Fig 1.
Multiple mechanisms for ideal symport.
The four “ideal” kinetic pathways of a hypothetical symporter that transports substrate using the available free energy of the driving ion are shown. This state-space contains eight states, and symport models include at least six connecting transitions.
Fig 2.
Exploration of model space using Markov-chain Monte Carlo.
The plot shows a ModelExplorer trajectory based on a symporter energy function during a 1e6 MC step simulation. Note that each point represents a different fully specified model. Energy minima correspond to models that are more fit, using EMC = −Jsubstrate as a fitness function in this case, where Jsubstrate is the flux of substrate. Models are initially at a high energy but quickly find local minima. A tempering schedule (see S1 Text) of alternating temperature increases and decreases prevents the simulation from being trapped in local minima.
Fig 3.
Ideal symport in a non-ideal (mixed) model.
The fluxes, J, of the driving ion and substrate of an example symporter model are plotted over a range of ion chemical potential differences. Note the 1:1 stoichiometry of the substrate to ion flux, indicating an ideal symporter with no leaks. The ion and substrate flux have been scaled by the maximum ion flux for visual clarity. Inset: kinetic pathway of the same symporter model at an ion chemical potential difference of −4kBT, as indicated by the vertical line.
Fig 4.
Dissection of a single model exhibiting enhanced selectivity into component pathways.
The full model is shown in (A) with the net probability flows scaled by the largest edge flow, while panels (B)—(D) are various continuous cycles abstracted from the full model. (B) An ion leak pathway in which the substrate and ion bind extracellularly, but only the ion is transported into the cell because the substrate unbinds on the extracellular side. (C) A (split) cotransport pathway in which the substrate and ion both are transported into the cell. (D) A second ion leak pathway, mirroring (B), in which the decoy and ion bind extracellularly, but only the ion is transported into the cell. Overall, the substrate and decoy are both driven to unbind in the outward-facing conformation, shown on the left, due to ion leak pathways. However, due to the difference in binding affinities between decoy and substrate, the substrate is more likely to rebind and be transported into the intracellular region. The full process employs the ion leaks to enhance selectivity.
Fig 5.
Enhanced discrimination driven by an ion leak.
(A) For the model of Fig 4, the substrate, ion, and decoy flux are shown for a varying chemical potential difference of the driving ion. Note the negligible decoy flux relative to the substrate flux near −4kBT. Inset is the kinetic pathway diagram of the model at a specific chemical potential difference (−4kBT, vertical line) of the ion. (B) The same discriminative model as in (A), but with the energy barrier between the ion-only bound states in the inward and outward conformations raised by 100kBT, effectively shutting off the ion leak. Both the substrate and decoy fluxes increase. Inset is the kinetic pathway diagram of the model with the leak removed, resulting in two symmetrical pathways for substrate and decoy transport. (C) Comparing the ratio of substrate to decoy flux (selectivity) for the same model with and without an ion leak. With the ion leak, the selectivity approaches infinity due to the negligible decoy flux. In contrast, removing the ion limits the selectivity to the expected equilibrium-like value of eΔΔG=1. Note that the sign change of the selectivity is due to the change in substrate and decoy flux direction.
Fig 6.
Pathway meta-analysis: Clustering analysis of the best-performing models based on model similarity.
The dendrogram was truncated to four levels for visual clarity, with the number of models below the truncation shown in parenthesis. Dendrogram created using Python/SciPy and Matplotlib.
Fig 7.
Kinetic pathways of the four model classes (A-D) found from clustering analysis (Fig 6).
Each of these models exhibits a high level of selectivity due to an ion leak, as discussed in the text. Note that the net flow values shown along edges are scaled by the maximum flow edge of the individual model.