S1. MATERIALS AND METHODS
S1.1 Transcription reactions
DNA oligonucleotides (see also Figure 9) were ordered from IDT (Coralville, IA),
and were gel-purified prior to use. The sequences for hairpins D1.3 and D3.1 were (5’CGGCCTTTGCCGGCTCTTACTCCTGATCTGACTGCTAATACGACCTCCCTATA
GTGAGTCGTATTAGCCCATTGACCTAGAAGACTTGCTAATACGACTCACTATA
GGGAG-3’ and 5’GCCCATTGACCTAGAAGACTTGCTAATACGACCTCCCTATAGTGAGTCGTATT
AGCTCTTACTCCTGATCTGACTGCTAATACGACTCACTATAGGGAG-3’,
respectively. Hairpins were dissolved in transcription buffer at a final concentration of 2
μM for the experiment shown in Figure 9 or 100nM for the experiment shown in Figure
11, and were conformationally equilibrated by heating to 85oC for 5 minutes then cooling
to 25oC over 5 minutes. RNA inhibitors InhD1 and InhD3 were transcribed using an
Ampliscribe TXN kit (Epicenter, Madison, WI) and were purified by denaturing PAGE.
Transcription reactions that incorporated radiolabels were carried out according to
Triana-Alonso et al. (1995) in a modified transcription buffer to minimize aberrant, selfencoded 3' extensions. The composition of the modified reaction mix was ATP, GTP,
and CTP at 3.75 mM, but UTP at 0.8 mM. An aliquot of 1 μL alpha 32P-GTP (3000
Ci/mmol, 10 Ci/ mL, Perkin Elmer, Boston, MA) was included in each 10μL reaction.
The reactions were initiated by adding a mixture of inhibitor (when present) and
polymerase. The final concentrations of inhibitors were 5μM for the experiment shown
in Figure 9 and 1μM for the experiment shown in Figure 11. Reactions were generally
carried out for 15 minutes at 37oC and were stopped by heating to 75oC for 5 minutes to
inactivate T7 RNA polymerase. Following DNase treatment, transcripts were sieved on
12% denaturing polyacrylamide gels. Radioactive bands were identified and quantitated
on a Phosphorimager (Molecular Dynamics, Sunnyvale, CA).
S1.2 Computational modeling
The NAND network was simulated by using an RK45 ODE solver. The NAND
gate output was modeled as a 2D sigmoid translated so that its inflection point was at 0.5
and using the arbitrarily chosen exponent of 15.
NAND(a,b) = 1 – ( 1/(1+exp(-15*(a-0.5) ) ) ) * ( 1/(1+exp(-15*(b-0.5) ) ) )
The gates were assumed to be complementary so that the output of a gate was
equally capable of pulling down a node or pulling it up. Thus, the differential change in
potential energy for each input capacitor is the sum of the difference between the
potential of that capacitor and the potential of the output of every gate scaled by the
conductance of that particular path (the matrix W).
The following pseudo-code was used to calculate values for dy, in which y is the
state vector (charge across each capacitor i), W is the 2N by N conductance matrix, D is
the 2N vector of diffusion constants, and y_left and y_right are the state vectors of the
left and right neighbors. See also Figure 5.
For each gate input i:
For each gate output j:
dy(i) += W(i,j) * ( output(j) – y(i) )
end
dy(i) += D(i) * ( y_left(i) – y(i) )
dy(i) += D(i) * ( y_right(i) – y(i) )
end
Figure S1 demonstrates that in dimensionless Monte-Carlo experiments with the
amorphous latch that the mean feature size varies to the half power of the diffusion
constant (Figure S1). Figure S2a plots the evolution of two 180 degree phase-separated
waves of two diffusively-coupled oscillators. While the two oscillators start 180 degrees
out of phase they later synchronize. Both the wavelength and the amplitude of the early
phase is reduced compared to the later, synchronized phase. Figure S2b demonstrates
the intuitive fact that the peak frequency of the early phase oscillators falls rapidly as
diffusion is increased and stronger coupling reduces the ability of the boundaries to
survive.
The transcriptional ring oscillator was simulated by using an RK45 ODE
solver. The switch output is modeled using the kinetic dynamics as described by Kim et
al. (2006). A switch consists of a double stranded DNA template with a nick in the
promoter region. The switch can be put in an ‘ON’ state by the addition of an activator, a
complementary single-stranded DNA that completes the promoter region. The switch
can be put in an ‘OFF’ state by the addition of an inhibitor, a single-stranded RNA that is
complementary to the activator. Each switch is assumed to be identical with the same
values for their parameters. The parameters of our model are listed in Table S1. In
addition, we model diffusion using Fick's 1st Law. Assuming that the viscosity of buffer
was 1cP and using the method presented by Fusco et al. (2003), we estimated that the
diffusion constant of our inhibitors was on the order of 10-6 cm2/s.
The following equations describe the dynamics of one switch in our system,
where T is the concentration of ‘OFF’ switches, TA1 is the concentration of ‘ON’
switches, A1 is the concentration of free activator that can bind to switch T, I1 is the
concentration of inhibitor produced by T, A2 is the concentration of activator that I1 can
bind to, TA2 is the concentration of ‘ON’ switches that I1 can bind to, I2 is the
concentration of inhibitor that can bind to the A1, AI2 is the concentration of bound A1,
and I1left and I1right refer to the concentration of I1 in the left and right neighbors. See also
Figure S3.
d[T]/dt = -kTA[T][A1]+kTAI[TA1][I2]
d[A1]/dt = -kAI[A1][I2]-kTAI[T][A1]+kcatH/KMH[RNASE][AI2]
d[I1]/dt = -kAI[A2][I1]-kTAI[TA2][I1]+kcatON/KMON[RNAP][TA1]
+kcatOFF/KMOFF[RNAP][T]+D/X(I1right+I1left-2I1)
Supplemental References
Fusco, D., Accornero, N., Lavoie, B., Shenoy, S. M., Blanchard, J., Singer, R. H. &
Bertrand, E. 2003 Single mRNA Molecule Demonstrate Probabilitic Movement in
Living Mammalian Cells. Curr. Biol. 13, 161-167. (doi:10.1016/S09609822(02)01436-7)
Triana-Alonso, F. J., Dabrowski, M., Wadzack, J., Nierhaus, K. H. 1995 Self-coded 3'Extension of Run-off Transcripts Produces Aberrant Products during in Vitro
Transcription with T7 RNA Polymerase. J. Biol. Chem. 270, 6298-6307.