ZIMMERMAN
BRIGGSPHOTOTROPICDOSAGERESPONSE
CURVES
Literature
Cited
1.
ASOMANING,
E.
J.
A.
A.
W.
GALSTON.
1961.
Comparative
study
of
phototropic
response
pig
ment
content
in
oat
barley
seedlings.
Plant
Physiol.
36:
453464.
2.
BLAAUW,
A.
H.
1909.
Die
Perzeption
des
Lichtes.
Rec.
Trav.
Botan.
Neerl.
5:
209372.
3.
BLAAUWJANSEN,
G.
1959.
The
influenceof
red
far
red
light
on
growth
phototropism
ofthe
Avena
seedling.
Acta
Botan.
Neerl.
8:
139.
4.
BRIGGS,
W.
R.
1960.
Lightdosage
the
photo
tropic
responses
of
corn
oat
coleoptiles.
Plant
Physiol.
35:
951962.
5.
BRIGGS,
WV.
R.
1963.
Red
light,
auxin
relation
ships,
the
phototropic
responses
of
corn
oat
coleoptiles.
Am.
J.
Botany
(in
press).
6.
BUNSEN,
R.
H.
RoscoE.
1862.
Photochemische
Untersuchungen.
Ann.
Phys.
Chem.
117:
529562.
7.
CURRY,
G.
M.
1957.Studies
on
the
spectralsensitivityof
phototropism.
Ph.D.
Dissertation.
Harvard
University,
Cambridge,
Mass.
8.
CURRY,
G.
M.,K.
V.
THIMANN,
P.
M.
RAY.
1956.
The
basecurvatureresponse
of
Avena
seedlings
to
the
ultraviolet.
Physiol.
Plantarum
9:
429440.
9.
Du
Buy,H.
G.
E.
NUERNBERGK.
1934.
Phototropismus
und
Wachstum
derPflanzen.
II.
Ergeb.
Biol.
10:
207322.
10.
FR6SCHEL,
P.1908.
Untersuchungen
iiberdie
heliotropische
Prasentationszeit.
I.
Sitzber,
Math
naturwiss.
Kl.
Kais.
Akad.
Wiss.
107:
235256.
11.
THIMANN,
K.V.
G.
M.
CURRY.
1960.
Photo
tropism
phototaxis.
In:
Comparative
Bio
chemistry
I:
243306.
Academic
Press,
N.Y.
12.
THIMANN,
K.V.
G.
M.
CURRY.
1961.
Photo
tropism.
In:
Lightand
Life,WV.
D.
MIcElroy
and
B.
Glass,
eds.
pp
646670.
Johns
Hopkins
Press,
Md.
13.
ZIMMERMAN,
B.
K.
W.
R.BRIGGS.
1963.
A
kinetic
model
forthe
phototropic
responses
of
oat
coleoptiles.
Plant
Physiol.
38:253261.
A
Kinetic
Model
for
Phototropic
Responses
of
Oat
Coleoptiles1
2
Burke
K.
Zimmerman3
Winslow
R.
Briggs4
BiophysicsLaboratory
Department
of
BiologicalSciences,
Stanford
University,
Stanford,
California
The
present
paper
is
an
attempt
to
account
ac
curately
for
phototropic
tip
curvature
in
the
Avena
coleoptile
as
a
function
of
the
intensityofunilateral
illumination
and
duration
of
exposure
to
monochro
matic
light
(4358
A).
Detailed
dosageresponse
curves
on
which
this
analysis
is
based
are
presented
in
the
preceding
paper
(22).
It
haslongbeen
as
sumed
that
the
direction
andmagnitude
of
the
light
gradient
across
the
coleoptile
tip
determine
respec
tively
thedirection
and
magnitude
ofthe
phototropic
response
(14,16).
Experimentsby
Buder
(9)
con
firm
that
thedirection
of
the
gradient
does
indeed
deterimine
thedirection
of
the
phototropic
response.
1
Received
July
6,
1962.
2
This
work
was
supported
by
a
National
Institutes
of
Health
Predoctoral
Fellowship
to
the
senior
author
and
by
grants
G8688andG21530
from
the
National
Science
Foundation
and
a
grant
from
ResearchCorpora
tion
to
the
junior
author.
3
Biophysics
Laboratory.Present
address:
Commit
tee
on
Biophysics,
University
of
Chicago,5640
Ellis
Avenue,Chicago
37,
Illinois.
4Department
of
Biological
Sciences.
By
using
narrow
light
pipes
and
illuminating
the
interior
of
the
tip
as
well
as
illuminating
small
por
tions
of
the
outside
of
the
tip
with
narrow
beams
of
light,
Buder
found
that
thedirection
of
curvature
was
always
in
thedirection
of
the
maximum
light
gradient,
not
in
the
direction
of
the
incident
radia
tion.
Hisexperiments
also
produced
gradients
many
times
larger
than
normally
occur
when
the
entire
tip
is
placed
in
a
beam
of
light,
but
approximately
the
same
amount
of
curvature
was
always
observed.
The
role
of
the
light
gradient
in
determining
the
magnitude
of
the
phototropic
response
has
been
questioned
by
Curry
(12).
He
pointed
out
that
the
light
gradient
across
the
tip
shouldbe
a
function
of
the
wavelength
of
theincident
light.
Thus
if
the
magnitude
of
curvature
depends
upon
the
magni
tude
of
the
light
gradient,
it
should
also
be
a
func
tion
of
wavelength,
something
not
observed
(17).
His
attempt
to
account
for
the
shape
of
the
dosage
response
curves
on
thebasis
of
the
light
gradient
was
also
unsuccessful.
Detailed
studies
of
tip
curvaturebased
on
the
assumption
that
both
the
direction
and
magnitude
of
the
gradient
are
necessary
for
determining
thephoto
253
PLA\NT
PHYSIOLOGY
tropic
response
(21)
did
not
successfully
account
for
the
(losageresponse
curves
previously
described
(21,
22
).
The
assumption
that
the
magnitude
of
the
gra(lient
(leterlniines
the
magnitude
of
the
phototropic
response
wvas
thits
rejected
for
the
following
reasons.
The
measured
gradienit
across
the
ti)
was
too
small
to
account
for
the
width
of
the
log
dosageresponse
curves.
EIven
assuming
arbitrary
gradients
much
larger
or
imtuclh
smiialler
than
that
observed,
the
dosageresp)olnse
curves
beyond
peak
first
positive
curvature
could
not
beaccounted
for.
Twto
extremes
were
considered,
a
uniforml
(listribution
of
pigment
over
tlle
surface
of
the
coleoptile
til.
and
a
homogge
neous
(listributioni
of
pigmiient
thlouglhouit
the
volume
of
the
tip,
xxitlh
theassumptiointhatthe
response
is
proportional
to
the
dlifference
between
the
numiiber
of
pigmiient
mlolecules
excite(d
on
lighted
and(I
shaded
sides.
Several
kinetic
mechanismiis
for
pigment
ac
tivation
xvere
assumedI
with
negative
results
in
all
cases.
Thu..
it
will
be
assumlled1
inthis
paper
that
xvhile
the
gradient
(letermiineis
the
(lirection
of
the
response.
it
has
no
bearing
on
tlle
mnagnituide.
The
approach
used
in
this
paper
is
to
(letermine
xhether
the
(losageresponise
curves
previutsly
de
scribedl
fol
oatts
22)
can
be
dlescrile(l
1v
an
algebraic
sum
of
simiplemiiathematical
fuiictions.
The
problenm
is
thein
to
findl
a
kinetic
mlo(lel
xviich
yields
futictionsof
the
forlml
obtained.
AnIy
finite,
continuouts,
singlevaluied
funiction
canI
be
dlescribedl
by
a
Fourier
series
(a
suIml
of
trigonometric
functions)
or
1y
a
poxver
series
(a
suI
of
algdebraic
functions)
if
one
takes
a
sufficient
numlhbel
of
termls
and
a(djusts
tlle
constants
appropri
atel]v.
Therefore,
any
combilnation
of
funlctions
Nhlich
cani
be
fitted
to
the
dosageresp)oise
curxves
nmust
be
kept
sim)ple.
i.e.,
xvith
as
fexv
termls
and
eimipirical
conistants
as
possible.
If
suichl
curvefitting
is
successful.
one
imiust
then
obtain
this
comnbination
of
simple
fulnctiolns
as
the
soluitions
of
(lifferential
equations
arising
from
thekinetic
mo(lel.
There
is
goo(d
evidence
that
tip
curvatuire
may
be
the
result
ot
three
independent
or
at
least
distinct
mechanism.s,
corresponding
roughly
to
first
positive,
first
negative,
andl
second
positive
curvature.
Each
of
these
medliates
the
lateral
translocation
of
auxin
(1.
7,
19,
20).
The
portion
of
the
dosageresponse
curve
involv
ing
first
positive
and(
first
negativecurvature
)beys
the
BuInsen1Roscoe
recil)rocitv
law
of
photoclhemiiical
equivaleince
(10),
i.e.,
that
it
is
a
functioniof
the
total
amount
of
radiation
absorbed
alone.
The..e
points
are
consideredl
in
cletail
else
where
22 .
Exposure
of
the
plants
to
red
light
de
creases
the
sensitivity
of
first
positive
and
first
nega
tive
curvature
to
blue
light,
therebyproducing
a
shift
of
the
enitire
(losageresponse
curves
to
higher
dosages
(log
einsteiins
cm
2
at
4358
A).
Secon(d
positive
curvature,
oni
theother
hand,
seems
to
be
a
function
only
of
the
length
of
exposuire.
Thus,
at
low
intensities
the
curvature
that
one
observes
in
certain
regions
is
the
sum
of
the
dosagerespoinse
curve
for
first
positive
and
first
negative
curvature
plus
that
for
secoind
positive
curvature.
Red
light
imakes
thislattersysteimi
more
sensitive
to
phototropicinduc
tion,
shifting
its
dosageresponse
curve
toward
lower
dosages
(22).
Fromii
the
above,
it
is
clearthat
at
least
secoindl
positive
curxature
is
quite
distinct
fromii
the
othermechanismi,s.
In
furtlher
support
ofthis
contentioni.
auxin
trainslocationl
experinments
(7,
r,.
Gillespie.
K.
V.
ThliiaiI
I)e
rsoaIlal
comimiiluinicationI)
have
shoxvn
that
the
capacity
for
lateral
transport
of
atuxini
ex
tends
at
least3
limi
basally
fronm
the
tip
for
liglht
dosages
eliciting
second
positixve
curvature.
It
oc
curs
only
in
the
top
fe
i
ntildre(d
miicrons
for
first
positive
curvature.
Stuclies
oIn
the
course
of
developmnent
of
the
vari
Ouis
tv
pes
of
curvature
and
theirfinal
appearaince
also
suggest
the
presenice
of
tlhree
separate
mlechani
isImls.
This
exidence
is
presentedl
and
(liscussed
else
xvhere
( 22 ).
Developmiieint
of
a
Kiinetic
AMolel
for
Phototrop
ismii.
On
the
basis
of
the
foregoing
exvidence,
it
shall
be
asstlume(d
that
three
(listiict
systemis
contribute
to
the
observed
(losageresponse
curves,
anid
the
nota
tion,
systemiis
I.
II,
ailcl
III,
slhall
replace
the
sonme
tilmies
confusinlg
terminology
of
first
positive,
first
negative,
anid
second
positive
curv
atuire.
The
ascendiing
portion
of
the
(losagerespoinse
curxes
obtai
ned
xvith
and(
ithoutred
liglht
treatmiient,
resemiiblesthe
curve
one
would
expect
if
pigmllentactivation
obeys
first
order
kinetics,
i.e.,
represenlts
the
simiiple
conxversion
of
one
molecular
species
to
anotlher
ith
tlle
rate
proportional
to
the
conicetrtia
tion
of
the
first.
Suclh
a
curve
is
expresse(d
b y
aIn
equation
of
the
formii
R
K(1

ekIt)
xvhere
R
is
phototropic
r
espolnse
(or
concentration
of
active
species),
K
aid
k
empirical
constants,
I
the
intensity
of
blue
radiation,
Iand
t
the
(luraItionl
of
the
exposure.
This
is
the
nmost
likely
process
as
it
is
tlle
simplest
mechainismi
by
xvhich
light
can
convert
an
inactix
e
molecular
species
to
an
active
forml.
It
wvould
tlhein
appear
reasonable
to
assumtiie
that
the
entile
dosageresp)onse
curve
for
systenms
I
and
II
could
be
(lescribedl
by
a
combiinatioin
of
similar
functionis
correspoin(liing
to
the
fornmation
of
active
formsand
to
the
subsequent
inactivation
of
these
activespecies.
Such
a
sequence
of
events
hlas
been
propose(l
earlier
oIn
strictly
qualitative
groutn(ls
(6).
Fiunctioins
of
tli
s
forml
\vere
successfullxy
fitted
to
the
existing
dosageiresponse
curves
(froml
2
2
for
svsteuis
I
and
(fig
1,
2,
lowermllost
curves)
with
thie
folloxillg
result:
R
K,
{
(
1
_e
klIt)
1
ekIt)]
RK
K
](1
kIt
(1
ek
t ]
}
I
This
equation
is
entirely
eiiipirical,
yet
corresponds
exactly
to
the
forml
one
would
expect
for
a
positive
curvature
mleclhanismii
consisting
of
fornmation
of
an
active
species
folloxved
by
inactivatioin
at
higher
doses,
anid(
a
separate
but
similar
negative
curvature
svsteni.
I lTere
are
only
four
enmpirical
conistaIlts,
sinlceK1
iS
normallization
constanlt
an(l
of
the
254
ZIMMERMAN
BRIGGSKINETIC
MODEL
FOR
PHOTOTROPIC
RESPONSES
LLJ
Un
z
0
0
Uf)
Lii
0r
LI
0
0
0
I
a
LOG
(Ixt),EINSTEINS
CM2
at
4358
A
1.6
Lu;_
I.TA
xIu
I=1. +XIU
U)
zO
2I
880
,,
0
U)
a_
l
0
4
r
0
a.8
~
LO
It,INTIS
M2at45
olnt
gie
ordlgtteatet
xprmna
H
H
I
1o.4
x
10
00
<~
0

Plus
Red
Light
.4
13
12
ii
10
9
8
LOG
(Ixt),EINSTEINS
CM2
at
4358
IG.
1
(top).
Theoretical
phototropic
dosageresponse
curves
at
three
intensities
of
monochromatic
blue
light.
Plantts
given
nso
red
light
treatment.
Experimental
pointsforthese
intensities
(22)
shown
for
comparison.
intensities
(I)
in
einsteins
CMn
2
seca
1
at
4358A.
FIG.
2
(bottohe).
Theoretical
phototropic
dosage
respoisecurves
at
three
intensities
of
monochromatic
blue
light.
Plants
given
2
hr
red
light.
Experimental
points
(22)
shown
for
comparison.
Lowermost
curve,
systems
I
and
II
alone.
Intensities
(I)
in
einsteins
CM2
sec1
at
4358
A.
snall
k s
is
arbitrary.
The
reason
onek
is
arbitrary
is
thatthe
absolute
position
of
the
dosageresponse
curve
depends
on
the
k s
which
are
in
part
determined
by
theeffectiveness
of
the
light
interacting
with
the
pigment.
While
this
effectiveness
might
be
of
con
siderable
interest,
it
has
no
bearing
on
the
kinetics.
Hence,
one
of
the
k s
may
be
adjusted
until
the
positions
of
the
experimental
and
theoretical
dosage
response
curves
along
the
abcissa
coincide,
theothersthen
being
determiined
in
relation
to
it.
One
may
now
writereasonable
kinetic
schemes
for
systems
I
and
II.
System
I:
hv
lhv
x
y
>z,
k1I
kI
hvhv
System
II:
x
>
y
z,
k3I
k41
where
y
is
a
phototropically
active
pigmentform
for
lateral
transport
of
auxin
away
from
thelighted
side
and
is
an
active
form
for
translocation
in
the
op
positedirection.
Inactive
forms
of
y
and
y
are
denoted
by
z
and
z .
The
unexcited
pigments
are
x
and
x .
Thus
the
total
response
for
systems
I
and
II
would
be
R
=
y

y .
Reverse
reactions,
recovery
following
inactiva
tion,
and
(legradation
and
synthesis
of
new
pigment
molecules
have
been
assumed
to
be
unimportant
over
the
period
of
the
experimentand
thus
neglected,
al
though
all
miglht
well
occur.
Briggs
(6)
exposed
corn
coleoptiles
to
a
high
dosage
of
unilaterallight
which
producedno
curvature
(presumably
because
the
pigments
for
systems
I
and
II
had
beenconverted
entirely
toz
and
z
respectively,
and
exposure
times
were
too
short
for
induction
of
significant
second
positive
curvature).
He
followed
this
exposure
at
various
time
intervals
with
a
second
exposurewhich,by
itself,
normally
resulted
in
maximum
first
positive
curvature.
If
the
second
exposure
followed
the
first
immediately,
no
curvature
resulted.
But,
as
the
de
lay
between
exposures
was
increased,
more
and
more
curvature
occurred,
with
20minutes
required
to
obtain
maximum
first
positive
curvature.
Curry
(12)observed
the
same
phenomenon
for
oatcoleop
tiles.
These
observations
stronglysuggest
thepres
ence
of
either
a
recovery
mechanism
or
de
novo
synthesis
of
pigment.
In
either
case,
the
overall
rate
constant
is
small
enough
to
be
neglectedwithout
in
troducing
significant
errors.
Hence,
systems
I
and
II
canbothbe
treated
as
closedorconservativesystems.
i.e.,
x
+
y
+
z
=
constant.
Forsystem
I,
one
obtains
the
following
set
of
simultanieous
differential
equations:
dx
=
klIx,
(it
=
k1Ix

k2Iy,
and
dIt
dz
=
k..Iy.
dt
II
III
IV
Solutions
are
readily
obtained
by
standard
procedures
(17).
*
(t)
=
x(0)
[k1/(k,k,)]
(ek2It

ek,It)
=
x(O)
[kl/(klk2)]
[(1

eklIt)

(1

ek,It)]
V
where
y(t)
is
the
concentration
of
y
at
time
t
after
the
light
is
turned
on,
and
x(0)
is
theconcentration
of
x
at
t
=
0.
Since
the
phototropic
response
is
as
sumed
to
beproportional
to
the
concentration
of
y,
the
aboveequation
is
precisely
the
form
dlesired
from
2
55
PLANT
PHYSIOLOGY
theempirical
curve
(I).
A
similar
set
of
e(quations
is
obtained
for
v (t).
Vith
a
curvatureof
300
nornlalizedl
to
1.0,
miieasuredin
einsteins
cm2
sec
,
an(l
t
in
seconds,the
following
values
of
the
constants
providle
the
best
fit
of
y
an(d
to
the
experimental
losageresponse
curves.
No
re(d
light
treatmiient:
x(0)
=
0.92,
k,
4.47
X
1011,
k.,
2.40
X
109,
x'(0)
=
0.82,
k,=
3.73
x
109,
k4
1.18
X
10 .
Two
hours
redl
light
precediilg
blue
exposure:
x(0)
=
0.92,
k,
1.12
X
1011,
k.,
6.0
x
108.
x'(0)
0.82,
k3
=
9.33
x
108,
k4
2.95
x
108.
The
third
phototropic
system
will
inow
be
con
si(leredl.
If
one
subtractsthe
theoretical
curves
for
svstemis
I
and(
II
alone
(i.e.,
lowermost
curve,
fig
2)
fromii
an
experimlental
curve
in
wlhiclh
system
III
oc
cuirs,
onie
obtains
for
system
III
siplyI
a
linear
func
tion
of
timle,
i.e.,
R
Kt.
This
dlependence
becomes
app)arent
when
the
subtracte(d
curves
for
several
in
tenisities
are
plotte(d
asa
funiction
of
t
(fig
3).
The
value
of
K
may
vary
slightly
with
intensity.
but
the
dlata
are
n,Ot
sufficiently
goo(d
to
w
arrant
this
con
eluision.If
the
plants
are
given
re(d
light
before
ex
posture
to
blue,
K
is
approximately
(louble(l.
It
is
implortant
to
note
that
since
tlis
system
can
result
in
very
large
curvatures
to
thepointwhere
factors
other
than
l)hototropic
in(luctioni
of
tlle
aux.in
translocation
syvstem
are
limiting,
the
formii
Kt
may.,
in
fact.
be
the
1.6
z
0
a
PLUS
RED
LIGHT
1.2
Kz8.27xI04
SEC

.8
0~
0~~~~~
0
MINUS
RED
LIGHT
z
0
K2.95xlO
4
SEC
z
500
1000
1500
EXPOSURE
TIME,
SECONDS
FIG.
3.
Net
phototropic
response
obtained
wxhen
curvature
predicted
for
systems
I
and
II
is
subtracted
from
total
experimental
curvature
(from
22)
and
plotted
as
a
function
of
exposure
time.Liniear
dependence
of
system
III
on
length
of
exposure
is
now
apparent.
In
tensities
(I)
in
einisteins
cm
2
sec
1
at
4358A.
Open
circles:
I
=
1.4
X
1012,2hrred
light.
Solid
circles:
I
=
1.4
X
10
1,
2
hrred
light.
Open
squares:
I
=
1.4
X
10
12,
Ino
red
light.
Solid
squares:
I
=
1.4
X
1011,
no
red
light.
K
=
u(0)
Laa,a/(a,
+
a1)]
liniting
case
of
A(1

e
at),
vlicll
for
small
(at
becomes
Aat
as
readilydetermiinied
by
a
j)over
serlies
expansion.
It
could
also
be
an
apl)roximation
of
a
variety
of
other
functions.
Howvever,
tlle
exl)olen
tial
formii
above
can
be
(lerive(l
fromii
a
kineticallv
reasonable
milodel
wvhich
cani
accounit
for
thle
red
liglht
in(luce(l
sshifts
of
all
three
systems.
CoInsider
now
the
followinIg
kinetic
miiodlel:
a1i
a.,
u
t
v

NV,
a1I
where
wv
is
a
phototropically
active
componenit
aln(d
u
and
v
areprecursors.
Again,
recovery,
synthesis,
an(l
(legradation
are
neglectedl.
The
lightinduce(d
conversion
of
u
to
v
is
considered
strongly
reversible,
whereas
wN,
is
formiied
byv
means
of
a
(lark
reaction
(le
pending
upon
the
concentration
of
v.
This
miiodel
leadls
(lirectly
tothe
followilng
set
of
(lifferential
eqtua
tiolls:
(III
=
a,Iv

a1Ju,
(It
=
allu

(aT1
+
av.
(It
1
(It
a
.,
.
VI
VII
VTIII
Solutions
are
again
readily
obtainedl
by
stan(lar(l
pro
ce(lures.
Since
both
v
and
wN
are
of
interest.
they
are
g,iven
below
v(t)
=u 0 l
)
et
tut
I
iX
antIl
w(
t)
=
u
0)t
)
A ,I,
a
n
I
N
t
u
I
I
l
e
A
t.
l
,\t
where
A
=

32
(a,I
+
a,I
+
a.,
4
12
r
I
+
a1I
+
a,
)2

4a,a.1J]
ad(
A.

2
(a,
I
+
a.,I
+
.
)

Y2
[(alI
+
a1I
+
a.
)2
_
4a,ai1]
Nowv
let
us
assumle
that
a..<<aI.
a,I
for
all
experimentallymeaningful
values
of
I.
Qualitative
ly
it
is
now
clear
that
an
equilibriumi
will
be
estab
lished
betwreen
u(t)
and
v(t),
the
ratioof
u
to
v
remlainiing
relatively
constant
while
v
is
converted
exponentially
to
w.
This
becomes
apparent
in
the
followinIg
approximate
solutionl
for
v
for
small
t,
imiiposing
the
above
conditioni.
256
ZIMMERMAN
BRIGGSKINETIC
MODEL
FOR
PHOTOTROPIC
RESPONSES
v(t)
=
u(O)[al/(al
+
a,)]
[1

e
(a,
+
a
)It]
xi
Thus
when
t
becomes
sufficiently
large,
v(t)
ap
proaches
aconstant
which
is
then
slowly
diminished
according
to
the
size
of
the
a s,
but
governed
by
a.,.
A
further
assumption
must
now
be
made
and
that
is
that
the
equilibrium
concentration
between
u
and
v
is
far
in
the
direction
of
tu.
Ifnot,
even
a
relatively
short
flashof
light
(but
long
enough
to
establish
equilibrium)
would
convert
a
substantial
part
of
u
to
v
which
would
thenbeconverted
spon
taneously
to
w
even
after
thelight
is
turned
off.
Thus,
relatively
large
curvatures
would
be
obtained
over
a
variety
of
light
conditions
and
the
empirical
linear
time
dependence
of
system
III
curvature
would
not
be
observed.
This
assumption
is
also
necessary
to
account
for
the
flashing
light
experiments
to
be
disctussed
later.
Thus.
it
is
specified
that
the
amount
of
v
remaining
immediately
after
the
light
is
turned
off
(
w hich
is
subsequently
converted
to
v)
is
sufficiently
small
to
contribute
little
to
phototropic
curvature.
Under
these
conditions,
the
amount
of
w
formedduring
an
experiment
can
be
represented
by
u
(t)
where
t
is
the
duration
of
time
the
light
is
left
on.
The
approximate
solution
of
w(t)
after
equilibrium
between
uand
v
has
been
established
[t
>
1
l/(al
+
a_1)I]
is
as
follows:
w(t)
=
u(O)
[1

e
a1a.t/(aj
+
a1)
u
[a,aC,/(al
+
a,)]t
XII
Thus
w(t),
the
quantity
observed
as
phototropic
response,
has
the
behavior
of
a
singleexponential
term
and
is
entirely
independent
of
intensity,
as
re
quired
by
the
empirical
curves.
If
u(O)
is
more
than
three
or
fourtimes
x(0),
then
only
small
values
of
[caia./(al
+
a1)]t
can
be
important,
and
thus
a
linear
timedependence
will
be
observed.
The
constant
u(0)
(ala2)/(a1l
+
aCY)
may
be
readily
foundfrom
the
experimental
curves
for
sys
tem
III
(fig
3).
The
portion
of
the
dosageresponse
curves
resulting
from
system
III
may
thus
be
pre
dicted
at
any
intensity.
As
a
further
experimental
test
of
the
model
pro
posed
for
system
III,
it
is
a
direct
consequence
of
equations
IX
through
XII
that
flashing
light
would
enhance
the
response
for
a
given
dose
and
intensity
over
the
responseobtained
from
continuous
exposure.
Since
dw/dt
is
equal
to
a9v(t).
then
w(t)
is
equal
to
the
a.)
f0tv(t)dt.
It
is
clear
that
if
the
light
is
left
on
long
enough
for
v
to
reach
equilibrium
con
centration
and
thenturned
off.
v
will
notrevert
to
u
butcan
only
disappear
by
spontaneous
conversion
to
w
via
the
dark
reaction
governed
by
a.,.
While
this
also
occurs
with
thelight
on,
it
is
not
necessary
to
leavethelight
on
for
the
concentration
ofv
(t)
to
remain
close
to
equilibrium
during
the
course
of
an
experiment.
Hence,
the
integral
of
v,
and
thus
v,
vouldbe
much
larger
for
agiven
dosage
than
if
the
light
were
applied
continuously.
The
condition
im
posed
above,
that
equilibrium
betweenuandv
be
far
in
the
direction
of
u
must
hold
in
order
for
the
enhancement
to
be
substantial.
Otherwise
one
flash
wouldl
be
sufficientto
convert
most
of
u
to
v,
and
essentially
the
same
response
would
be
obtained
over
a
variety
of
conditions.
Graphically
the
situation
is
illustrated
in
figure
4.
The
total
area
under
thecurve
is
proportional
to
the
phototropic
response.
In
principle,
one
shouldbe
able
to
evaluatethe
a
s
from
flashing
light
experiments
based
on
equa
tions
IX
andl
X.
In
practice,
many
difficulties
areencountered.
It
may
be
possible,
however,
to
deter
mine
the
constant
a,
without
elaborate
experiments.
Suppose
thatthe
length
and
number
of
light
flashes
are
fixed,that
the
length
ofa
single
light
flash
is
sufficiently
long
for
equilibrium
to
be
reached,
and
that
total
illumination
is
high
enough
for
systems
I
and
II
to
be
inactivated.
If
the
length
of
the
dark
period
is
nov
varied,
one
would
expect
enhance
ment
of
curvature
which
eventually
levels
off
as
the
clark
period
is
lengthened.
The
response
one
would
expect
wvhile
the
light
is
on
vill
benearly
equal
whether
or
not
the
light
is
flashed
provided
that
the
length
of
the
experiment
is
sufficiently
short
so
that
the
equilibriumconcentration
of
v
will
not
be
seri
ously
altered.
Howvever,
it
must
be
long
enough
for
an
accurately
measurable
amount
of
curvature
to
be
obtained,
say
5°.
The
magnitude
of
enhance
ment
is
now
given
as
follows:
E
=
u(0)n
a
_)
f
T
e
a2,t
a,
+
a ,
u(0)n
a
a,l
fa1
)
(1

e
a,)
(It
=
XIIIwhere
E
is
the
magnitude
of
enhancement,
u
(0)
the
initial
concentration
of
u
(t),
n
the
number
of
flashes,
and
T
the
duration
of
the
dark
period.
While
this
is
only
an
approximation,
it
shouldgive
an
enhancement
curve
similar
to
that
in
figure
5.
Let
Emax
be
the
maximum
possible
amount
of
en
hancement
obtainable
with
a
particular
number
and
length
of
flashes.
Thus.
a.,
and
u(O)
a1
(a,
+
a1)
may
be
determined
both
from
Emax
and
the
time
constant
by
xvhiclh
the
curve
rises.It
is
clear
that
a
plotof
log(E

Emax)
versus
t
will
give
a.,
as
the
slope
of
a
straight
line.
It
shouldbe
borne
in
mind
that
n,
the
number
offlashes,
must
be
kept
small.
If
n
is
too
large.
one
may
still
obtain
a
curve
similar
to
that
in
figure
5,
but
looking
more
like
a
straight
line
thanan
ex
ponential.
This
is
because
so
much
curvature
is
ob
tained
that
factors
other
than
the
concentration
of
w
become
limiting
long
before
the
dark
period
may
be
made
long
enough
for
the
concentration
of
v
to
ap
proach
zero
after
each
flash.
Thus,
in
order
tore
duce
the
number
of
flashes
and
still
remain
beyond
the
range
of
systems
I
and
II,
one
may
increase
257