
s
-1
[30,
49]
k
K
2
N
on
76
μ
M
-1
s
-1
[30,
49]
k
K
2
N
off
32.68
s
-1
[30,
49]
CaMKII
binding
to
CaM-Ca
MKII
�
k
CaMKII
on
50
μ
M
-1
s
-1
[30,
53]
k
CaMKII
off
60
s
-1
[27,
30,
54]
CaMKII
phosphor
ylation
k
CaM
1
C
p
0.032
s
-1
[21,
30]
k
CaM
2
C
p
0.064
s
-1
[21,
30]
k
CaM
1
C
1
N
p
0.094
s
-1
[21,
30]
k
CaM
2
C
1
N
p
0.124
s
-1
[21,
30]
k
CaM
1
N
p
0.061
s
-1
[21,
30]
k
CaM
2
N
p
0.12
s
-1
[21,
30]
k
CaM
1
C
2
N
p
0.154
s
-1
[21,
30]
k
CaM
4
p
0.96
s
-1
[21,
30]
CaM
binding
to
phosphory
lated
CaMKI
I
k
CaM
0
p
;
on
1.27
�
10
−
3
μ
M
-1
s
-1
[30,
52]
k
CaM
1
C
p
;
on
19.7
μ
M
-1
s
-1
[30,
52]
k
CaM
2
C
p
;
on
0.3
μ
M
-1
s
-1
[30,
52]
k
CaM
1
C
1
N
p
;
on
1.1
μ
M
-1
s
-1
[30,
52]
k
CaM
2
C
1
N
p
;
on
1.73
μ
M
-1
s
-1
[30,
52]
k
CaM
1
N
p
;
on
7.3
μ
M
-1
s
-1
[30,
52]
k
CaM
2
N
p
;
on
0.03
μ
M
-1
s
-1
[30,
52]
k
CaM
1
C
2
N
p
;
on
0.63
μ
M
-1
s
-1
[30,
52]
k
CaM
4
p
;
on
10
μ
M
-1
s
-1
[30,
52]
k
CaM
p
;
off
0.07
s
-1
[52]
Ng
binding
to
CaM
k
Ng
on
5
μ
M
-1
s
-1
[55]
k
Ng
off
1 s
-1
[55]
CaMKII
dephospho
rylation
by
PP1
(Micha
elis-Menten
constants)
k
cat
0.41
s
-1
[56]
K
m
11
μ
M
[56]
All
the
numbers
with
the
exception
of
“CaMKII
binding
to
CaM-Ca
MKII”
are
used
in
both
monomers
and
holoenz
yme
models.
The
difference
between
the
2 models
are
the
initial
conditions:
we
either
start
the
simulations
with
only
monom
ers
or
only
holoenz
yme,
and
do
not
allow
the
disintegra
tion
of
the
latter.
�
only
present
in
the
model
for
monomers
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ournal.pc
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t001
PLOS COMP
UTATIONAL
BIOLOGY
Interactio
ns
between
calmoduli
n and
neurogran
in govern
the
dynamics
of CaMKI
I
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17,
2020
5 / 29
Model
of
CaMKII
holoenzyme.
Assumptions
specific
to
the
holoenzyme
model
:
It has
been
shown
that
while
the
kinase
domains
of
individual
subunits
are
attached
to
the
rigid
hub
domain
with
a highly
flexible
linker
domain,
�
20%
of
subunits
form
dimers,
and
�
3%
of
them
are
in
a compact
conformation,
both
of
which
render
CaM-binding-domain
inaccessible
[115].
Here,
we
do
not
include
such
detailed
interactions
between
linker
domains
and
flexibil-
ity
of
the
individual
subunits
within
the
holoenzyme.
Rather,
we
assume
that
the
kinase
domains
are
positioned
rigidly
within
the
2 hexameric
rings,
such
that
each
subunit
is in
a
position
to
phosphorylate
only
one
of
its
neighbors
as
depicted
in
Fig
1D.
We
further
assume
that
the
CaMKII
subunits
within
the
holoenzyme
have
the
same
binding
rates
to
different
species
of
Ca
2+
/
CaM
as
the
CaMKII
monomers,
and
once
activated
their
phosphorylation
rate
is the
same
as
that
of
the
corresponding
monomers.
Any
possible
Table
2.
Reactions
implemen
ted
in
our
models.
Ca
2
þ
þ
CaM
ðð
n
1
Þ
Ca
2
þ
Þ
Ð
k
n
on
k
n
off
CaM
ð
nCa
2
þ
Þ
Ng
þ
CaM
ð
0
Ca
2
þ
Þ
Ð
k
ng
on
k
ng
off
CaM
�
Ng
CaMKII
þ
CaM
ð
nCa
2
þ
Þ
Ð
k
CaMnCa
on
k
CaMnCa
off
CaMKII
�
CaM
ð
nCa
2
þ
Þ
CaMKII
�
CaM
ðð
n
1
Þ
Ca
2
þ
Þþ
Ca
2
þ
Ð
k
KnCa
on
k
KnCa
off
CaMKII
�
CaM
ð
nCa
2
þ
Þ
CaMKII
�
CaM
ð
nCa
2
þ
Þþ
CaMKII
�
CaM
ð
mCa
2
þ
Þ
Ð
k
CaMKII
on
k
CaMKII
off
CaMKII
�
CaM
ð
nCa
2
þ
Þ�
CaMKII
�
CaM
ð
mCa
2
þ
Þ
CaMKII
�
CaM
ð
nCa
2
þ
Þ�
CaMKII
�
CaM
ð
mCa
2
þ
Þ
!
k
CaMmCa
p
CaMKII
�
CaM
ð
nCa
2
þ
Þ�
pCaMKII
�
CaM
ð
mCa
2
þ
Þ
pCaMKII
�
CaM
ð
nCa
2
þ
Þ
Ð
k
CaM
p
;
off
k
CaMnCa
p
;
on
pCaMKII
þ
CaM
ð
nCa
2
þ
Þ
PP
1
þ
pCaMKII
�
CaM
ð
nCa
2
þ
Þ
Ð
k
M
PP
1
�
pCaMKII
�
CaM
ð
nCa
2
þ
Þ !
k
cat
PP
1
þ
CaMKII
�
CaM
ð
nCa
2
þ
Þ
Listed
are
the
reactions
implement
ed
in
both
models
with
the
exception
of
the
5th
reaction
between
2 CaMKII
monom
ers,
that
is only
present
in
the
mCaMKII
model
since
in
the
hCaMKI
I model
the
the
subunit
s are
already
linked
to
one
another
within
the
holoenzym
e. Here
n stands
for
the
number
and
different
configuratio
ns
of
Ca
2+
ions
bound
to
CaM
(thus
nCa
2+
should
be
read
as
‘aCbN’,
where
a and
b can
range
from
0 to
2,
see
Table
1.)
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ournal.pc
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t002
PLOS COMP
UTATIONAL
BIOLOGY
Interactio
ns
between
calmoduli
n and
neurogran
in govern
the
dynamics
of CaMKI
I
PLOS
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71/journal.p
cbi.1008015
July
17,
2020
6 / 29
allosteric
interactions
are
ignored.
Based
on
these
assumptions,
the
model
for
the
CaMKII
holoenzyme
does
not
include
the
reaction
for
2 CaMKII
monomers
binding
to
one
another.
Rather,
once
the
appropriate
neighbor
of
an
active
CaMKII
subunit
binds
Ca
2+
/
CaM
,
the
phosphorylation
rule
is invoked
with
the
corresponding
reaction
rate
(Table
1).
Model
development.
To
model
the
dynamics
of
CaMKII
holoenzyme,
we
changed
the
types
of
molecules
that
we
start
the
simulation
with.
Instead
of
CaMKII
monomers,
we
initi-
ated
the
simulation
with
CaMKII
holoenzymes,
and
defined
the
phosphorylation
rules
for
individual
subunits
such
that
a given
subunit
can
only
phosphorylate
one
of
its
neighbors
(see
model
assumptions).
The
rules
are
the
same
as
those
of
the
monomers,
with
the
exception
of
2
CaMKII
monomers
binding
each
other
to
phosphorylate
one
another,
since
they
are
already
bound
within
the
holoenzyme.
In
this
case,
the
reaction
rules
apply
to
the
CaMKII
subunits
within
the
holoenzyme
rather
than
CaMKII
monomers.
This
model
does
not
contain
any
CaMKII
monomers
outside
of
the
holoenzyme.
We
calcu-
lated
the
total
number
of
the
holoenzymes
to
keep
the
concentration
of
the
[
mCaMKII
]
= 80
μ
M
consistent
with
our
model
of
the
monomers.
For
our
simulated
PSD
volume
V
= 0.0156
μ
m
3
this
concentration
corresponds
to
80
μ
M
= 80
�
(6.022
�
10
23
�
0.0156
�
10
−
15
)
�
10
−
6
monomers
which
constitutes
752/12
�
63
holoenzymes.
Rule-based
modeling
and
BioNetGen
Since
each
CaMKII
monomer
can
bind
a calmodulin
molecule,
which
can
adopt
9 distinct
states
(assuming
that
the
2
Ca
2+
binding
sites
on
each
of
the
lobes
are
indistinguishable),
and
be
phosphorylated
or
unphosphorylate
d,
there
are
more
than
18
12
states
the
dodecameric
holoenzyme
could
adopt.
To
avoid
a combinatorial
explosion
of
states
we
built
our
model
in
a
rule-based
modeling
language
BioNetGen
[116],
which
is briefly
described
here.
A powerful
idea
at
the
heart
of
rule-based
modeling
is colloquially
referred
as
“don’t
care,
don’t
write”
[117,
118].
Here,
for
a given
reaction
(or
rule),
we
only
specify
the
states
of
the
reactants
that
are
relevant
for
the
said
reaction,
and
leave
the
rest
unspecified.
For
example,
a
specific
CaMKII
subunit
will
bind
a given
species
of
Ca
2+
/
CaM
with
the
same
rate
regardless
of
the
conformation
of
its
neighbors.
Thus,
a rule
to
bind
a CaM
molecule
that
has
2
Ca
2+
ions
bound
to
its
C-terminus
would
look
like:
CaMKII
(286
�
U
,
cam
)
+
CaM
(
camkii
,
C
�
2,
N
�
0)
()
CaMKII
(286
�
U
,
cam
!1).
CaM
(
camkii
!1,
C
�
2,
N
�
0),
k
CaM
2
C
on
;
k
CaM
2
C
off
.
This
rule
indicates
that
regardless
of
the
presence
or
conformations
of
a non-phosphory-
lated
(indicated
by
286
�
U
)
CaMKII
subunits’
neighbors
it binds
CaM,
with
2
Ca
2+
ions
at
its
C lobe,
reversibly
with
the
reaction
rates
k
CaM
2
C
on
and
k
CaM
2
C
off
(Table
1).
This
dramatically
reduces
the
number
of
reactions
that
need
to
be
written.
In
the
case
of
our
CaMKII
model,
we
only
end
up
with
�
40
reactions
despite
the
huge
number
of
possible
states.
In
BioNetGen,
once
the
reactions
have
been
specified,
if there
is no
danger
of
running
into
combinatorial
explosion
(the
numbers
of
possible
states
and
reactions
are
manageable),
a full
network
of
reactions
can
be
generated,
and
the
simulations
can
be
run
with
ordinary
differen-
tial
equations
(ODEs)
[118].
If the
number
of
states
of
a given
enzyme
is too
large
however,
a
stochastic
agent-based
approach
is adopted,
in
which
case
only
the
reactions
between
existing
discrete
molecules
that
occur
during
the
simulation
need
to
be
tracked
by
the
program
(net-
work
free
simulation
[119,
120]).
In
this
case,
the
important
quantity
is the
number
of
different
states
present
which
can
be
much
smaller
than
the
number
of
possible
states.
In
this
work,
we
conduct
the
simulations
for
the
monomer
model
with
ODEs,
and
for
the
holoenzyme
model
with
the
network
free
method.
PLOS COMP
UTATIONAL
BIOLOGY
Interactio
ns
between
calmoduli
n and
neurogran
in govern
the
dynamics
of CaMKI
I
PLOS
Computationa
l Biology
| https:/
/doi.org/10.13
71/journal.p
cbi.1008015
July
17,
2020
7 / 29