Potassium Starvation in Yeast: Mechanisms of Homeostasis Revealed by Mathematical Modeling

The intrinsic ability of cells to adapt to a wide range of environmental conditions is a fundamental process required for survival. Potassium is the most abundant cation in living cells and is required for essential cellular processes, including the regulation of cell volume, pH and protein synthesis. Yeast cells can grow from low micromolar to molar potassium concentrations and utilize sophisticated control mechanisms to keep the internal potassium concentration in a viable range. We developed a mathematical model for Saccharomyces cerevisiae to explore the complex interplay between biophysical forces and molecular regulation facilitating potassium homeostasis. By using a novel inference method (“the reverse tracking algorithm”) we predicted and then verified experimentally that the main regulators under conditions of potassium starvation are proton fluxes responding to changes of potassium concentrations. In contrast to the prevailing view, we show that regulation of the main potassium transport systems (Trk1,2 and Nha1) in the plasma membrane is not sufficient to achieve homeostasis.


Mathematical Model
The model describes ion transporters and channels as resistors, the membrane as a capacitor and the proton pump as a battery in an electric circuit [1][2][3][4][5]. Transport of ions across the plasma membrane changes their respective intracellular concentrations, the volume and the membrane voltage. Apart from the H + -ATPase Pma1, the transport systems were modeled according to Ohm's law using either constant or voltage dependent conductivities. In the latter case, the conductivity as a function of voltage is given by the maximum conductivity times the probability for the transporter to be active. This probability can be derived from a two state voltage gated model for the activity of the transporter [1].
The core model comprises the transport systems Trk1,2, Pma1, Nha1, Tok1 as well as potassium and proton leakage currents. We used information from the literature to model the voltage and concentration dependent kinetics of these transport systems. For Trk1 and Trk2 we used electro-physiological measurements [6] to fit a two state model as described above; see Section 2.6. Both proteins were subsumed as one transport system since there is no data for the individual transporters. In addition, the low affinity transporter Trk2 is considered to be less relevant. For Tok1 we used a model from the literature [7]. For the potassium leak we assumed a constant conductivity and used published data [6] (Figure 9c). Measurements in Neurospora crassa were used for the constant conductivity of the proton leak [8]. The proton pump Pma1 was described by a simple thermodynamic approach [9] using values for the maximum pump current from electro-physiological measurements [10].
For the antiporter Nha1 there is currently no electro-physiological data available. We decided to describe Nha1 as a voltage gated channel, see Section 2.4 for details. The membrane potential is modeled according to a capacitor equation, which satisfies the requirements of the cell's electro-neutrality. For the dynamics of the cell volume we used a model from the literature [11] which is based on the balance of internal (Π i ), external (Π o ) and turgor pressure (Π t ).  • I K the subscript denotes the ion related to the current (or flux)

Units, conventions and variables
• I T rk1,2 the superscript denotes the transport system related to current (or flux) • E equilibrium potential of a transporter (Nernst potential in case of a uniport) • E K the subscript indicate the respecting ion for the equilibrium potential • z valence of a respecting ion, unless otherwise stated z = 1 4π r 2 = r 3 ≈ 7.5 × 10 −5 cm, for a spherical cell with r = 2.28 × 10 −4 cm and a volume of 50 fL A , initial surface to actual surface ratio Sign convention

Kinetic model of bicarbonate reaction scheme
Transport, conversion and production of CO 2 was described by a kinetic model [12]. The bicarbonate reaction is catalyzed by carbonic anhydrase. Carbon dioxide is expected to turn quickly into carbonic acid (pK A = 6.35), which dissociates into bicarbonate and protons. Since CO 2 is either present in carbonic acid or hydrogen carbonate the concentration of CO 2 can be written as The dynamics of CO 2 is governed by where we have defined the reaction rates The production terms J prod.
are determined by J prod. CO2 , which is an input to the model and describes the metabolic production of CO 2 . This time dependent input function was inferred in the reverse tracking approach, see Section 1.4.
Collecting these terms, the dynamics of the internal pH reads

Complete model equations
The full model reads The membrane potential is expressed by the charge balance equation [13] with Faradays constant F , specific membrane capacitance c m , cell surface A and [Z] sums up all fixed charges and is defined by the choice of [K + ] i (0), pH i (0), V (0), [CO 2 ] i (0) and V m (0). (23)

H2CO3
(24) The production rate J prod. CO2 is an input to the model, see Sections 1.2 and 1.4. Note that the production changes with the volume even if the amount of produced CO 2 does not change. This is taken into account by the relationship where V (0) denotes the initial volume. Similarly, the factor S = A(0) A(t) was introduced to correctly cover the surface dependence of the conductance parameters.
The equilibrium potentials are given as The starvation process (reduction of external KCl) is modeled as where µ = 0 and σ = 0.04 × 60 2 s.

Reverse Tracking Approach
The model equations in Section 1.3 are thermodynamically consistent but do not account for any regulatory events (e.g. gene regulation or signal transduction) which might modulate the activity of the transport channels and other components of the system. We observed that extensive parameter scans did not yield a satisfactory fit to the observed time course for intracellular potassium (see Figure 1A in the main text and Section 2). On the one hand this indicates that the model is not "overfitted" but on the other hand it shows that important aspects of the homeostatic response are not covered by the model. Potassium homeostasis involves a variety of cellular and metabolic processes. Exhaustive modeling of all these processes is currently impossible due to lack of detailed knowledge and lack of experimental data. Instead we decided to treat parameters of the models as unknown input functions which are modulated by these unmodeled processes. We used the parameters determining the activity of the transport systems and the carbon dioxide production rate as potentially altered by unmodeled external processes. Instead of estimating a number for this parameter we tried to infer a time course for this input function ("time dependent parameter") by minimizing an objective function which measures the difference between the data and the simulation. In particular, we solved the minimization problem min p(t) Here, , θ) denote the experimental and the simulated potassium concentration respectively. The latter depends on the time dependent parameter p(t) which is tested and further constant parameters θ. Each transport system or component in the model for which we were able to find a p(t) with a satisfactory minimum were considered as potential actuators of potassium homeostasis.
The numerical problem to estimate a time course, which tracks the simulated potassium along a given path can be solved by the implementation of an I-controller with an appropriate amplification factor η. It was tentatively chosen in each individual case to reduce oscillations possibly produced by the I-controller. The specific reverse tracking approaches for carbon dioxide production and the regulation of transport systems are given below.

Carbon dioxide production
For the carbon dioxide production rate we used d dtJ prod.
where the volume dependent production rate was scaled by the initial volume to the actual volume ratio.

Transport systems
In case of the transport systems we used p(t) as a kind of regulation function for a constant conductivity parameterg: where g(t) gives the time dependent conductance parameter.

Trk1,2 is not a potential actuator
A priory we expected the main potassium uptake systems Trk1,2 to be potential actuators of potassium homeostasis.
For the reverse tracking approach, three different cases were analyzed: • Trk1,2 as a K + uniport (model standard) • Trk1,2 as a 2K + symport • Trk1,2 as a 1H + /1K + symport The respective parameter values for the tested approaches are given in Supplementary Table 2.7.
The results for each case were very similar (see Supplementary Figure 1a). The loss of external potassium (traces left were 15 µM ) at time point zero induces a tremendous increase (magnitude 10 4 ) of the conductance. However, the increased uptake rate can not compensate for the rapid loss of internal potassium ( Supplementary Figure 1 b). Surprisingly, the system is not very sensitive to changes in g T rk1,2 . The incapability of tracking the potassium data led to the exclusion of Trk1,2 as a potential actuator of potassium homeostasis.
As an explanation of this finding, we suggest, that the micro-molar concentration of potassium in the medium is not enough to ensure the potassium supply. Literally spoken: opening the doors is of no use, if nothing is outside.

Nha1 is not a potential actuator
For Nha1 four different possibilities were tested with the reverse tracking approach: • Nha1 as a 2H + :1K + anti-porter described by Ohm's law with a voltage gating (model standard) • Nha1 as a 2H + :1K + anti-porter described by Ohm's law without gating • Nha1 as a 2H + :1K + anti-porter described due to Ohm's law with a potassium gating applied • Nha1 as a 1H + :1K + anti-porter.
The potassium gating was modeled as a potassium dependent activation term which is multiplied by the constant maximum conductivity g N ha1 . The parameter q N ha1 determines the half maximal activation. The values used in Supplementary Figure 2 are given in Supplementary Table 5.
For each of the four tested model variants we found a tracking signal. However, the estimated control signal is not very plausible. At the moment of the external potassium drop, Nha1 is rapidly deactivated to prevent the potassium loss. This is followed by a phase where Nha1 is reactivated resulting in increased loss of potassium (see Supplementary Figure 2).
The time course for the wild-type, the trk1,2 and the nha1 mutant have the same characteristics ( Figure 1A in the main text): There is an initial phase of rapid potassium loss (approx. 1h) which is followed by a second phase of slow potassium loss before a new stationary state is attained. These characteristic phases remain visible upon deletion of Nha1. From this we inferred that a supposed deactivation of Nha1 is unlikely to be the actuating process for potassium homeostasis.
In addition, the predicted increase in Nha1-activity under starvation is in contradiction to experimental findings. It was reported that the potassium content in starved cells for the wild-type, nha1 mutants and a strain over-expressing Nha1 attains the same value after six hours of starvation [14]. This implies a deactivation of Nha1 under potassium starvation.

Simulated membrane potential
In the main text we presented estimated CO 2 production time courses for the wild-type and for trk1,2 mutants (see Figure 2C). The time course of the membrane potential for this estimate is shown in Supplementary Figure 3. In the simulation the removal of external potassium causes a hyperpolarization of the membrane potential. After some hours the membrane potential attains a new steady state value which is higher in magnitude compared to the pre-starvation level. In addition, the membrane potential of the trk1,2 mutant is higher (in magnitude) than the membrane potential of the wild type. For comparison with original data see [15].

Effect of Nha1 voltage gating
Electro-physiological data for Nha1 was not available. In the course of model development we started with a constant conductivity for this transporter. It turned turned out that under this assumption all potassium is lost completely, see Supplementary Figure 4. Thus we concluded, that Nha1 must be deactivated under potassium starvation in order to prevent an excessive potassium efflux. This assumption is supported by experimental observations [14] that Nha1 has an influence on the potassium content in later stages of starvation. The deactivation of Nha1 was modeled as voltage dependent gating. We observed positive stationary concentrations after starvation. Nevertheless, deactivation of Nha1 is not sufficient to reproduce the slow dynamics of potassium loss after starvation. In case of the Nha1 voltage gating the current finally reaches a lower stationary value compared to a constant conductivity model. Voltage gating can limit the potassium efflux, but it is not sufficient to reproduce the slow dynamics of potassium loss after starvation.

Simulated volume experiment
The potassium starvation experiment of [15] also comprises the measurement of the cell volume. Supplemental Figure 5 gives the respective measurements and model simulations for the wildtype and the trk1,2 mutant. In both cases, the volume decreases after the shift to the potassium free medium. However, the volume decrease is higher in the wildtype. As it was shown in Figure 1A the trk1,2 mutant maintains a higher potassium concentration compared to the wildtype. Thus, the volume measurement highlights the role of potassium as a major osmotic substance. However, the high variability of the cell volume in the measurements indicate the involvement of other processes like cell growth and division or volume regulation. These processes are not characterized for the given data and are not included in the volume description. Supplementary Figure 5 shows the simulated volume for the reverse tracking approach from Figure 2B. The simulated volume is only in qualitative agreement with the measurements. This indicates, that the volume model needs further improvements. However, we found the volume of minor relevance since even the assumption of a constant volume does not lead to a visible change in the overall result.  [15] show a high natural variability. The simulated volume is in qualitative agreement with the data.

Data fit of Trk1,2
In the simplest case a transport systems behaves like an ohmic resistance and electrophysiological measurements show a liner dependence between membrane voltage and current density. A related approach is the voltage gating, where the conductance parameter itself is dependent on the membrane voltage. The so called Boltzmann function or Boltzmann expression can be derived from the voltage dependent transition between an active and inactive state of the transporter [1][2][3][4][5] g(V m ) = g with constant conductivity g, the parameter d controlling the steepness of the transition and V 1/2 is the membrane potential of half maximal activation. This expression can be interpreted as the multiplication of a constant conductivity with a voltage dependent probability for the transporter to be active. The paper from [6] Figure 9c, gives the current-voltage relationship for the main potassium uptake system Trk1,2. The measurements are performed for both Trk proteins together and do not allow for a distinction between Trk1 and Trk2; raw data see Figure 6. The electro-physiological measurements were obtained under four different conditions. The external KCl concentration was either 150 mM, 10 mM, 1 mM or 50 µM and the internal potassium concentrations was 30 mM in all four cases.
Using Matlab and its lsqnonlin procedure, we fitted the expression to the data of [6] The nonlinear fit was robust to different initial values or parameter boundaries of the fitting routine.  Figure 1A in the main text shows two repetitions of a potassium starvation experiment with nha1 mutants.

Nha1 data
To distinguish between the two experiments we present this data again in Supplementary Figure 7. In tendency, the potassium reduction is fast in the first hour of starvation and slows down later on.