Yamada T , Nishiyama M , Oba S , Jimbo HC , Ikeda K , Ishii S , Hong K , Sakumura Y .

	Fig 1. Conversion of a biomolecular signal to an electrophysiological signal in a nerve growth cone. (a) Signaling pathways from an extracellular guidance cue, e.g. Sema3A, to membrane potential (MP) shifts in a growth cone. Sema3A, upon binding to its receptor on the plasma membrane (PM), increases the intracellular cGMP level. The effector proteins, Cyclic nucleotide-gated ion channels (CNGC) and cGMP-dependent protein kinases (PKG) that directly bind to cGMP (solid arrows) trigger, respectively, the activation of chloride channels (ClC) and sodium channels (NaC), which induce either hyperpolarization or depolarization. The potential cross interactions between the downstream effectors, which are unknown, are indicated by dashed blue lines. (b) Growth cone recording stimulated by a stimulant, 10 μM 8-Br-cGMP, a membrane permeable cGMP analogue (Top). Samples of recorded growth cone MP time series (MPTS; control: n = 7). (c) The MPTS contains over 10,000 data points (black). Outlier noises, such as spikes, are removed by sampling the data points at a 1 sec interval (green dots; see upper inset, enlargement of dashed square region; Supplementary Fig. 1). (d) The schematic diagram illustrates the mathematical model that functions as an alternative system and is designed in mesoscopic scale that expresses effective molecular signal flows. The control model contains the signaling pathway from the cGMP stimulation to MP shifts, via CNGC- and PKG-downstream factors (DFs), corresponding to the pathways in (a). The unknown signal flows between CNGC-DF, PKG-DF, ClC, and NaC in the core system are indicated within the dashed grey box. The parameters of the unknown flows and known factors, i.e., CNGC-DF to ClC and PKG-DF to NaC, are involved in the blue box and characterized by the core system parameter set, ��. The 8-Br-cGMP stimulation and MP, which are outside of the core system, are characterized by cell-dependent peripheral parameter set, ��. We modeled the diversified data as a core system with peripheral elements representing cell-to-cell variability. Two pharmacological experimental conditions were also modeled, which represent complete inhibition of ClC, blocked by DNDS (n = 5) and NaC, blocked by STX (n = 4) (see Supplementary Fig. 2). A total of 16 datasets (control: n = 7; DNDS: n = 5; STX: n = 4) were analyzed.
	Fig 2. Model candidates and Bayesian system identification procedure. (a) The matrix shows the possible interactions in the gray region in Fig. 1d; pathways from CNGC-DF to PKG-DF, CNGC-DF to NaC, PKG-DF to ClC, and PKG-DF to CNGC-DF in the clockwise direction. Each of four pathways has three potential signals, i.e. activation (green), inhibition (red), or no interaction, as modeled by the core system parameters (��). Thus, there are 34=81 model candidates labeled as M1 to M81. (b) Schematic illustration of prior distributions of model parameters that models the core system uncertainty and cell-dependent variability, expressing that extremely large values are biologically implausible and result in overfitting to the data (left; left-truncated Gaussians). Parameter values are applied to each of the models listed in a to generate model time series (TS) (��̂ (��); blue line in the right). (c) Schematic illustration of model fitness, which depends on the parameters (left). The model fitness is the product of all the model likelihood (10 μM 8-Br-cGMP; control/DNDS/STX; n = 16), and the i-th likelihood with the parameter values in b was computed as the product of Gaussian probability density functions of the i-th MPTS (black; ��(��)) for the entire time points (��=1,⋯,����) with the mean ��̂ (��) (blue) and the s.d. ���� (right). (d) Illustrates the model plausibility (left; Bayesian evidence), which evaluates both model fitness and parameter plausibility. The model plausibility was computed by integrating the product of the model fitness and the parameter plausibility over the entire ranges of all the parameters (red area in the left). The logarithmic evidences for all the 81 models were displayed in the matrix corresponding to that in a with red-white color (right; higher evidence is colored by red). (e) The core system parameters were estimated from all the datasets (��������; n = 16) and cell-dependent peripheral parameters of the i-th cell (����; n = 1) were from only the i-th cell’s MPTS, as posterior probability distributions. When we use representative specific parameter values instead of distributions during validation steps, we selected mean a posteriori (MAP) values (black circle in d) and denote the parameter set with superscript “MAP” like ��������������.
	Fig 3. Most plausible model and the generality test by leave-one-out cross validation. (a) Computed model evidences (logarithm evidences; Fig. 2d) are expressed as a matrix representation following Fig. 2a (10 μM 8-Br-cGMP; control/DNDS/STX; n = 16). A model with large log evidence (toward greater red) indicates greater plausibility than that with smaller log evidence (toward white). Models in left six columns are labeled as model group a; those in right three columns are labeled as model group b. Models indicated by M1 and M7 (insets) contain, respectively, the fewest and common interactions in groups a and b, representing minimal models within the respective model groups. (b) Histograms of log evidence in the model groups a and b, as shown in a. Vertical dashed lines indicate the log evidence of the two representative models, M1 and M7. (c) Schematic procedure of the generality test of the core system parameter estimations by applying the leave-one-out (LOO) data selection to the models. Black and gray horizontal bars represent experimental MPTS. When the ��-th MPTS is left out from the 10 μM 8-Br-cGMP datasets (n = 16), it is used to estimate peripheral parameters (����) and validate model fitness (product of all the likelihood in white box), whereas the remaining MPTS (n = 15) are completely separated from the left-out MPTS to estimate the MAP core system parameters (����������). Repeating LOO for all the datasets, 16 system parameter sets are obtained (����������;��=1,⋯,16). The model fitness to the left-out MPTS was taken the logarithm and averaged over all the data points. (d) Matrix representation of the averaged model fitness by LOO-derived parameter (����������) (left) and the difference between the fitness and that computed from all the dataset (��������������) (right). (e) Distributions of the estimated MAP core system parameter values given by LOO in a (����������; black point) compared with those by all the datasets (��������������; green circle; n = 16). The parameters exhibited in the horizontal axis and their values are represented in the vertical axis with relative representation with respect to the mean (actual values are listed in upper axis with the corresponding color). Most of the estimations are within the mean ± s.d. (gray region).
	Fig 4. Specificity of identified core system to 8-Br-cGMP-induced MPTS. (a) Schematic procedure of model specificity test using MPTSs in different conditions using MAP core system parameters estimated from all the 10 μM 8-Br-cGMP datasets (��������������; n = 16; black horizontal bars in the left side of the shaded band). The model likelihood to the ��-th MPTS from other testing datasets (white bars) was computed while computing the cell-dependent parameters, ����, and by repeating this the model fitness to all the n datasets were obtained. Testing datasets are the datasets untrained by the core system (5 μM 8-Br-cGMP-induced MPTS, PKG-inhibited condition (KT5823), and netrin-1-induced MPTS), those with noise (10 μM 8-Br-cGMP-induced MPTS with randomly labeled channel inhibitor). (b and c) Matrix representations of model fitness by the datasets derived by the control core system; 10 μM 8-Br-cGMP-induced MPTS with mixed labels of channel conditions (control, DNDS, STX) (b; n = 16) by randomly selecting channel blocker conditions in the model (see Supplementary Methods), and 5 μM 8-Br-cGMP-induced MPTS (c; total n = 11; control: n = 2; with DNDS blocking ClC: n = 5; with STX blocking NaC: n = 4). (d and e) Same as b and c, but the datasets derived by the different core systems; 10 μM 8-Br-cGMP-induced MPTS under the PKG-inhibited condition (KT5823) (d), and netrin-1-induced MPTS (e).
	Fig 5. Model predictability of untrained 8-Br-cGMP-induced MPTS. (a) Schematic illustration of the procedure for testing model predictability for the late-phase of MPTS induced by 10 μM 8-Br-cGMP. The initial-phase (250 sec) of the left-out ��-th MPTS was extracted to estimate the MAP peripheral parameters (����), and the remaining ��-th Msas used to test the model predictability. The MAP core system parameters were estimated from the initial and late phase of the remaining MPTS (����������; n = 15). (b) Sample MPTS predicted by model M1 (blue dotted line) and M7 (blue solid line) by the prediction procedure in a. (c) Matrix representation of log-likelihood of the predicted MPTS when repeating ��=1,⋯,16 and summing them. (d) Same as a, but 5 μM 8-Br-cGMP MPTS was predicted with ���� given the initial-phase MPTS (up to 250 sec) and �������������� by all the 10 μM 8-Br-cGMP MPTS. Black and white bars represent experimental MPTS. MPTS in the white regions (after 250 sec, the late-phase) were never used for the parameter estimation; these MPTSs were completely untrained data sets even for the peripheral parameters. (e) Sample MPTS predicted by model M1 (blue dotted line) and M7 (blue solid line) by the prediction procedure in d. (f) Matrix representation of log-likelihood of the predicted MPTS when repeating ��=1,⋯,11 and summing them.
	Fig 6. Reproducibility of cGMP-dependent MP shifts. (a) Model MPTS by the model M7 induced by different 8-Br-cGMP concentration: (i) 0.5, (ii) 6, and (iii) 20 μM with �������������� and �������������� (Supplementary Table 1). Each MPTS reaches the steady-state at about 300 sec. We defined the model’s steady-state MP shift as the difference between the MPs at times zero and infinity, namely, ��̂ (��→∞)−��̂ (��=0), where the core system parameters (��������������) from the 10 μM 8-Br-cGMP dataset were used. For the peripheral parameters, ��K is a mere constant bias in ��̂ (��) (see Supplementary Methods) and, therefore, was canceled out, ���� was ignorable at times zero and infinity, and ���� and ���� were replaced by their means in ��������������. (b) Model of 8-Br-cGMP concentration dependent MP shifts. Prediction of the cGMP-dependency of the steady-state MP shifts induced by the direct pipette stimulation (inject 8-Br-cGMP through the recording pipette; blue dashed line). Mimicking of the bath application condition of 8-Br-cGMP (blue solid line) by correcting the 8-Br-cGMP permeation process from outside to inside of the cell (Supplementary Methods). Experimental data of steady-state MP shifts under the bath application condition (black square with lower x-axis; ref.16) are shown for comparison. Normal condition indicates a cGMP level when Sema3A was applied in the turning assay; ODQ (1H-(1,2,4) oxadiazolo(4,3-a)quinoxalin-1-one, a soluble guanylyl cyclase inhibitor) condition reduces growth-cone cGMP levels.
	S1. Preprocessing of membrane potential time series The recorded raw data (black) was sampled at one sec intervals (green) to remove most of spike‐like noise. The step‐like noise, which was manually detected, was replaced with a straight line (blue). To estimate the size of the observation noise, the sampled data (green) was smoothed by applying a moving average filter with a ± 5 sec window (red), and the differences between the sampled and the smoothed at the same time point were obtained (see Methods in the text).
	S2. MPTS recorded from growth cone of Xenopus spinal neurons (a‐b) MPTS induced by 10 M 8‐Br‐cGMP injected from a pipette under control conditions (a) (same as Fig. 1b), in the presence of the chloride channel inhibitor, DNDS, (b) and sodium channel inhibitor, STX, (c). (d‐f) MPTS induced by 5 M 8‐Br‐cGMP under control conditions (d), DNDS (e), and STX (f). Sampled and raw MPTSs are in the upper and lower rows, respectively.
	S3. MPTS induced under different conditions (a) MPTS induced by 10 M 8‐Br‐cGMP in the presence of PKG inhibitor (KT5823). (b) MPTS induce by Netrin‐1.
	S4. Computation of the 8‐Br‐cGMP diffusion rate (a) Growth cone model for computing the time series of diffusion of 8‐Br‐cGMP, which is composed of three shell‐shaped compartments. The 0‐th compartment is the pipette, which releases 10 M 8‐Br‐cGMP. To express the size variation of growth cone, Gaussian noise was added to the thickness of the outmost shell compartment (compartment #3). (b) The time series of 8‐Br‐cGMP concentration in the compartment #3 (red line) as calculated by the diffusion equation (Eq. (S2) in Supplementary Methods). The black line is the exponential function (Eq. (S1) in Supplementary Methods) fitted to the red line.
	S5. Pharmacological application during MP induction DNDS, STX, and KT5823 inhibit specific molecules, chloride channel (ClC), sodium channel (NaC), and PKG, respectively
	S6. Model for 8‐Br‐cGMP permeation into the cytoplasm via plasma membrane (a) Bath‐applied extracellular 8‐Br‐cGMPs permeates through the plasma membrane into the cytoplasm with a fixed diffusion rate. Simultaneously, the level of the intracellular free 8‐Br‐cGMPs decreases as the uptake by bio‐molecular reactions increases. (b) Computed intracellular 8‐Br‐cGMP concentration using the permeation model in (a).
	S7. Comparison of model selection criteria (a) Akaike Information Criterion (AIC). (b) Bayesian Information Criterion (BIC). (c) Maximum log likelihood. When Maximum log likelihood is denoted by ܮ ,AIC = െ2ܮ ൅ 2݇, BIC = െ2ܮ݇ ൅ lnሺ݊ሻ, respectively, where ݇ is the number of core system parameters of the corresponding model and ݊ is the number of data (=16). Upper panels show matrix representation for each criterion and lower panels show corresponding histogram as in Fig. 3 a and b.
	S8. Maximum log likelihood of each core system training step during leave one out (LOO) procedure. Matrix representation of maximum log likelihood of each core system training step in LOO cross validation. Each panel shows the maximum log likelihood for one of the LOO dataset combination.
	S9. Normalized RMSE of model predictability of untrained 8‐Br‐cGMP‐induced MPTS (a) Matrix representation of normalized root mean squared error (RMSE) corresponding to Fig. 5c. (b) Same as a, but 5 M 8‐Br‐cGMP MPTS was predicted corresponding to Fig. 5d.
	Figure 1. Conversion of a biomolecular signal to an electrophysiological signal in a nerve growth cone. (a) Signaling pathways from an extracellular guidance cue, e.g. Sema3A, to membrane potential (MP) shifts in a growth cone. Sema3A, upon binding to its receptor on the plasma membrane (PM), increases the intracellular cGMP level. The effector proteins, Cyclic nucleotide-gated ion channels (CNGC) and cGMP-dependent protein kinases (PKG) that directly bind to cGMP (solid arrows) trigger, respectively, the activation of chloride channels (ClC) and sodium channels (NaC), which induce either hyperpolarization or depolarization. The potential cross interactions between the downstream effectors, which are unknown, are indicated by dashed blue lines. (b) Growth cone recording stimulated by a stimulant, 10 μM 8-Br-cGMP, a membrane permeable cGMP analogue (Top). Samples of recorded growth cone MP time series (MPTS; control: n = 7). (c) The MPTS contains over 10,000 data points (black). Outlier noises, such as spikes, are removed by sampling the data points at a 1 sec interval (green dots; see upper inset, enlargement of dashed square region; Supplementary Fig. 1). (d) The schematic diagram illustrates the mathematical model that functions as an alternative system and is designed in mesoscopic scale that expresses effective molecular signal flows. The control model contains the signaling pathway from the cGMP stimulation to MP shifts, via CNGC- and PKG-downstream factors (DFs), corresponding to the pathways in (a). The unknown signal flows between CNGC-DF, PKG-DF, ClC, and NaC in the core system are indicated within the dashed grey box. The parameters of the unknown flows and known factors, i.e., CNGC-DF to ClC and PKG-DF to NaC, are involved in the blue box and characterized by the core system parameter set, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}θ. The 8-Br-cGMP stimulation and MP, which are outside of the core system, are characterized by cell-dependent peripheral parameter set, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document}φ. We modeled the diversified data as a core system with peripheral elements representing cell-to-cell variability. Two pharmacological experimental conditions were also modeled, which represent complete inhibition of ClC, blocked by DNDS (n = 5) and NaC, blocked by STX (n = 4) (see Supplementary Fig. 2). A total of 16 datasets (control: n = 7; DNDS: n = 5; STX: n = 4) were analyzed.
	Figure 2. Model candidates and Bayesian system identification procedure. (a) The matrix shows the possible interactions in the gray region in Fig. 1d; pathways from CNGC-DF to PKG-DF, CNGC-DF to NaC, PKG-DF to ClC, and PKG-DF to CNGC-DF in the clockwise direction. Each of four pathways has three potential signals, i.e. activation (green), inhibition (red), or no interaction, as modeled by the core system parameters (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}θ). Thus, there are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${3}^{4}=81$$\end{document}34=81 model candidates labeled as M1 to M81. (b) Schematic illustration of prior distributions of model parameters that models the core system uncertainty and cell-dependent variability, expressing that extremely large values are biologically implausible and result in overfitting to the data (left; left-truncated Gaussians). Parameter values are applied to each of the models listed in a to generate model time series (TS) (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{V}(t)$$\end{document}Vˆ(t); blue line in the right). (c) Schematic illustration of model fitness, which depends on the parameters (left). The model fitness is the product of all the model likelihood (10 μM 8-Br-cGMP; control/DNDS/STX; n = 16), and the i-th likelihood with the parameter values in b was computed as the product of Gaussian probability density functions of the i-th MPTS (black; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(t)$$\end{document}V(t)) for the entire time points (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=1,\cdots ,{T}_{i}$$\end{document}t=1,⋯,Ti) with the mean \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{V}(t)$$\end{document}Vˆ(t) (blue) and the s.d. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma }_{i}$$\end{document}σi (right). (d) Illustrates the model plausibility (left; Bayesian evidence), which evaluates both model fitness and parameter plausibility. The model plausibility was computed by integrating the product of the model fitness and the parameter plausibility over the entire ranges of all the parameters (red area in the left). The logarithmic evidences for all the 81 models were displayed in the matrix corresponding to that in a with red-white color (right; higher evidence is colored by red). (e) The core system parameters were estimated from all the datasets (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta }_{all}$$\end{document}θall; n = 16) and cell-dependent peripheral parameters of the i-th cell (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi }_{i}$$\end{document}φi; n = 1) were from only the i-th cell’s MPTS, as posterior probability distributions. When we use representative specific parameter values instead of distributions during validation steps, we selected mean a posteriori (MAP) values (black circle in d) and denote the parameter set with superscript “MAP” like \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta }_{all}^{MAP}$$\end{document}θallMAP.
	Figure 3. Most plausible model and the generality test by leave-one-out cross validation. (a) Computed model evidences (logarithm evidences; Fig. 2d) are expressed as a matrix representation following Fig. 2a (10 μM 8-Br-cGMP; control/DNDS/STX; n = 16). A model with large log evidence (toward greater red) indicates greater plausibility than that with smaller log evidence (toward white). Models in left six columns are labeled as model group a; those in right three columns are labeled as model group b. Models indicated by M1 and M7 (insets) contain, respectively, the fewest and common interactions in groups a and b, representing minimal models within the respective model groups. (b) Histograms of log evidence in the model groups a and b, as shown in a. Vertical dashed lines indicate the log evidence of the two representative models, M1 and M7. (c) Schematic procedure of the generality test of the core system parameter estimations by applying the leave-one-out (LOO) data selection to the models. Black and gray horizontal bars represent experimental MPTS. When the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i$$\end{document}i-th MPTS is left out from the 10 μM 8-Br-cGMP datasets (n = 16), it is used to estimate peripheral parameters (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi }_{i}$$\end{document}φi) and validate model fitness (product of all the likelihood in white box), whereas the remaining MPTS (n = 15) are completely separated from the left-out MPTS to estimate the MAP core system parameters (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta }_{i}^{MAP}$$\end{document}θiMAP). Repeating LOO for all the datasets, 16 system parameter sets are obtained (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta }_{i}^{MAP};i=1,\,\cdots ,\,16$$\end{document}θiMAP;i=1,⋯,16). The model fitness to the left-out MPTS was taken the logarithm and averaged over all the data points. (d) Matrix representation of the averaged model fitness by LOO-derived parameter (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta }_{i}^{MAP}$$\end{document}θiMAP) (left) and the difference between the fitness and that computed from all the dataset (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta }_{all}^{MAP}$$\end{document}θallMAP) (right). (e) Distributions of the estimated MAP core system parameter values given by LOO in a (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta }_{i}^{MAP}$$\end{document}θiMAP; black point) compared with those by all the datasets (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta }_{all}^{MAP}$$\end{document}θallMAP; green circle; n = 16). The parameters exhibited in the horizontal axis and their values are represented in the vertical axis with relative representation with respect to the mean (actual values are listed in upper axis with the corresponding color). Most of the estimations are within the mean ± s.d. (gray region).
	Figure 4. Specificity of identified core system to 8-Br-cGMP-induced MPTS. (a) Schematic procedure of model specificity test using MPTSs in different conditions using MAP core system parameters estimated from all the 10 μM 8-Br-cGMP datasets (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta }_{all}^{MAP}$$\end{document}θallMAP; n = 16; black horizontal bars in the left side of the shaded band). The model likelihood to the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i$$\end{document}i-th MPTS from other testing datasets (white bars) was computed while computing the cell-dependent parameters, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi }_{i}$$\end{document}φi, and by repeating this the model fitness to all the n datasets were obtained. Testing datasets are the datasets untrained by the core system (5 μM 8-Br-cGMP-induced MPTS, PKG-inhibited condition (KT5823), and netrin-1-induced MPTS), those with noise (10 μM 8-Br-cGMP-induced MPTS with randomly labeled channel inhibitor). (b and c) Matrix representations of model fitness by the datasets derived by the control core system; 10 μM 8-Br-cGMP-induced MPTS with mixed labels of channel conditions (control, DNDS, STX) (b; n = 16) by randomly selecting channel blocker conditions in the model (see Supplementary Methods), and 5 μM 8-Br-cGMP-induced MPTS (c; total n = 11; control: n = 2; with DNDS blocking ClC: n = 5; with STX blocking NaC: n = 4). (d and e) Same as b and c, but the datasets derived by the different core systems; 10 μM 8-Br-cGMP-induced MPTS under the PKG-inhibited condition (KT5823) (d), and netrin-1-induced MPTS (e).
	Figure 5. Model predictability of untrained 8-Br-cGMP-induced MPTS. (a) Schematic illustration of the procedure for testing model predictability for the late-phase of MPTS induced by 10 μM 8-Br-cGMP. The initial-phase (250 sec) of the left-out \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i$$\end{document}i-th MPTS was extracted to estimate the MAP peripheral parameters (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi }_{i}$$\end{document}φi), and the remaining \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i$$\end{document}i-th Msas used to test the model predictability. The MAP core system parameters were estimated from the initial and late phase of the remaining MPTS (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta }_{i}^{MAP}$$\end{document}θiMAP; n = 15). (b) Sample MPTS predicted by model M1 (blue dotted line) and M7 (blue solid line) by the prediction procedure in a. (c) Matrix representation of log-likelihood of the predicted MPTS when repeating \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,\cdots ,16$$\end{document}i=1,⋯,16 and summing them. (d) Same as a, but 5 μM 8-Br-cGMP MPTS was predicted with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi }_{i}$$\end{document}φi given the initial-phase MPTS (up to 250 sec) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta }_{all}^{MAP}$$\end{document}θallMAP by all the 10 μM 8-Br-cGMP MPTS. Black and white bars represent experimental MPTS. MPTS in the white regions (after 250 sec, the late-phase) were never used for the parameter estimation; these MPTSs were completely untrained data sets even for the peripheral parameters. (e) Sample MPTS predicted by model M1 (blue dotted line) and M7 (blue solid line) by the prediction procedure in d. (f) Matrix representation of log-likelihood of the predicted MPTS when repeating \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,\cdots ,11$$\end{document}i=1,⋯,11 and summing them.
	Figure 6. Reproducibility of cGMP-dependent MP shifts. (a) Model MPTS by the model M7 induced by different 8-Br-cGMP concentration: (i) 0.5, (ii) 6, and (iii) 20 μM with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta }_{all}^{MAP}$$\end{document}θallMAP and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi }_{all}^{MAP}$$\end{document}φallMAP (Supplementary Table 1). Each MPTS reaches the steady-state at about 300 sec. We defined the model’s steady-state MP shift as the difference between the MPs at times zero and infinity, namely, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{V}(t\to \infty )-\hat{V}(t=0)$$\end{document}Vˆ(t→∞)−Vˆ(t=0), where the core system parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\theta }_{all}^{MAP})$$\end{document}(θallMAP) from the 10 μM 8-Br-cGMP dataset were used. For the peripheral parameters, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V}_{{\rm{K}}}$$\end{document}VK is a mere constant bias in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{V}(t)$$\end{document}Vˆ(t) (see Supplementary Methods) and, therefore, was canceled out, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau }_{S}$$\end{document}τS was ignorable at times zero and infinity, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${A}_{Z}$$\end{document}AZ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${A}_{W}$$\end{document}AW were replaced by their means in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi }_{all}^{MAP}$$\end{document}φallMAP. (b) Model of 8-Br-cGMP concentration dependent MP shifts. Prediction of the cGMP-dependency of the steady-state MP shifts induced by the direct pipette stimulation (inject 8-Br-cGMP through the recording pipette; blue dashed line). Mimicking of the bath application condition of 8-Br-cGMP (blue solid line) by correcting the 8-Br-cGMP permeation process from outside to inside of the cell (Supplementary Methods). Experimental data of steady-state MP shifts under the bath application condition (black square with lower x-axis; ref.16) are shown for comparison. Normal condition indicates a cGMP level when Sema3A was applied in the turning assay; ODQ (1H-(1,2,4) oxadiazolo(4,3-a)quinoxalin-1-one, a soluble guanylyl cyclase inhibitor) condition reduces growth-cone cGMP levels.

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