Fig 1.
Problem description and modular implementation of SYSMOLE.
(A) Example of classic voltage clamp experimental traces, which can be used as an input (red) and output (black) trace pair for the SYStems-based MOLecular kinetic scheme Extractor (SYSMOLE) method. In voltage-clamp (VC) experiments to study ion-channel currents, one would define an input-output system in which the scalar input signal u(t) is the depolarizing voltage step (top), and the scalar output signal y(t) is the actual current trace elicited by the depolarization (bottom). (B) Example of two molecular kinetic schemes describing ion channel dynamics in response to a voltage step that yield similar macroscopic traces. C indicates the closed-channel state, O the open-channel state, I the inactivated-channel state, CI the closed-inactivated-channel state, and OI the open-inactivated-channel state. (C) Modular organization of SYSMOLE and workflow of the method. Briefly, the Identifier Module uses the input u(t) and output y(t) traces to obtain the transfer function G(s), which can be characterized by the time constants of its poles (τpoles) and zeros (τzeros), and a gain. The Classifier Module finds the configuration or block diagram associated with the transfer function G(s). The Molecular Kinetic Converter Module uses the block diagram together with the transfer function G(s) to derive the molecular kinetic scheme.
Fig 2.
Second-order canonical configurations.
(A) Example of two first-order processes, a and b, described by first-order transfer functions Ga(s) and Gb(s) with parameters τa = 5 ms and ka = -1, and τb = 100 ms and kb = 1, and their responses in time (outputs ya(t) and yb(t)) to an ideal step input at t = 30 ms (u(t)). (B) Block diagram (left) and corresponding output y(t) (right) in response to a step input u(t) of the first-order processes Ga(s) and Gb(s) combined following the three canonical configurations: cascade, feedback and parallel.
Fig 3.
Classifier Module implementation.
Classifier Module flow chart implementation of the comparison between G(s), the transfer function obtained by the Identifier Module, and the transfer functions associated with the canonical configurations. This module tests sequentially whether G(s) incorporates the features of a transfer function describing a higher-order system, a first-order system, or a second-order system in each of the canonical configurations: cascade, feedback or parallel. The cost function values for the parallel and feedback optimization problems, fvalp and fvalf respectively, are used to discern between these two canonical configurations as explained in the Results section.
Fig 4.
Molecular Kinetic Converter Module implementation.
Molecular kinetic schemes derived from block diagrams for first-order systems (A), second-order systems in cascade (B), second-order systems in feedback (C), second-order in parallel addition (D), and second-order systems in parallel subtraction (E) configurations. Red circles indicate the observable state for each configuration. Black and blue circles depict non-observable states. The supporting material (S1 Text) provides, based on the steps described in the main text, a complete derivation of the molecular kinetic scheme corresponding to each configuration. It also includes, for each configuration, the equations that describe the transition rates between states (σk) and the observable proportionality constant (γ) as a function of the pole and zero time constants (τpole1, τpole2, and τzero), and the gain of the system.
Fig 5.
Similar traces from different molecular kinetic schemes.
Application of SYSMOLE to two similar synthetic traces in response to an ideal step at t = 30 ms. The top trace (A) was generated with the Synthetic Trace Simulator (see Results section) by combining in parallel two processes with parameters ka = - 5, kb = 3, τa = 5, ms and τb = 100 ms. The correct molecular kinetic scheme was derived by SYSMOLE and the kinetic parameters obtained. The bottom trace (B) was also generated with the Synthetic Trace Simulator by combining in feedback two processes with parameters ka = - 5, kb = 2, τa = 5 ms and τb = 200 ms. The correct molecular kinetic scheme was derived by SYSMOLE and the kinetic parameters obtained. Boundary conditions used in the Classifier Module are τa ∈ [1, 9] ms, τb ∈ [50, 250] ms ka ∈ [−20, 20], and kb ∈ [−20, 20] for the parallel problem, and kb ∈ [0, 20] for the feedback problem since combinations in feedback with kb < 0 are unstable.
Fig 6.
Effect of noise on kinetic parameters and probability of error in classification.
(A) Representative traces of output signal y(t) used to test the effects of noise. Traces were generated with the Synthetic Trace Module (see Results section) combining two first-order processes with parameters ka = - 5, kb = 3, τa = 5 ms and τb = 100 ms in parallel with different levels of white Gaussian noise added. (B) Relative error in transition-rate kinetic parameters (σ1, σ2, and σ3) and probability of error in detection for parallel and feedback configurations. Each point in the graph depicts the mean and standard deviation of 100 simulations. Boundary conditions used in the Classifier Module are τa ∈ [1, 9] ms, τb ∈ [50, 250] ms, ka ∈ [−20, 20], and kb ∈ [−20, 20] for the parallel problem, and kb ∈ [0, 20] for the feedback problem since combinations in feedback with kb < 0 are unstable.
Fig 7.
Molecular kinetic scheme and effects of nifedipine on L-type calcium channel traces.
(A) Representative L-type calcium ion-channel current (output) elicited by voltage steps from -90 to 0 mV (input), first in the absence of nifedipine, and after addition of nifedipine at 100 nM and 1 μM concentrations. (B) Molecular kinetic scheme obtained by applying SYSMOLE to the traces and expected effects of nifedipine on transition rates. (C) Fractional occupancies as a function of time of the closed-channel (black), open-channel (red), and inactivated-channel (blue) states resulting from simulating the molecular kinetic scheme. Simulations were generated with the mean value of the kinetic parameters (N = 3–6 traces per treatment). Kinetic parameter values can be found on Table 1.
Table 1.
Kinetic Parameters Ion-channel Traces.
Fig 8.
Molecular kinetic scheme of crosstalk signaling in the mGluR2/5-HT2AR heteromeric complex in response to glutamate antipsychotics.
(A) (Top) Scheme of crosstalk signaling through the mGluR2/5-HT2AR complex. In response to the endogenous ligands glutamate and serotonin, the mGluR2/5-HT2AR complex signals through Gi and Gq signaling pathways respectively. Certain ligands such as LY379268, a dominant mGluR2 agonist with antipsychotic properties, are able to crosstalk through the complex and activate both Gi and Gq. (Bottom) Molecular kinetic schemes associated with the cis-activation and the trans-activation theories of crosstalk. (B) (Left) Representative G-protein sensitive ion-channel (GIRK4*) current trace (output) elicited by a step in LY379268 concentration from 0 to 1 μM (input). (Right) Molecular kinetic scheme obtained by applying SYSMOLE to the traces. (C) Fractional occupancies as a function of time of the four states in the molecular kinetic scheme: Gi OFF / Gq OFF, Gi ON / Gq OFF, Gi ON / Gq ON, and Gi OFF / Gq ON. The molecular kinetic scheme postulates a new state Gi OFF / Gq ON and supports the trans-activation theory for crosstalk through the complex (see Results section). Simulations were generated with the mean value of the kinetic parameters (N = 5 traces). Kinetic parameter values can be found on Table 2.
Table 2.
Kinetic Parameters mGluR2/5-HT2AR Complex Traces.