Development 143: doi:10.1242/dev.125971: Supplementary information
Supplementary Methods - mathematical derivations
Joseph D. Barry, Erika Donà, Darren Gilmour, Wolfgang Huber
Contents
1 Model definition
1
2 Model solutions
1
3 Timer ratio with FRET
2
4 Time to reach steady state
2
5 Timer signal
2
6 Timer signal and FRET
3
1
Model definition
The fluorophore maturation kinetics was described as a one-step process. Each fluorescence channel was modelled
separately but due to tandem timer design each channel shared the same constant protein production rate p and constant
degradation rate k. The time-dependent rate equations used are therefore
X˙i0 (t) = p − (k + mi )Xi0 (t)
Ẋi (t) = mi Xi0 (t) − kXi (t)
(1)
2
Model solutions
The time-dependent solutions to eq. 1 with the boundary conditions Xi0 (0) = 0 and Xi (0) = 0 were calculated to be
Xi0 (t)
Xi (t)
= p(1 − e−(k+mi )t )/(k + mi )
= pe−(k+mi )t (k − emi t (k + mi − ekt mi ))/(k(k + mi )) .
(2)
The corresponding steady state solutions to eq. 2 are
limt→∞ Xi0 (t) = p/(k + mi )
limt→∞ Xi (t) = pmi /(k(k + mi )) .
(3)
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where Xi0 (t) and Xi (t) are the molecular populations of the non-mature and mature fluorophore populations respectively
at time t for the ith fluorescence channel with i ∈ {1, 2}. We chose the convention that i = 1 is the fast-maturing
fluorescence channel and i = 2 is the slow-maturing fluorescence channel.
Development 143: doi:10.1242/dev.125971: Supplementary information
3
Timer ratio with FRET
Fluorescence intensity Ii is proportional to the number of mature fluorescent molecules Xi . If FRET occurs between
channel 1 and channel 2 the fluorescence intensity of channel 1 will be reduced by an amount proportional to the FRET
efficiency E and the proportion b of channel 2 fluorophores available as acceptors. In the time-dependent model this was
described as
I1 (t) = f1 X1 (t)(1 − b(t)E)
(4)
I2 (t) = f2 X2 (t)
where the proportionality constant fi incorporates multiplicative effects such as fluorophore brightness and quantum yield,
and b(t) = X2 (t)/(X20 (t) + X2 (t)). We did not consider FRET from channel 2 to channel 1 since it is physiologically
improbable to encounter such cases as slower-maturing fluorophores tend to have longer wavelengths than faster-maturing
fluorophores.
The time-dependent timer ratio R incorporating FRET was defined as
R(t) = I2 (t)/I1 (t) = f X2 (t)/(X1 (t)(1 − b(t)E))
(5)
where f = f2 /f1 .
In full the the timer ratio is therefore
R(t) = f
e(m1 −m2 )t (k + m1 )(k − em2 t (k + m2 − ekt m2 ))
−m t
kt
2 k+m2 −e m2
(k − em1 t (k + m1 − ekt m1 ))(k + m2 )(1 − E k−e(1−e
)
kt )(k+m )
2
(6)
, which in steady state reduces to
lim R(t) = f
t→∞
4
m2 (k + m1 )
.
m1 (k + m2 − Em2 )
(7)
Time to reach steady state
The time to reach steady state for the ratio was determined from the kinetics of the slower maturing fluorophore, FP2.
Since FP2 is not affected by FRET from FP1 we may calculate this either from the fluorescence intensity in eq. 4 or the
molecular population in eq. 2. Here we focus on the latter. We choose to define the time to reach steady-state as the
point of intersection chose the line tangent to the point of inflection and the steady state value (see Fig. S4A) given by
eq. 3. The time coordinate t∗ at the point of inflection was calculated to be
(8)
The slope s of the tangent to the FP2 profile is given by
s = pm2 (1 + m2 /k)k/m2 /(k + m2 ) .
(9)
Calculating the point of intersection between the tangent line and the steady state value led to the following definition
for the time to reach steady state, Tss .
Tss = 1/k + 1/(k + m2 ) + log(1 + m2 /k)/m2
5
(10)
Timer signal
A
B
B
A
The timer signal was defined between two protein half-lives, T1/2
and T1/2
where T1/2
> T1/2
. We considered the log
B
B
B
A
A
A
fold-change between the corresponding timer ratios R = I2 /I1 and R = I2 /I1 .
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t∗ = (1/m2 ) log(1 + m2 /k) .
Development 143: doi:10.1242/dev.125971: Supplementary information
To investigate the effect of background noise on our ability to detect differences in timer signal we defined the following
additive error model.
S = log2
I2B + B
2
I1B + B
1
I2A + A
2
I1A + A
1
(11)
A B B
2
where A
1 , 2 , 1 , 2 ∼ N (0, σ ) are independent. Computer simulations were used to obtain an estimate of the population
mean µS and standard deviation σS . From this we formed the coefficient of variation term
CV = σS /µS .
6
(12)
Timer signal and FRET
To explain why an increase in FRET increases timer timer signal we considered timer signal without additive noise and
denoted timer signal without FRET (E = 0) as D0 and timer signal with positive FRET (E > 0) as DE . We calculated
that
1
DE /D0 = 1 +
log2
D0
1 − bA E
1 − bB E
(13)
where bA and bB are the proportions of FP2 fluorophores available as acceptors for the shorter-living and longer-living
proteins, respectively. Since the population of mature FP2 fluorophores is relatively more abundant for the long-living
protein, bB > bA , which implies that 1 − bB E < 1 − bA E. Therefore log2 ((1 − bA E)/(1 − bB E)) is a positive quantity
and eq. 13 is greater than one.
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Development 143: doi:10.1242/dev.125971: Supplementary information
Supplementary materials and methods
DNA plasmids and mRNA synthesis
All new plasmids used in this study and listed in Table S2 were created using
Multisite Gateway cloning (Invitrogen). Capped mRNA for injection was obtained
through in-vitro transcription using the SP6 mMessage mMachine kit (Ambion).
Fluorophore maturation curves
One-cell stage zebrafish embryos were injected with mRNA (200 ng/µl) encoding for
nuclear localization signal (NLS) tagged timers (NLS-mKate2-sfGFP, NLS-tdTomsfGFP and NLS-TagRFP-sfGFP). Up to 5 embryos were mounted in a fluorinated
ethylene propylene (FEP) tube as previously described (Schmid et al., 2013).
Embryos were imaged at 30°C from ~2 hours post injection on a ZEISS Lightsheet
Z.1 microscope using a ZEISS 20× / 1.0 water-immersion Plan Apochromat objective
lens with 0.36 zoom. Dual-colour imaging was performed switching channel every
plane, illuminating from two sides (online fusion) using 5% of the maximum
intensities of 488 nm and 561 nm laser lines and ~50 msec exposure. Stacks were
development form early stages embryos were not immobilized. Four views per
embryo were acquired to increase the chances of imaging at least some parts of the
sample. Time interval was 10 min. Image analysis was done using Fiji. After
background subtraction a global embryo mask was obtained from the green channel.
The mean green and red fluorescence intensities within the mask were measured for
each z-plane and the average across the stack represented one timepoint for 1 view.
For each timepoint the sample intensity is the average value across the views.
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taken with z-steps of 15 µm to minimize photobleaching. To allow proper
Development 143: doi:10.1242/dev.125971: Supplementary information
FRET efficiency measurements
For each timer, FRET efficiency measurements were obtained from acceptor
photobleaching experiments. Embryos were injected with mRNA encoding for NLStdTom-sfGFP and NLS-TagRFP-sfGFP. For control experiments, individual FPs
(NLS-sfGFP, NLS-TagRFP) or their combinations (NLS-sfGFP plus NLS-TagRFP)
were injected. For tdTomato,
transgenic embryos
expressing NLS-tdTom
(cxcr4b:NLS-tdTomato) were used instead of embryos injected with NLS-tdTom
mRNA. To allow maturation of slow-maturing RFPs, photobleaching was performed
at 36 h.p.f. using a PerkinElmer Improvision Ultraview VOX spinning disk confocal
microscope equipped with a Zeiss 40x 1.2 NA water immersion objective. An area of
100 x 100 µm was bleached using the 561 nm laser line. Pre- and post-bleaching
single plane images were acquired using sequential acquisition.
Pre- and post-bleaching mean fluorescence intensities within manually-defined
nuclear ROIs in the bleached area, or outside as control, were measured in Fiji.
Subsequent analysis was done using MATLAB (MathWorks) and plots were
generated in R. To correct for background the mean intensity of a sample-free area
regression line fitting the mean intensity data of the two fluorescence channels in
embryos expressing only one FP at the time. For our imaging settings cross-excitation
of sfGFP in the red channel was negligible, while both RFPs were faintly detectable
in the green channel (m tdTom = 0.0211, m TagRFP = 0.0031). This was taken into account
while calculating the FRET efficiency.
Bleaching efficiency (Eb) for each fluorophore was calculated as
Eb = (Ipre-Ipost) / Ipre
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was subtracted. Cross-excitation efficiency was calculated as the slope (m) of the
Development 143: doi:10.1242/dev.125971: Supplementary information
where Ipre and Ipost are respectively the ROI mean intensities pre- and post-bleaching
(Eb-TagRFP = 0.790±0.084, Eb-tdTom = 0.856±0.093). The bleaching protocol did not
cause any detectable sfGFP bleaching.
FRET efficiency (EFRET) was calculated as
EFRET = (Ipost-G* - Ipre-G*) / Ipost-G*
with
Ipre-G* = Ipre-G - (m * Ipre-R) and Ipost-G* = Ipost-G - (m * Ipost-R)
where Ipre-G* and Ipost-G* are the cross-excitation corrected mean intensities in the green
channel pre- and post-bleaching. Ipre-G and Ipost-G are the green mean intensities prior to
correction, while Ipre-R and Ipost-R are the red ones. FRET efficiencies calculated this
way are 0.442 ± 0.036 for tdTom-sfGFP and 0.142 ± 0.035 for TagRFP-sfGFP.
References
Development • Supplementary information
Schmid, B., Shah, G., Scherf, N., Weber, M., Thierbach, K., Campos, C. P.,
Roeder, I., Aanstad, P. and Huisken, J. (2013). High-speed panoramic
light-sheet microscopy reveals global endodermal cell dynamics. In Nat
Commun, pp. 2207.
Development 143: doi:10.1242/dev.125971: Supplementary information
Development • Supplementary information
Fig. S1. Verification of the previously reported ordering of RFP maturation rates
in the zebrafish embryo. Maturation profiles for tdTom-sfGFP, TagRFP-sfGFP and
mKate2-sfGFP are shown for both GFP and RFP channels. Profiles were averaged
across 4 views per embryo and up to 5 embryos per timer. The signal in both GFP and
RFP channels for each embryo view was divided by their respective values at 500
minutes. The ordering of the normalized fluorescence intensities in the RFP channel
indicates the ordering of the maturation times of the fluorophores.
Fig. S2. The effect of FRET on timer signal. (A) Timer signal increases as FRET
from FP1 to FP2 increases. For the red profiles FP1 maturation time was fixed to 5
minutes while FP2 maturation time was varied. For the green profiles FP2 maturation
time was set to 100 minutes while FP1 maturation time was varied. Half-lives were
set to 30 and 180 minutes for the short- and long-living proteins, respectively.
Production rate was set to 2 molecules per minute and the additive noise was held
constant. Variability does not increase with FRET since only a subpopulation of the
timers has a mature FP2 fluorophore available as an acceptor. Therefore even with a
FRET efficiency of 1, total FP1 intensity does not tend to zero. See Supplementary
Mathematical Methods for a full description of the incorporation of FRET into the
model. (B) The coefficient of variation (CV) of timer signal decreases as FRET is
increased from 0 (uppermost line) to 1 (lowermost line) in steps of 0.25. The FP2
maturation time at which the CV attains a minimum decreases with increasing FRET
but by a small amount (note the narrow distance between the dashed vertical lines).
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Development 143: doi:10.1242/dev.125971: Supplementary information
Development 143: doi:10.1242/dev.125971: Supplementary information
Development • Supplementary information
Fig. S3. Time to reach steady-state. (A) An analytic estimate for the time to reach
steady-state was determined by finding the point at which the tangent to the point of
inflection intersects the steady-state value for FP2 fluorescence intensity (horizontal
line). (B) The time to reach steady-state is plotted as a function of protein half-life and
FP2 maturation time.
Fig. S4. FRET does not qualitatively affect the results of Fig. 3. Here the
simulations performed in Fig. 3 were repeated with a FRET efficiency of 0.5. As can
be seen by direct comparison with the profiles in Fig. 3, while FP1 intensities
decrease and FP2/FP1 intensity ratios increase, the qualitative shape of the profiles
remains unaffected.
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Development 143: doi:10.1242/dev.125971: Supplementary information
Development 143: doi:10.1242/dev.125971: Supplementary information
Table S1. Primers used for BAC modification
Target Orientation Homology
Sequence
arm
tdTomForward
Left
GATGCTGACCAAGAAAAGGGGGCCTA
sfGFP
TATCATCTGTATCTACTGAATCAGAGT
CGTCCAGTGCACTGACGAGTATGGTGT
CTAAGGGCGAAG
mKate2- Forward
Left
GATGCTGACCAAGAAAAGGGGGCCTA
sfGFP
TATCATCTGTATCTACTGAATCAGAGT
CGTCCAGTGCACTGACGAGTGTGAGCG
AGCTGATTAAGG
mKate2- Reverse
Right
ACACAAAAATACTTTACAATGTACAAA
sfGFP
AACTGTAGTAAAGTCCTGTTTTTATAA
and
GCTTAATCATCCATGTGGGCACCCGTG
tdTomGCCGTATCT
sfGFP
Primers used for amplification of the selectable targeting cassettes encoding for
the fluorescent timers. Underlined are the sequences homologous to PCR
templates.
Table S2. List of new plasmids generated for this study
pME constructs
pME-NLS
SP6 constructs for in vitro mRNA transcription
SP6:NLS-sfGFP
SP6:NLS-TagRFP
SP6:NLS-mKate2
SP6:NLS-TagRFP-sfGFP
SP6:NLS-tdTom-sfGFP
SP6:NLS-mKate2-sfGFP
Development • Supplementary information
p3’E constructs
p3’E-sfGFP
p3’E-tdTom-sfGFP
p3’E-mKate2-sfGFP