►
Description
Talk from materials presented at various Dynamics Days conferences on Dynamical Representations of Heterochrony and applications to theoretical morphospace.
A
Hello,
my
name
is
bradley
alicia,
I'm
from
the
orthogonal
research
and
education
lab
in
the
openworm
foundation.
There's
my
url
at
the
bottom
of
the
screen,
and
today
I'm
here
to
talk
to
you
about
dynamical
representations
of
head
of
our
crony
in
the
developmental
process,
and
this
is
something
that's
been
presented
at
a
number
of
dynamics.
Days,
conferences,
including
dynamics,
knees,
2019
and
dynamics
days,
digital
2020..
A
So
I'm
going
to
start
this
with
a
question,
and
that
is
how
does
a
developmental
phenotype
grow
and
change
shape
from
a
spherical
egg
to
a
complex
organism?
And
you
can
see
in
this
example
that
we
are
using
the
c
elegans
model
organism
to
look
at
this
problem.
We
have
a
very
early
stage
egg
and
it
transforms
into
this
adult
phenotype.
A
But
in
this
presentation
we're
going
to
talk
about
how
to
put
some
mathematical
flesh
on
the
bones
of
the
idea
so
to
demonstrate
this
we're
going
to
start
with
three
different
graphs
by
various
graphs,
and
we
ask
the
question:
given
the
same
length
of
development,
one
of
the
following
three
conditions
may
hold,
so
the
first
case
is
the
one
on
the
left,
which
is
that
growth
starts
at
the
same
time,
but
slows
your
speeds
up.
So
this
blue
function
is
the
sort
of
control
function.
A
This
is
the
original
growth
rate
it
starts
here
and
it
ends
up
here
if
we
take
if
we
change
the
growth
rate
so
that
it's
slower,
we
end
up
with
this
red
function.
So
it
starts
at
the
same
time,
but
it
grows
more
slowly
and
so
the
end
of
the
function
is
exhibits,
less
growth,
so
this
y-axis
represents
some
facet
of
growth
and
this
x-axis
represents
developmental
time.
So
if
we
go
up
to
the
upper
right,
we
ask
if
growth
starts
later
and
proceeds
at
the
same
speed.
What
does
that
look
like?
A
And
so
again
we
have
this
blue
function.
That
is
the
same
as
here:
it's
the
original
function
and
what
we
can
do
is
take.
This
blue
function
start
growth,
the
growth
process
later
in
time
and
then
proceed
at
the
same
rate,
and
when
we
do
that,
we
end
up
with
less
growth
because
we
run
out
of
time
to
grow
any
longer,
and
so
we
can
go
down
to
the
upper
right
hand
corner
of
this
graph
and
look
at
a
different
scenario.
A
In
this
case.
The
growth
starts
later,
but
it
speeds
it
speeds
up
so
that
it
catches
up
with
the
original
function.
So
in
this
blue
function
we
can
start
growth
later,
so
we
can
delay
the
growth
onset
of
growth,
but
then
growth
is
faster
and
it
speeds
up
and
catches.
This
blue
function
with
the
offset
of
growth,
and
so
why
is
this
important?
A
Well,
it
describes
how
long
different
parts
of
the
body
can
grow
and
that
describes
a
lot
of
the
aspects
of
growth
and
form
and
shape,
and
so
how
do
we
provide
a
quantitative
description
of
this
process?
There
exists
a
quantitative
description
already,
but
it's
largely
based
on
that
bivariate
type
of
analysis,
linear
using
linear
functions
and
that's
the
description.
A
Now
we
know
from
biology
that
this
bivariate
relationship
has
is
very
complex.
For
example,
changes
in
timing
and
development.
We
know
are
controlled
by
something
called
heterochronic
genes
which
are
genes
that
are
expressed
that
change
the
timing,
the
onset
and
offset
of
growth,
which
provide
an
input
to
developmental
modules,
which
are
parts
of
the
phenotype
that
evolve
or
develop
at
different
rates.
A
They
provide
an
input
to
developmental
modules,
which
are
differentially
growing
in
development
and
shaping,
and
then
the
phenotypic.
This
results
in
phenotypic
modules,
which
we
observe
in
the
adult
phenotype
the
developmental
modules
being
the
inputs
to
those
over
time,
and
this
all
acts
to
shape
this
differential
growth.
A
And
so
we
can
go
through
a
couple
hypotheses
here
to
sort
of
get
a
sense
of
what's
going
on
and
we
can
use
some
mathematical
representations
to
describe
these
different
hypotheses
of
heterochromy.
I
don't
want
to
call
them
components
because
we
want
to
be
open-minded
about
it,
so
the
growth
hypothesis
of
heterocrony
suggests
that
using
a
linear
model,
the
proportion
of
time
between
onset
alpha
and
offset
beta
of
growth
determines
this
growth
trajectory.
So
this
is
a
simple
linear
model
like
we
saw
with
the
bivariate
case.
A
This
goes
back
to
albert,
and
this
is
these:
are
the
conditions
for
this
linear
model?
You
have
an
onset
and
offset
the
onset
is
alpha,
the
offset
is
beta
and
you
just
simply
turn
those
on
and
off
at
different
times,
and
so
you
have
this
happens
in
developments
very
linear
and
we
have
a
mathematical
representation
of
it.
So
this
is
what
it
looks
like
you
have
this
alpha
this
beta,
you
change
alpha.
You
change
beta.
A
So
in
this
case
I've
drawn
out
the
k
or
the
slope
for
these
two
red
functions
and
these
two
red
functions
are
different,
but
the
black
and
the
red
functions
are
also
different,
so
the
growth
hypothesis
would
suggest
that,
when
the
relative
values
of
alpha
and
beta
change,
the
slope
k
changes
as
well,
and
so
this
difference
between
k1
and
k2
are
equivalent
to
something
called
tau,
which
we'll
talk
about
several
slides
down
the
line.
So
keep
that
in
mind,
but
this
is
basically
this
change
in
slope
is
something
greater
than
just
something
trivial.
A
So
the
growth
hypothesis
will
suggest
suggests
that
you
can
have
more
than
one
trajectory
in
the
same
organism
or
in
the
same
developmental
duration
or
the
developmental
duration
for
a
single
organism.
So
the
growth
hypothesis
suggests
that
we
can
have
two
growth,
trajectories,
y1
and
y2,
which
are
nominally
independent,
so
they
might
be
different
genetic
modules
where
they
might
be
different
parts
of
life
history
and
these
different
growth
functions
are
nominally
independent,
but
they
can
also
be
combined
in
different
ways,
and
so
in
this
case
we
have
alpha
1
and
beta
1.
A
So
this
is
the
first
part
of
the
growth
trajectory
and
then
at
some
point
in
life,
you
switch
to
this
red
growth
trajectory
which
continues
out
a
little
farther,
but
this
in
this
case
we're
just
showing
a
continuous
path
from
alpha
1
to
b
to
2..
So
this
this
first
group,
this
black
trajectory,
there's
a
switch
here
when
it
intersects
at
the
red
trajectory
that
flips
it
to
this
earth
trajectory.
A
If
this
growth
trajectory
were
in
a
different
organism,
say
this
red
function,
it
would
behave
like
this.
It
would
be
almost
sigmoido
in
its
expression,
but
since
they're
combined
there's
a
continuous
curve
that
includes
this
whole.
All
of
this
black
earth
trajectory
in
this
part
of
the
red
group
trajectory,
and
so
we
can
model
these
sort
of
hidden
factors
within
growth
in
an
organism.
A
Now,
in
some
organisms
you
have
something
called
discontinuous
development,
which
is
where
you
have
this
black
function,
and
then
you
have
this
gray
dotted
line,
and
then
you
have
this
red
function,
so
this
gray
dotted
line
might
represent
something
like
metamorphosis,
which
we'll
talk
about
in
a
minute
that
sort
of
separate
these
two
functions,
and
you
have
this
function
that
grows.
You
know,
there's
growth
and
then
the
growth
ends,
and
then
something
happens
at
the
phenotype
or
transforms,
and
you
need
a
new
growth
trajectory
and
you
pick
up
in
a
discontinuous
fashion.
A
A
And
so
the
takeaway
from
this
slide
is
that
a
continuous
trajectory
represented
by
something
like
this,
where
you
have
a
linear
function,
can
shift
to
something
non-linear,
and
so
we
can
model
these
things.
We're
not
going
to
have
to
necessarily
model
it
from
a
real
system.
We
can
work
theoretically
and
figure
out
what
kinds
of
things
are
behind
these
sort
of
shifts
from
continuous
to
discontinuous.
A
So
this
is
an
example
of
discontinuous
development.
This
is
actually
metamorphosis.
This
is
the
example
of
a
caterpillar
metamorphosing
into
a
butterfly,
and
you
see
that
the
caterpillar
forms
this
cocoon
at
some
stage
in
development,
there's
decellularization
that
goes
on
and
some
other
differentiation
of
cells,
the
differentiation
and
redifferentiation,
and
eventually
you
end
up
with
this
butterfly.
So
this
is
a
close-up
shot
of
the
process
where
you
have
the
caterpillar
establishing
a
cocoon
changes
in
the
cellularization
of
the
organism,
and
then
you
end
up
with
this
butterfly.
A
So
this
is
an
example
here
of
this
discontinuous
development.
But
what
else
can
we
learn
from
this
example-
and
I
mentioned
before
that
sequences
are
also
important.
So
there's
this
thing
called
sequence,
the
sequence
hypothesis
of
heterochrony-
and
this
is
an
actually
a
qualitative
description
of
this
process.
But
this
is
basically
the
idea
that
development
exists
as
a
series
of
stages
and
that
these
stages
represent
a
sequence,
so
you
have
sequence,
a
to
your
sequence,
c
or
sequence,
a
through
sequence,
e
or
stage
a
through
stage
e,
and
they
form
the
sequence.
A
A
Well,
you
end
up
with
this
nonlinear
sequence
of
events.
You
end
up
with
things
going
on
at
the
in
within
each
stage,
but
since
the
stages
are
permutated,
you
end
up
with
a
nonlinear
growth
function.
This
is
the
same
growth
function
as
this.
It's
just
that
you've
taken
these
stages
of
development
and
you've
mixed
them
around.
A
So
this
is
actually
interesting
when
you
think
about
this
discontinuous
development,
because
you
think
about
these
different
growth
trajectories,
you
think
about
how
they
combine,
and
we
can
see
that
sequence
hypoth.
The
sequence
hypothesis
of
heterochronic
might
actually
explain
some
of
this
as
well,
but
we
also
talked
about
delays
or
where
we
talked
about
it
in
terms
of
differences
in
slope
and
k,
and
then
that's
related
to
something
called
tau.
So
this
is
the
delay
hypothesis
of
heterocoding.
A
So
in
this
case
we're
using
something
called
delay.
Differential
equations
and
these
dd's
can
be
used
to
characterize
both
absolute
changes
and
relative
rates
of
change
in
growth.
So
we
have
these
ddes,
which
are
characterized
by
time
delay
systems,
and
this
is
the
form
that
they
take,
and
then
this
represents
the
trajectory
of
solutions
in
the
past
and
then
tries
to
figure
out
how
those
things
are
delayed
and
where
can
be
delayed,
and
so
we
end
up
with
in
both
dds
and
time
delay
systems.
A
We
can
solve
them
using
what
they
call
the
method
of
steps,
and
so
we,
this
is
the
formulation
for
a
dd
with
a
single
delay.
We
have
this
tau
parameter,
which
is,
of
course,
what
we're
going
to
use
for
our
time
delays
in
lieu
of
the
change
in
k,
and
so
then
we
can
combine
this
with
alpha
and
beta
and
come
up
with
a
model
where
we
have
an
onset
and
offset
and
then
a
delay
in
the
onset
and
offset
or
delays
within
that
trajectory.
A
A
So
a
concrete
example
of
developmental
delay,
as
opposed
to
something
like
metamorphosis,
or
this
discontinuity
is
diapause
in
drosophila.
So
drosophila
is
a
well-known
model.
Organism
and
insects
have
this
thing
where,
if
you
raise
their
eggs
at
a
certain
temperature,
it
has
phenotypic
effects
both
in
development
and
adulthood.
A
A
If
you,
by
contrast,
raise
the
egg
from
15
celsius
and
then
you
place
it
at
12
celsius.
This
has
a
negative
effect
on
diapause
and
a
negative
effect
on
chill
survival,
so
fewer
drosophilas
survive
showing
to
negative
20
c.
When
you
have
this
treatment
here,
and
so
these
are
examples
of
developmental
delay
and
how
development
unfolds
to
form
an
adult.
Phenotype
and
it's
an
example
also
of
how
we
might
model
this
mathematically,
but
delays
are
important
in
heterochromia
and
in
development
in
general.
A
Another
case
here
is
where
we're
looking
sort
of
at
something
we
call
a
divergence
measure,
and
so
we
might
want
to
look
at
two
different
developmental
trajectories
and
we
might
want
to
look
at
the
heterocrony
or
change
in
growth
between
them,
but
we
might
want
to
do
this
in
continuous
fashion.
So
we
want
to
assess
these
functions
all
the
way
from
the
origin
point
all
the
way
down
to
the
offset
point,
and
so
we
can
do
that
here
with,
and
in
this
case
we
have
a
partial
delay.
A
So
we
have
a
delay
in
this
phase
of
development,
but
not
this
phase,
and
so
since
this
is
a
compound
thing,
we
want
to
know
the
divergence
across
this
entire
function,
and
so
we
can
do
this
with
by
looking
at
the
rate
of
divergence
between
both
trajectories
using
this
measure
here,
and
so
this
measure
here
is
a
divergence
measure
that
we
can
use
and
characterize
this
over
developmental
time.
At
any
point
of
elemental
time,
we
should
be
able
to
detect
where
the
delays
begin
and
end
and
where
the
the
trajectories
are
isomorphic.
A
A
Actually,
they're
not
directed
trees
or
semi-directed
trees,
because
you
have
things
that
go
in
both
directions
and
some
of
these
trees,
but
they
provide
a
mechanism
for
modeling
gene
gene
interactions
and
that
they
facilitate
pattern
formation.
So
this
is
a
binary
gene
expression
network.
It
starts
at
this
root
and
it
either
acts
and
trans
or
acts
and
cysts.
A
It
acts
in
trans
by
triggering
another
type
of
regulatory
element
and
it
acts
in
cis
by
operating
at
its
same
level
of
the
same
type
of
regulatory
element,
and
so
this
is
the
way
this
works.
And
then
you
have
these
identities
for
the
nodes
and
they're
things
that
go
up
and
down
the
nodes
and
they're
exchanging
information,
and
one
of
these
triangular
state
machines.
A
They
generally
exist
at
the
bottom
of
the
tree
towards
the
tips,
and
there
are
these
triangular
relationships
where
you
have
things
going
from
1
to
10,
10
to
11
and
11
to
1,
and
you
have
these
triangular
relationships,
these
these
circuits
or
these
loops
in
the
tree.
That
then,
facilitate
pattern
formation
through
interactions
between
nodes
at
the
same
level,
and
so
we
can
do
this
and
generate
all
sorts
of
different
patterns
and
we
can
control
the
rate
of
information
flow
through
these
networks
using
our
original
parameters.
A
We
can
also
overlap
these
trees
so
that
you
know
this
is
an
example
of
two
developmental
programs
that
are
overlapping
so
that
they
can
refine
the
spatial
scale
of
pattern
formation
at
the
tips
of
these
trees.
So
each
of
these
trees
they
generate
things
that
go
down
to
the
tips
and
then
there's
something
that's
emitted
from
the
tips
in
terms
of
a
phenotype.
A
And
so,
if
we
look
at
this
just
using
a
simple
binary
tree,
we
can
see
how
this
works,
and
so
this
is
something
called
a
galton
board,
which
is,
if
you've
ever
seen
the
game.
Plinko-
and
you
can
look
this
up
on
google
you'll
know
what
this
is
immediately.
But
you
have
something
that's
emitted
from
the
root
of
the
tree
and
it
goes
down
to
each
of
these
nodes
and
in
each
node
the
node
can
either-
and
this
is
a
probabilistic
decision
by
the
node.
A
It
can
either
sort
the
emission
in
this
direction
or
pass
it
down
to
the
next
node,
and
so,
if
it
sorts
in
this
direction,
it
accumulates
down
at
the
tip
of
the
tree
here.
If
it
passes
down
to
this
node,
then
this
node
decides
whether
it
is
going
to
emit
or
it's
going
to
pass
on
to
this
node
and
so
forth.
A
This
is
something
called
an
expression
tree,
so
it
starts
from
the
center
of
the
tree
and
it
emits
or
it
it
radiates
outward,
and
it
forms
these
tissues
at
different
layers,
and
these
tissues
are
the
centroid
of
each
of
these
tissues
of
one
of
these
nodes
and
these
tissues
grow
different
rates
and
they-
and
so
these
are
the
four
parameters
that
control
a
lot
of
what's
going
on
in
this
model,
and
so
we
have
like
an
angle
of
differentiation.
So
how
far
apart?
A
A
A
Finally,
we
have
something
called
lagrangian
growth
in
form.
This
is
something
that
richard
gordon
from
the
diva
room
group
came
up
with.
This
is
thinking
about
embryos
as
lagrangian
structures,
and
so
we
go
back
to
our
trajectory
divergence,
measures
and
convergence.
I
guess
we
use
that
measurement
and
we
can
use
other
parameters
to
understand.
The
embryo
is
sort
of
a
lagrangian
system
where
we
have
these
spatial
flows.
A
So
now
we're
going
from
this
bivariate
space
of
linear
functions
to
a
spatial
flow,
and
so
this
is
something
we
can
use
to
look
at
the
divergence
of
someone
edges
in
space,
calcium
waves
and
embryos
or
other
differential
spatial
phenomena
as
a
function
of
time.
So
here
we
have
things
that
are
flowing
outward
and
things
that
are
flowing
inward
in
a
sphere
in
a
spherical
volume,
and
these
are
things
we
can
measure
using
this
lagrangian
approach.
A
We
wouldn't
necessarily
be
able
to
measure
different
aspects
of
changes
in
growth.
So,
finally,
I
invite
you
to
check
out
our
preprint
here.
Developmental
incongruity
is
a
dynamic
representation
of
heterochromia,
so
I've
talked
about
the
last
part
of
this,
but
the
developmental
incongruity
part
is
something
that
is
a
sort
of
a
case
study
for
what
we're
looking
at
here,
kind
of
rooted
in
this
idea
of
developmental
incongruity
or
development,
where
you
have
multiple
competing
developmental
programs
or
discontinuous
growth
and
development.
A
Things
like
that,
and
so
that's
what
we
focus
on
in
this
preprint.
It's
on
bioarchive
also
check
out
our
representational
brains
and
phenotypes
group.
It's
a
group
of
people
interested
in
development
and
growth
and
form,
and
then
also
cognition
and
modeling,
all
these
things.
So
it's
a
very
eclectic
group
but
we're
interested
in
this
issue
along
with
other
issues.
A
I
haven't
mentioned
it
here,
but
we
also
have
a
group
called
the
divorm
group.
That
is
also
interested
in
more
of
the
developmental
biology
aspect
of
this
only
and
mapping
that
to
data
and
so
check
that
out-
and
here
are
the
references
again
like
I
said,
the
idea
of
heteroprotein
goes
back
quite
a
ways,
but
these
references
start
in
1979..
A
We,
you
know
their
references
on
sequence,
heterocrony
on
time,
delay
systems,
which
is
purely
a
mathematical
concept
that
we've
applied
to
this
work.
The
albert
paper,
which
kind
of
lays
out
the
idea
of
heterocrony
as
a
quantitative
model
and
then
finally
don
williamson,
who
did
a
lot
of
you,
know,
sort
of
original
thinking
on
larvae
and
how
development
and
evolution
interface
and
how
you
know.