Fluid Flow and Interlinked Feedback Loops Establish Left-Right Asymmetric Decay of Cerl2 mrna in the Mouse Embryo

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1 Fluid Flow and Interlinked Feedback Loops Establish Left-Right Asymmetric Decay of Cerl2 mrna in the Mouse Embryo Tetsuya Nakamura, 1 Daisuke Saito, 2 Aiko Kawasumi, 1 Kyosuke Shinohara, 1 Yasuko Asai, 1 Katsuyoshi Takaoka, 1 Fenglan Dong, 1 Atsuko Takamatsu, 3 Jose Antonio Belo, 4 Atsushi Mochizuki, 2 and Hiroshi Hamada 1 1

2 Supplementary Figure S1. Quantitative FISH Analysis of Cerl2 mrna with an Intron robe in Individual Nuclei of Crown Cells. Each data point indicates the level of the Cerl2 intron signal in one nucleus. Four, three, and seven embryos were examined at 1, 2, or 3pss, respectively. No L-R difference in Cerl2 expression was apparent at any stage. 2

3 Supplementary Figure S2. Statistical analysis of apical localization of Cerl2 mrna. The node was divided into 10 areas from its anterior end to the posterior end, and apical/basal localization of Cer2 mrna in each area was examined by observing a transverse section of each area. When Cer2 mrna was detected in the apical side of crown cells, apical localization score of 1.0 was given (otherwise, the score was 0). After investigating 10 embryos at 1pss, 6 embryos at 2pss and 10 embryos at 3pss, the apical localization score for each area was averaged. Upon T-test, the difference in apical/basal localization between left and right sides is small at 1 pss (p=0.016), becomes larger (p=0.0018) at 2 pss, and is obvious at 3 pss (p= ). 3

4 Supplementary Figure S3. Whole-Mount FISH analysis of Nodal and Cerl2 Expression in Mouse Embryos at 2pss. Each set of embryos was processed at the same time in the same tube with either a Nodal or Cerl2 exonic probe. The expression of each gene varied among embryos in terms of both absolute level and L-R asymmetry. 4

5 Supplementary Figure S4. Quantitative Analysis of Nodal and Cerl2 Expression. (a) Relation between the FISH signal for Nodal and Lucifer yellow fluorescence in fertilized mouse eggs. See Supplemental Experimental rocedures for details. The FISH signal obtained for Nodal expression at the node of mouse embryos is within the linear range. (b) The data shown in Figures 3C through 3E are presented here with identification of the embryos corresponding to each point. 5

6 Supplementary Figure S5. Effect of Wnt signal on the position of the Basal Body and Cerl2 mrna expression, Degradation of Wnt3 and Wnt3A roteins Induced by Cerl2. (a-d) Control (ROSA CreERT2,βcatenin +/+ ) (a) and ROSA CreERT2,βcatenin flox/δ (c) embryos exposed to tamoxifen during gestation were subjected to immunofluorescence staining with antibodies to Odf2 to identify the basal body; green) and to E-cadherin (to identify the pit cell membrane; red) at E8.5. Higher magnification images are shown in (b) and (d), respectively. AB score (average basal body position score: Hashimoto et al., 2010) is 0.34 and 0.28 in the control and the mutant embryos, respectively, suggesting that the posterior tilt of node cilia is maintained in ROSA CreERT2,βcatenin flox/δ embryos. (e-h) Control (ROSA CreERT2,βcatenin +/+ ) (e) and ROSA CreERT2,βcatenin flox/δ (f-h) embryos exposed to tamoxifen during gestation were subjected to immunofluorescence staining with an antibody to β catenin (red), fluorescent in situ hybridization for Cerl2 mrna(green) and DAIstaining (blue). Three ROSA CreERT2,βcatenin flox/δ embryos are shown (f-h). When β catenin was relatively maintained (f), Cer2 mrna showed relatively normal R>L asymmetry. When β catenin was severely lost in crown cells (g, h), Cer2 mrna lost asymmetry (g) or was downregulated (h). (i,j) Node explants obtained from embryos at the late headfold stage were cultured in the presence of 1% methylcellulose, with (j) or without (i) Wnt3A protein(0.8 µg/ml) for 5 hours, and were examined for Cer2 expression. Notre that Cer2 mrna is reduced by Wnt3A. (k) Immunoblot analysis with antibodies to Wnt3A for lysates (1/10 and 1/100 dilutions) and culture supernatants (Sup.) of Cos7 cells transfected with expression vectors for Wnt3A (42 kda) and either LacZ or Cerl2-3 HA. (l) Immunoblot analysis with antibodies to Wnt3A for lysates and culture supernatants (Sup.) of Xenopus oocytes injected with mrnas for Wnt3A and either LacZ or Cerl2-3 HA. 6

7 Supplementary Figure S6. Expression of Nodal, GDF1, Brachury and mnkd1 after various pharmalogical treatments. (a-h) Embryos cultured without (a,c,e,g), with IWR (b, f) or with BIO (d, h) were examined for Nodal and Gdf1 expression. Note that expression of these markers was maintained in IWR-treated and BIO-treated embryos. (i,j) Embryos cultured with Actinomycin alone (i) or with Actinomycin plus BIO (j) were examined for Brachury expression. Note that BIO does not affect Brachury expression. (k-n) Embryos cultured without (k, m), with IWR (l) or with BIO (n) were examined for mnkd1 expression. Note that mnkd1 expression is down-regulated by IWR while it was upregulated by BIO. 7

8 Supplementary Figure S7. Density lot of Q wt + Q on the Two-Dimensional arameter Space. The horizontal and vertical axes represent C and D, respectively. The other parameters are fixed as A = , B = 0.07, E = , F = , G = , K 1 = , K 2 = and K 3 = The red curves indicate the parameter region where the system has bistable steady states before (solid) and after (broken) reception of the signal, as determined by mathematical analysis. 8

9 Supplementary Figure S8. Nullcline Analysis for the Mathematical Model. The horizontal and vertical axes correspond to and, respectively. The red and blue curves are the nullclines for dynamics of and in system <S3>, respectively. The arrows show the direction of trajectories. Black and white circles indicate stable and unstable equilibria, respectively. In (a), the red and blue curves have a single intersection, indicating that the system has a single equilibrium. The directions of the arrows show that the equilibrium is stable. This property of the system is referred to as single stable. In (b), the red and blue curves have three intersections. The directions of the arrows indicate that two equilibria are stable and the remaining one is unstable. This property of the system is referred to as bistable. The nullcline for after reception of the signal is shown by the broken curve. Results for the system under the mechanism of bistable switch are shown in (c). An increase in the signal results in a change in the shape of the nullcline of and transition of the system from bistable to single stable. The two stable states correspond to expression patterns on the right and left sides of the node in the wild-type embryo. (d)the intersection between the blue curve and the horizontal axis corresponds to the expression pattern of Wnt in the Cerl2 / embryo. The intersection between the red curves and the vertical axis correspond to the expression patterns of Cerl2 on the right (solid) and left (broken) sides in the Wnt / embryo. Results for the system under the mechanism of temporal transition to asymmetric state are shown in (d). An increase in the signal results in disappearance of the initial single stable state and the appearance of a new single stable state. The two stable states correspond to expression patterns on the right and left sides of the node in the wild-type embryo. 9

10 Supplementary Figure S9. Analysis of arameter Dependencies According to Signal Amplitude. (a)the change in nullclines induced by an increase in the value of each parameter is shown, with the arrows indicating the direction of the shift. (b)the vertical axis shows the number of successful results obtained with either bistable switch (red) or temporal transition to asymmetric state (blue) and with randomly chosen parameter sets. The horizontal axis shows signal amplitude (γ), which is the difference between the minimal and maximal values of the signal. 10

11 Supplementary Figure S10. article Image Velocimetry Analysis of the Flow in Methylcellulose-treated embryos. Mouse embryos at 3pss were transferred to culture medium containing 0 % (a) or 1.0 % (b) methylcellulose, and the nodal flow was monitored. Black arrows indicate the direction and velocity of the flow at each point in the node. Note that the flow is not detected in the presence of 1.0 % methylcellulose. 11

12 Supplementary Figure S11. Temporal Fluctuation of Nodal Flow Velocity. A wild-type mouse (ICR) embryo was isolated at E8.0 (1pss) and cultured for 100 min with the use of a whole-embryo culture method. Temporal changes in the velocity of nodal flow were analyzed by IV (see Supplemental Experimental rocedures). The velocity of leftward flow continued to fluctuate between 2 and 5 µm/s during this time. Red circles, blue open circles, green open triangles, and orange asterisks denote leftward, rightward, anterior-directed, and posterior-directed velocity, respectively. 12

13 Supplementary Methods Quantitative FISH Analysis Quantitative whole-mount FISH analysis was applied to measure the levels of Nodal and Cerl2 expression (Figures 3a 3e, Supplementary Fig. S4). To determine first whether FISH is suitable for quantitative analysis, we examined the linearity of color development by injecting fertilized mouse eggs with a solution containing Lucifer yellow (250 ng/ml) and a Nodal sense riboprobe (0.8 µg/µl). The eggs were then immediately fixed with 4% paraformaldehyde for 5 h. Each egg was placed in a well of a 24-well plastic dish and was subjected, together with 3pss mouse embryos placed in other wells, to FISH with a Nodal exon probe. The FISH signal from each egg was captured with a CCD camera (C , Hamamatsu hotonics) attached to a Axiophoto microscope (Zeiss) and was compared with the fluorescence signal of Lucifer yellow (Supplementary Fig. S4). We also plotted the Nodal signal from the node of each embryo converted to the same area as that of the fertilized eggs. We confirmed that signals from the embryonic node are within the region that shows linearity of the fluorescence signal. Signals obtained with Cerl2 exon and intron probes as well as Nodal intron probes were also within the linear range. For quantitation of the level of Cerl2 mrna in the nucleus (Fig. 1b, c), FISH images obtained with a Cerl2 intron probe were processed with ImageJ software. Arbitrary thresholds were set for background fluorescence and particle signals of the intron probe. We picked up only background fluorescence or particle signals from original images and converted binary images. Binary images were divided by 255 (maximum bit number) and multiply with original images, yielding background images or signal images. The background images were processed by Gaussian blur (sigma = 8) and subtracted from signal images. After subtraction, the signal images were subjected to particle analysis with ImageJ. The results for multiple embryos according to developmental stage are plotted in Supplementary Fig. S4. Quantification of Nodal Flow Velocity For time-lapse observations of the node in living intact embryos, embryos dissected at E8.0 were first cultured under 5% CO 2 for 30 min at 37 C in phenol red free DMEM supplemented with 75% rat serum. We then transferred each embryo to 3 ml of the same medium in a glass-bottomed dish (Iwaki) by hanging its ectoplacental cone from a glass pipette with an inner diameter of 0.3 mm. For proper orientation of the node relative to the objective lens during observation, the embryo was positioned in a hole of a membrane filter (Millicell ICM01250, Millipore). Each embryo was then maintained 13

14 at 37 C under 5% CO 2 in a culture chamber (TOKAI HIT) placed on the stage of an inverted microscope (Leica). To characterize the flow fields of nodal flow, we used multipoint scanning confocal microscopy and particle image velocimetry (IV) analysis. The node cavity was filled with DMEM supplemented with 10% fetal bovine serum and fluorescent microbeads (diameter of 0.2 µm with excitation and emission wavelengths of 505 and 515 nm, respectively; Invitrogen). Motion of the beads was observed at planes positioned 8 µm below the apical cell surface of the node cavity for 10 s (30 frames/s) with the use of a CSU-X confocal unit (Yokogawa) and an ixon EMCCD camera (Andor Technology) connected to a DMI6000B microscope (Leica) that was equipped with a 63 glyin-immersion objective lens. Time-series images for IV analysis were captured at a resolution of 512 by 512 pixels and were processed with interrogation windows of 16 by 16 pixels with 50% overlap, corresponding to a spatial resolution of 4.3 by 4.3 µm. The time-averaged velocity distributions were calculated for 10 s. For analysis of directional flow and its curve fitting, leftward (V l ), rightward (V r ), upward (V a ), and downward (V p ) directional flows were calculated in the area that includes all directional flows and vortices (Supplementary Fig. S11). The velocities V l, V r, V a, and V p were defined from the 90 th and 10 th percentile values of the velocity vector components in the positive or negative directions of the x-axis and y-axis, respectively. Mathematical Model We analyzed the dynamic properties of the gene regulatory circuit underlying L-R asymmetry of gene expression in mouse embryos by mathematical modeling. There were two objectives for this aspect of the study: (1) To determine unknown components of the system from analysis of the mathematical model as the parameter values are changed. (2) To understand the mechanism by which the system generates asymmetric patterns of gene expression. We first developed a mathematical model based on knowledge of the reciprocal regulation of Cerl2 and Wnt expression. We then examined the mathematical model by numerical simulation to confirm that it is able to reproduce the observed expression patterns if the parameters are assigned appropriate values. Next, the condition of the parameters was examined by numerical analysis in an exhaustive search. And, finally, we analyzed the model by simple mathematical analysis and determined the principal mechanism of the system that underlies the generation of left-right asymmetry. We developed an ordinary differential equation (ODE) model for the dynamics of the 14

15 concentrations of four molecules: Cerl2 mrna, Cerl2 protein, Wnt mrna, and Wnt protein. We used Hill functions for unknown forms of regulatory functions with parameterized Hill coefficients in order to take into account possible nonlinearity. The model includes the following aspects of gene regulation identified experimentally in the present study: (1) enhancement of the degradation of Cerl2 mrna by Wnt protein; (2) enhancement of the degradation of Cerl2 mrna by the nodal flow signal; (3) enhancement of Wnt transcription by Wnt protein; and (4) enhancement of the degradation of Wnt protein by Cerl2 protein. We consider the dynamics on the left and right sides of the node separately by assigning different signal inputs; that is, we calculate two ODE systems (right and left systems) separately, with only one (the left) system receiving input of the increasing nodal flow signal. Each ODE system is written as follows: p a1 b1 + b2 + b3 + σ1 ξ1( t) p + k1 n1 d r = n1 n1 signal r dt d p = c1 r d1 p + σ 2 ξ2( t) dt d r dt p = a a b ( t) n n n2 r + σ ξ p + k2 <S1> p + σ ξ ( t) n3 d p = c r d + d n3 n 3 p dt p + k3 where r and p are the concentrations of Cerl2 mrna and Cerl2 protein, respectively, and r and p are the concentrations of Wnt mrna and Wnt protein, respectively. The a i are the transcription rates of the corresponding genes, the the degradation coefficients for the mrnas, the corresponding mrnas, and the The c i are translation rates for the b i are d i are the degradation coefficients for the proteins. n i are Hill coefficients, and ξ i ( t) and σ i are the white noise and the magnitude of the noise, respectively. The noise terms are used only for calculation of the dynamics under tain conditions; they are not used for the analyses below. The nondimensionalized system is given by the following: 15

16 n1 d 1 R = S1 1( t) n1 n1 B signal R + ξ dt A + K1 d dt d R dt = R + S2 ξ2( t) C n2 R = 1 + D + S3 ξ3( t) n K E n 2 <S2> d n3 1 1 = R + S4 4( t) n3 n + ξ 3 dt F G + K 3 where b b b b t = b2t R = r = p R = r = p a1 a1 c1 a2 a2 c2 b b b a b b b A = B = C = D = E = F = G = b b d a b d d and b k a c K K K = 2 = 2 3 = a2 c2 b2 k2 a1 c a1 a1 c1 a2 a2 c2 b k σ b σ σ b σ S = S = S = S = The number of parameters is 18 after nondimensionalization. In the following analysis, we ignore the fluctuation of the internal system by setting S1 = S2 = S3 = S4 = 0. Our aim was therefore to determine the condition of values for 14 parameters for reproducing the observed patterns of gene expression. In numerical analysis, we calculated the dynamics of the system from t = 0 to t = 400, which seemed to be a sufficiently large window. The value for signal is fixed at 0 in the right system and is given as follows for the left system: 0 ( t ' t1 ) 2 ( ' 1) β ( ' 1) ( 1 ' 2 ) 1 ( t t ') < signal = α t t t t t t < t 2 < where t 1 = 100 is the start time of the signal, and β 2 = and 5 = are determined from time-series measurements of the flow. The signal is an increasing function of time with saturation in the defined region of time 16 α

17 ( t t ' t 250) < =. The initial state of the system is given as 1 2 * * * * ( r ( 0 ), p ( 0 ), r ( 0 ), p ( 0 )) ( R,, R, ) =, where R *, *, * R, and * with minimum * being chosen from the set for equilibrium of system <2>. The initial state corresponds to the gene expression observed on the right side of the node. We used Eular s difference method with the time step t = Some examples of dynamics by numerical calculation are shown in Fig.7. We examined the dynamics of the model by numerical analysis to determine the condition of parameter values for reproducing the observed patterns of gene expression. We first introduced two indices to qualify the extent to which the results of computer simulation reproduce the experimentally determined gene expression patterns. Quality for wild-type embryos: Q Min +, R +, L +, L +, R wt = [, ] Where,R + and,l + are the concentrations of Cerl2 protein and,l + are the concentrations of Wnt protein in the left or right systems of the wild-type simulation. These protein concentrations are measured at the final time, t = 400, of and,r + each numerical calculation. Higher values of this index reflect larger differences in the concentrations of Cerl2 and Wnt proteins between right and left systems. Quality for Wnt mutant embryos: Q +, R +, L =, R, L where,r and,l are the concentrations of Cerl2 protein in the right and left systems of the Wnt / simulation, respectively. Higher values of this index reflect smaller differences in the concentration of Cerl2 between right and left systems in the Wnt mutant relative to those in the wild type. The symmetric expression of Wnt in the model system for the Cerl2 / mutant is trivial. We confirmed that the levels of Wnt mrna or Wnt protein are the same in the right and left systems if R and are fixed at 0 in system <S2>. We therefore did not introduce another index for the Cerl2 mutant simulation. In numerical analysis, we repeated random choices of parameters, numerical calculations with the chosen parameters, and evaluation of the obtained expression patterns using Q wt and Q. We assigned criteria for successful simulations as 17

18 Q > 0.5 and Q > 2, where the threshold values are determined arbitrarily. We wt chose values for each parameter randomly, with a real number between 0 and 4 for A, C ~ G, K 1, K 2, K 3 and a natural number between 1 and 6 for n i. We calculated the dynamics of system <S2> with the chosen parameter set. After numerical calculation, we examined Q wt and Q with the final values of protein concentrations. If these indices had higher values than the threshold, we added the used parameter set to a set of successful parameters. We examined 140,000 different parameter sets, among which 4,253 sets satisfied both criteria. We examined the distribution of these successful parameter sets and considered the condition of each for reproduction of the observed gene expression patterns. We examined the distribution Q wt and Q in detail on the two-dimensional parameter space ( C,D ), with C and D being chosen as a pair of important parameters from results of numerical calculations. The region of high and that of high Q wt Q overlapped in many cases of numerical simulation. The density plot of Qwt + Q is shown in Supplementary Fig. S7, with the horizontal and vertical axes corresponding to C and D, respectively, and the other parameters being fixed. A balance between C and D values is apparent in the distribution of successful sets. A balance between C and D is thus important for reproduction of observed gene expression patterns. We examined different choices for pairs of parameters, and found that many pairs are balanced in reproductions of observed expression patterns. The parameter dependency of the system is explained in more detail by mathematical analysis below. Mathematical analysis Here we consider equilibria of the system, which are obtained by solving an equation system given by equating the right side of ODE system <S2> to zero. Before considering equilibrium of the total system, we equate only the first and third equations of system <S2> to zero and substitute the obtained expressions into the second and fourth ODEs of system <S2>. We obtained a closed ODE system including only two variables, the concentrations of the two proteins. d dt 1 = n1 1 C + + B signal n1 n 1 A + K1 n2 n3 d 1 1 = E 1+ D n n + n n dt + 1 K F G + K <S3> 18

19 We do not consider the dynamics of the above system but use the form to facilitate discussion of the equilibria; that is, we analyze the equilibria of system <S3> visually using nullcline analysis on the two-dimensional space (, ) Supplementary Fig. S8a and b shows two examples of the analysis. The red and blue curves are the nullclines for the dynamics of 19 and in system <S3>, respectively. The right side of the first ODE of system<s3> is equal to zero in the red curve, and that of the second ODE of system <S3> is equal to zero in the blue curve. The intersections of the red and blue curves are equilibria of the system. The arrows indicate the direction of the dynamics on the plane, from which we can determine the stability of the equilibria. The system has a single stable equilibrium in Supplementary Fig. S8a and two stable equilibria in Supplementary Fig. S8b. Supplementary Fig. S8c and d shows typical nullclines of system <S3> obtained from a successful parameter set. As we showed in the computer simulation, the system manifests large changes in gene expression with an increase in the signal under the appropriate values of parameters. The large changes in gene expression can be understood from a bifurcation of steady states of the system. Supplementary Fig. S8c shows the case for the system under the mechanism of bistable switch. Before receiving the signal, the system has two stable equilibria, one characterized by high Cerl2 and low Wnt, and the other by low Cerl2 and high Wnt. The first steady state corresponds to expression on the right side of the node or to that on the left side before reception of the signal. As the signal level increases, one nullcline changes its shape and position. The equilibrium corresponding to the initial state disappears by saddle-node bifurcation, and the system must transition to the remaining stable equilibrium, resulting in large changes in gene expression for Cerl2 and Wnt. The number of equilibria changes from two to one as a result of the increase in the signal. We therefore conclude that the large changes in gene expression represent transition of the system from bistable to single stable from the dynamics point of view. We can also understand expression patterns in mutant embryos using Supplementary Fig. S8c. The state of Cerl2 knockout mice is determined from system <S3> by fixing to zero. It is represented by the intersection of the blue curve with the horizontal axis, where the expression level of Wnt is high and close to that for the left state of the wild type. The symmetric expression pattern in the Cerl2 mutant is consistent with the blue curve not being dependent on signal. Similarly, the state of Wnt knockout mice is determined by fixing to zero. It is represented by the intersection of the red curves with the vertical axis, where the expression level of Cerl2 is high before reception of the

20 signal and slightly smaller thereafter. The figure explains that the expression pattern is almost symmetric, but with the level of Cerl2 on the right being a little higher than that on the left, in the Wnt mutant. The mathematical analysis also explains the distribution of successful parameters. We can calculate the number of stable equilibria from system <S3> for a given parameter value. The red curves in Supplementary Fig. S7 show the regions on the parameter space (C, D) where the number of stable equilibria is two. The region surrounded by the solid curves corresponds to the condition before reception of the signal, and that surrounded by the broken curves corresponds to the condition after signal reception. We can explain the large part of the successful parameter set as the region where the system is bistable before the signal and single stable after the signal. Supplementary Fig. S8d shows the case when the system manifests asymmetric gene expression under the second mechanism of temporal transition to asymmetric state. Before reception of the signal, the system has a single stable equilibrium, characterized by high Cerl2 and low Wnt and corresponding to expression on right side of the node or to that on the left side before signal reception. As the signal level increases, one of the nullclines changes its shape and position. The equilibrium corresponding to the initial state disappears and is replaced with a new stable state characterized by a lower value for Cerl2 and a higher value for Wnt. In some cases of parameter choice, the first state does not disappear but just moves toward the lower right on the (, ) space. We should point out the possibility that the system may have two stable equilibria as the signal increases, but it has only one equilibrium both before and after reception of the signal. This mechanism is possible theoretically, but such behavior of the system is observed rarely with randomly chosen parameters. arameter dependence We examined properties of the nullclines. The nullcline for the dynamics of curves in Supplementary Fig. S8) decreases with 20 on the (, (red ) plane in a manner independent of other parameters. In addition, the nullcline crosses the vertical axis with a finite value of, whereas it does not cross the horizontal axis because > 0 when. Similarly, the nullcline for the dynamics of (blue curves in Supplementary Fig. S8) is infinitely large at + 0 and it crosses the horizontal axis with a finite value of in this system.. There should thus be at least one equilibrium We examined the parameter dependence of the nullclines in Supplementary Fig. S9a. There are 14 parameters in the system, each of which affects either of the two nullclines of system <S3> and most of which shift the position of the nullclines. The

21 parameters A, C, and K 1 have similar effects in that the red curve moves toward the upper right as the value of each of these parameters increases. The parameters may thus be categorized in the same class. arameters D, E, F, G, K 2, and K 3 also have similar effects in that the blue curve moves toward the upper right as the value of each of these parameters increases; these parameters can thus be categorized in another class. We have already analyzed in detail the condition for the pair of parameters C and D (Supplementary Fig. S7), finding that balance between C and D is important for reproduction of the observed patterns of gene expression. The reason for this is now clear. From the mathematical analysis above, the positions of the two nullclines are important for the induction of large changes in gene expressions by a small change in the signal. The parameters C and D move the nullcline for the dynamics of that for the dynamics of and, respectively. The values for C and D thus need to be tuned and balanced in order for the two nullclines to be positioned at an appropriate distance. We found that many pairs of parameters must be similarly balanced for reproduction of the observed expression patterns, consistent with the two classes of parameters, { A, C, K } and {,,,,, } 1 D E F G K K, the members of each of which have 2 3 similar effects on the position of nullclines. We therefore conclude that the condition of parameters for reproduction of the observed patterns of gene expression is that { A, C, K } and {,,,,, } 1 D E F G K K should be balanced. 2 3 Effect of Signal Amplitude We examined the effect of reducing the amplitude of the signal. This is important because L-R asymmetry can be established even if the flow signal is reduced to a low level by reducing the number of rotating cilia to only two (Shinohara et al., 2012). We examined which mechanism, bistable switch or temporal transition to asymmetric state, is able to better explain biological phenomena in the presence of a small-amplitude signal. We performed numerical analysis, with the value of parameter B being set to 0.01, 0.04, 0.07, 0.1, 0.13, or For each amplitude value, the other parameter values were chosen randomly as before, with a real number between 0 and 4 for A, C ~ G, K 1, K 2, K 3 and a natural number between 1 and 6 for n i. The number of examined parameter sets was 140,000 for each value of signal amplitude. The dynamics were then calculated with the chosen parameter sets. The successful parameter sets were selected with the use of the same indices and criteria as above ( Q wt > 0.5 and Q > 2 ). For each successful parameter set, we counted the number of equilibria by 21

22 nullcline analysis and distinguished the mechanisms for asymmetric gene expression (that is, number of steady states = 1, "temporal transition to asymmetric state"; number of steady states > 1, "bistable switch"). Supplementary Fig. S9b shows the number of successful simulations realized by "bistable switch" and by "temporal transition to asymmetric state" according to signal amplitude, with the number for the former being much greater than that for the latter. The frequency of successful simulations for "bistable switch" showed a clear peak corresponding to the optimal signal amplitude for the mechanism. This optimal value for "bistable switch" (B = 0.1) was smaller than that for "temporal transition to asymmetric state" (B = 0.13). When the signal amplitude is small, it is almost impossible to realize asymmetric gene expression with the mechanism temporal transition to asymmetric state. Given that L-R asymmetry of gene expression is generated even when the signal amplitude is low, we conclude that bistable switch is more plausible than temporal transition to asymmetric state as a mechanism to explain biological phenomena. 22