A Mathematical Model of Lysosomal Ion Homeostasis Points to Differential Effects of Cl− Transport in Ca2+ Dynamics

The establishment and maintenance of ion gradients between the interior of lysosomes and the cytosol are crucial for numerous cellular and organismal functions. Numerous ion transport proteins ensure the required variation in luminal concentrations of the different ions along the endocytic pathway to fit the needs of the organelles. Failures in keeping proper ion homeostasis have pathological consequences. Accordingly, several human diseases are caused by the dysfunction of ion transporters. These include osteopetrosis, caused by the dysfunction of Cl−/H+ exchange by the lysosomal transporter ClC-7. To better understand how chloride transport affects lysosomal ion homeostasis and how its disruption impinges on lysosomal function, we developed a mathematical model of lysosomal ion homeostasis including Ca2+ dynamics. The model recapitulates known biophysical properties of ClC-7 and enables the investigation of its differential activation kinetics on lysosomal ion homeostasis. We show that normal functioning of ClC-7 supports the acidification process, is associated with increased luminal concentrations of sodium, potassium, and chloride, and leads to a higher Ca2+ uptake and release. Our model highlights the role of ClC-7 in lysosomal acidification and shows the existence of differential Ca2+ dynamics upon perturbations of Cl−/H+ exchange and its activation kinetics, with possible pathological consequences.


Mathematical model of lysosomal ion homeostasis
The equations written in purple are newly developed, the other elements were retrieved from a previous mathematical model as indicated above. All parameters and variables are listed in Tables S1 and S2, respectively.

Membrane potential
Our model includes a description of the membrane potential [1,2], which depends on the concentration of ions within the lysosomal lumen: By convention, the membrane potential (Δψ) is negative if the number of anions inside the lysosome (luminal) is higher relative to the outside (cytosol). The total membrane potential (ΔψT) accounts for the intrinsic charge on the outer (ψout) and inner leaflets (ψin) of the lysosomal membrane [2]: For each simulation, the initial value for the concentration of Donnan particles (B) was adjusted to set a null initial total membrane potential (ΔψT = 0).

Modified ion concentrations
To account for the effects of leaflet potential, the cytoplasmic and luminal concentrations were modified by a Boltzmann factor, leading to cytoplasmic and luminal surface concentrations: ) (S12) ) (S13) The subscripts i and e indicate internal (luminal) and external (cytosolic), respectively. R is the gas constant and is the Faraday's constant. At room temperature (T = 25 ⁰C), R . T/F = 25.69 mV. This value was used in all the simulations.

Number of ions
The number of luminal ions is calculated based on the concentration, the lysosome volume (V), and the Avogadro's number (NA): rCa 2+ is the ratio of total to free calcium. As the cytoplasmic values of these ions change very little during acidification, the cytosolic concentrations were considered to be constant.

Rate of change of the ions within the lysosomal lumen
The rate of change of lysosomal pH (pHL) is determined by the change of luminal proton concentration and the buffering capacity of the lumen (): In the equation above, the change in the number of luminal protons is determined by JVATP is the proton pumping rate of the V-ATPase pump (positive for proton influx), JClC-7 WT and JClC-7 fast are the turnover rates (positive for proton efflux) for ClC-7 WT and ClC-7 fast , respectively. JCAX is the turnover rate of a calcium/proton exchanger CAX (positive for proton efflux), and JH + is the passive, non-voltage activated proton flux through channel (positive for proton influx). The stoichiometries of ClC-7 and CAX for proton counter-transport are specified by + −7 and + , respectively.
The rate of change of luminal chloride ions is described by The number of potassium ions within the lysosomal lumen varies due to their passive flow across the lysosomal membrane: JK + is the turnover rate of the non-voltage activated potassium channel (positive for potassium influx).
Similarly, the rate of change in luminal sodium ions is determined by JNa + is the turnover rate of the sodium channel (positive for sodium influx).
The change in total calcium ions is described as JCAX is the turnover rate of CAX (positive for calcium influx), 2+ is the CAX stoichiometry for calcium, JCa 2+ is the passive flow through a calcium channel (positive for calcium influx), and JTRPML1 is the voltageand pH-dependent flow through TRPML1 channel (positive for calcium influx).
Due to calcium buffering, the rate in luminal free calcium within the lumen is determined by the variation in total luminal calcium and the ratio (rCa 2+ ) of total to free calcium:

Turnover rates
The pumping rate of the V-ATPase (JVATP) is given by a detailed mechanochemical model, which was calibrated against experimental data for current voltage [3]. JVATP which depends on the luminal pH (pHL) and on the membrane potential (Δψ): NVATP is the number of V-ATPase pumps located in the lysosomal membrane, and JVATP1 is the proton pumping rate of a single V-ATPase under different membrane potentials and pH gradients. The proton pumping profile was generated using the model of Grabe et al. [3] with values for the membrane potential The model assigns boundary values of pumping rate if the input value for the membrane potential is lower than -200 mV or higher than 500 mV.
The equation for the ClC-7 turnover rate formulated by Ishida et al. [2] is time-independent and represents an instantaneous (de)activation of the antiporter. Therefore, we used the same mathematical description to represent the ClC-7 fast turnover rate, but modified the equation to have an explicit term for the ClC-7 activity (A). Thus, the ClC-7 fast turnover rate is given by NClC-7 is the number of ClC-7 antiporters, ΔµClC-7 is the turnover driving force [2]: and A is the activity The switching function x [2] varies between zero at negative membrane potentials (Δψ) and 1 at positive Δψ: In the equations above, We described the ClC-7 WT turnover rate as a function of the effective activity Aeff : NCLC-7 is the number of ClC-7 antiporters, ΔµClC-7 is the driving force (Equation S30), and Aeff is defined as: with A the ClC-7 activity (Equation S31). In the equation above, if A is higher (lower) than Aeff, then Aeff increases (decreases) according to the activation (deactivation) time τ = τact (τ = τdeact) until it reaches the activity A. We define "activity" as a variable related to an open probability, which determines the (de)activation kinetics of the ClC-7 turnover rate. The effective activity Aeff reaches the activity A after a certain amount of time determined by the activation (τact) or deactivation time (τdeact). For simplicity, we considered the deactivation time τdeact to be proportional to the activation time τact: With rτ the deactivation to activation ratio.
The uncoupled transport of chloride was simulated as a passive chloride flux through a "channel-like" ClC-7 antiporter. Therefore, we describe the ClC-7 unc turnover rate as PClis the permeability per unit area for chloride ions, S is the lysosome surface area, NA is the Avogadro's The turnover rate for the ClC-7 knockout (JClC-7 ko ) was calculated with Equation S33 setting NClC-7 = 0.
The turnover rate for CAX is described by with the driving force ΔµCAX defined as The previous equation was created based on the driving force for ClC-7 (Equation S30).

Passive, non-voltage activated ion fluxes
The proton flux through the channel is described by [2] With PH+ the permeability per unit area for protons, and U = (Δψ . F)/(R . T) the reduced membrane potential.
Similarly, the passive flows for potassium, sodium and calcium ions through their corresponding channels are described by Equation S40, S41, and S42, respectively.

Voltage and pH activated calcium flux
The calcium flux through the TRPML1-like channel was described as where the permeability ( 1 ) depends on the luminal pH and on the membrane potential [4,5]: The function goes from 1 at Δψ < -40 mV, to 0 at Δψ ≥ -40 mV. Hence, for membrane potentials lower than -40 mV, the permeability of the TRPML1 channel is directly proportional to the membrane potential and does not depend on the luminal pH [4]. For membrane potentials higher than -40 mV, the

Conditions for the (de)activation of the ClC-7 antiporter and differentiation between fast and WT scenarios
This section provides a detailed description of the conditions needed for the (de)activation of the ClC-7 antiporter, according to our mathematical model. We provide an explanation for the equivalent behaviour of the ClC-7 fast and ClC-7 WT antiporter as depicted in Figures 3, 4, and 6, and for the differential behaviour between these two scenarios observed in Figure 7, and Figure S3. As the fast scenario mimics a ClC-7 antiporter with instantaneous (de)activation kinetics, the activation A is instantaneously achieved, and therefore is directly used for the computation of the turnover rate of the fast ClC-7 (JClC-7 fast , Equation S29).
The relationship between the activity A and the driving force ΔµClC-7 was calculated from Equation S31 and S32 by varying the value of the driving force from -500 mV to 500 mV ( Figure S1). The value of the activity is set to 0.3 for values higher than -155 mV. As the activity does not change for driving forces between -155 mV and +∞, an activation or deactivation of the ClC-7 will not occur in this range. Under these conditions, ClC-7 fast and ClC-7 WT display the same turnover rate (JClC-7), i.e. the same behaviour. The initial effective activity was set to Aeff,0 = 0.3 (Table S1), equivalent to the value of activity A for an initial null driving force.
Hence, differences between ClC-7 fast and ClC-7 WT are observed only for driving forces lower than -155 mV, i.e. in a domain in which the value of the activity is not constant, inducing a change in the effective activity from its initial value. In this scenario, a slow (for ClC-7 WT ) or instantaneous (for ClC-7 fast ) activation would be induced.
In the simulations of Figures 3, 4, and 6 in the main text, the driving forces of ClC-7 fast and ClC-7 WT antiporters did not reach values lower than -155 mV (Figure S1b, c, and d, respectively). Therefore, the    n.a n.a n.a n.a 0 0 0 0

Sensitivity analysis
We investigated the robustness of our model by varying every input parameter by ±10%. The reference scenario was simulated with the initial conditions specified in Table S1. Every input parameter of the model listed in Table S1 was varied in ±10%. Therefore, 66 test scenarios were simulated. For each test scenario we analysed disturbances on the steady-state output values of variables, which are shown in Table   S4. We calculated the relative difference between the output value obtained from the test simulation ( test ) and from the reference simulation ( ref ): The resulting relative differences are shown in Figure S8.  Figure S1. The (de)activation of the ClC-7 antiporter is determined by its driving force. (a) Activity as a function of the driving force. Temporal evolution of the driving force during the simulations shown in (b) Figure 3, (c) Figure 4, (d) Figure 6, (e) Figure 7, and (f) Figure S3.  (Table S3). From t = 0 s, the lysosomal membrane was permeable to calcium ions ( 2+ = 8.9 × 10 −5 cm/s) and impermeable to sodium and potassium ions ( + = + = 0).  Table S3) and the cytosolic calcium concentration was set to 100 nM. From t = 0 s, the lysosomal membrane was permeable to calcium ions ( 2+ = 1.5 × 10 −7 cm/s), and 10 CAX with 3H + :1Ca 2+ stoichiometry were turned on.  Table S3) and the cytosolic calcium concentration was set to 100 nM. From t = 0 s, the lysosomal membrane was permeable to calcium ions ( 2+ = 2 × 10 −7 cm/s), and 20 CAX with 2H + :1Ca 2+ stoichiometry were turned on. Depicted for the different ClC-7 scenarios during triggered calcium uptake (ClC-7 WT , dashed black line; ClC-7 fast , red; ClC-7 unc , blue; ClC-7 ko , green) are luminal free calcium concentration (b), turnover rate of CAX (c), calcium flux (d), luminal pH (e), total membrane potential (f), luminal concentration of potassium (g), sodium (h), and chloride ions (i), as well as the turnover rates of ClC-7 WT and ClC-7 fast (j), and ClC-7 unc (k). The initial conditions were set to the steady-state values of Figure 4 (i.e., after lysosomal calcium release, Table S3) and the cytosolic calcium concentration was set to 100 nM. From t = 0 s, the lysosomal membrane was permeable to calcium ions ( 2+ = 1.4 × 10 −7 cm/s), and 30 CAX with 1H + :1Ca 2+ stoichiometry were turned on.  Table S3) and from t = 0 s, the lysosomal membrane was permeable to calcium ions (PCa 2+ = 5.7 x 10 -4 cm/s) representing the opening of the uptake channel.  Table  S1 was varied in (a) -10% (b) +10%. The colour maps show the relative difference (Equation S43) between the steady state value of each variable on the vertical axis under a variation in the input parameter of the horizontal axis and the steady state output for the reference scenario (Table S4).