Fig 1.
Schematic representation of vesicular acidification model.
(A) Cartoon representation of the model with all elements and reactions explicitly shown. (B) Chemical equations and rate constants for each model component.
Table 1.
Model parameters.
Fig 2.
Discrete, stochastic simulations of a small vesicle.
(A) The instantaneous pH from the DS model simulation (orange) plotted over an experimental trace obtained from a 58 nm radius vesicle (black). The same experimental trace is shown in all three panels, and the times when ATP are added and the pump stops are indicated with arrows. (B) Time averaged DS model results using a 0.5 s window to produce (orange). (C) pH determined from fluorescent reporter (pHhν) calculated from the modeled photon count emitted from the pHrodo dye (orange). The inset shows the corresponding experimental (black) and simulated (orange) photon count, Nhν, over time.
Fig 3.
Liposome acidification dynamics using the ODE and DS models.
ODE model fits along with representative simulations from the DS model with pH interpreted from free proton counts or photon counts (indicated by hν) for the four experimental traces in Table 2. Each row is a different experimental trace plotted in black, and all simulations are orange. The first, second, and third columns contain the ODE, DS model, and DS model using photon count results, respectively. In the last column, the calculated pH is pHhν. All parameter values can be found in Table 2.
Table 2.
Parameter estimates for the DS and ODE models.
Fig 4.
ODE model fit is average of many DS simulations.
The ODE and DS model fits for Trace 2 from Table 2, r = 145 nm, is displayed. An average of 10 and 100 DS simulations with the DS method predicted parameter values is displayed as well. As the number of averaged DS simulation increases, the ODE model fit is approached.
Fig 5.
Effects of vesicle radius and steady state pH on H+ and pH fluctuations as predicted from our DS model.
(A&B) The standard deviation in the free proton count of DS model simulations (σ(NH)) as a function of vesicle radius with the lumenal/bath pH set to 5.5 (panel A) or as as a function of lumenal/bath pH with the radius fixed at 150 nm (panel B). (C) The instantaneous free proton count over time and corresponding count histogram for the common DS model simulation to panels A-D (diamonds) with r = 150 nm and pH = 5.5. (D&E) The standard deviation in pH of DS model simulations (σ(pH)) as a function of vesicle radius with the lumenal/bath pH set to 5.5 (panel D) or as as a function of lumenal/bath pH with the radius fixed at 150 nm (panel E). Dashed lines in A, B, D, and E scale as expected based on free proton count and pH scaling with volume and pH as described in the main text. (F) Standard deviation in steady state pH as a function of lumenal pH for vesicles undergoing active proton pumping. The proton pump rate was adjusted to achieve different internal pH values, each curve corresponds to a different size vesicle (r = 100, 150, 200, and 250 nm). The inset is a histogram of free proton counts for the r = 150 nm, pH = 5.45 data point (diamond). For all panels, the simulations of the DS model were run for 750 s at equilibrium with the bath pH, with no active pumping (except panel F), and PH = 4.6 × 10−5 cm/s. All other parameters can be found in Table 1.
Fig 6.
Vesicle pH fluctuations are constant over a range of pH values.
(A) DS simulation of a lysosome sized vesicle (r = 340 nm), with constant passive membrane permeability PH = 4.6 × 10−5 cm/s, and different numbers of proton pumps N = 40, 160, and 640. The pumps stochastically cycle between active and inactive pumping states with a mean dwell time of 273 s in each state and a constant pumping rate of IP = 100 H+/s when active. The inset shows the total active pump population in time. In each simulation, the extracellular pH and the starting initial pH are 6.0. (B) Distribution of pH values at steady state for the three simulations in panel A plus three more simulations for N = 20, 80, and 320. A normal curve with mean and standard deviation equal to that of each distribution is plotted overtop of each histogram. The pH was computed from the instantaneous proton count generated by the DS model in all panels. (C) Distribution of steady state pH values for a populations of vesicles, as in panel A, with pump numbers pulled from a Poission distribution with mean pump number N = 640. The five colors, right to left, correspond to the pH values produced by vesicles with the 5th (red), 25th (yellow), 50th (green), 75th (cyan), and 95th (blue) percentile of the distribution, scaled by their respective probability.