A Tool for Computation of Changes in Na, K, Cl Channels and Transporters Due to Apoptosis by Data on Cell Ion and Water Content Alteration

Abstract

Monovalent ions are involved in a vast array of cellular processes. Their movement across the cell membrane is regulated by numerous channels and transporters. Identification of the pathways responsible for redistribution of ions and cell water in living cells is hampered by their strong interdependence. This difficulty can be overcome by computational analysis of the whole cell flux balance. Our previous computational studies were concerned with monovalent ion fluxes in cells under the conditions of balanced ion distribution or during transition processes after stopping the Na⁺/K⁺ pump. Here we analyze a more complex case—redistribution of ions during cell apoptosis when the parameters keep changing during the process. New experimental data for staurosporine-induced apoptosis of human lymphoma cells U937 have been obtained: the time course of changes in cellular K⁺, Na⁺, Cl⁻, and water content, as well as Rb⁺ fluxes as a marker of the Na/K pump activity. Using a newly developed computational tool, we found that alteration of ion and water balance was associated with a 55% decrease in the Na⁺/K⁺-ATPase rate coefficient over a 4-h period, with a time-dependent increase in potassium channel permeability, and a decrease in sodium channel permeability. The early decrease in [Cl⁻]i and cell volume were associated with an ~5-fold increase in chloride channel permeability. The developed approach and the presented executable file can be used to identify the channels and transporters responsible for alterations of cell ion and water balance not only during apoptosis but in other physiological scenarios.

Full text

METHODS
published: 17 April 2019
doi: 10.3389/fcell.2019.00058
Frontiers in Cell and Developmental Biology | www.frontiersin.org 1April 2019 | Volume 7 | Article 58
Edited by:
Christoph Fahlke,
Julich Research Centre, Helmholtz
Association of German Research
Centers (HZ), Germany
Reviewed by:
Osvaldo Alvarez,
Universidad de Chile, Chile
Karl Kunzelmann,
University of Regensburg, Germany
Verena Untiet,
University of Copenhagen, Denmark
*Correspondence:
Alexey A. Vereninov
verenino@gmail.com
†These authors have contributed
equally to this work
Specialty section:
This article was submitted to
Membrane Physiology and Membrane
Biophysics,
a section of the journal
Frontiers in Cell and Developmental
Biology
Received: 17 January 2019
Accepted: 29 March 2019
Published: 17 April 2019
Citation:
Yurinskaya VE, Vereninov IA and
Vereninov AA (2019) A Tool for
Computation of Changes in Na+, K+,
Cl−Channels and Transporters Due
to Apoptosis by Data on Cell Ion and
Water Content Alteration.
Front. Cell Dev. Biol. 7:58.
doi: 10.3389/fcell.2019.00058
A Tool for Computation of Changes in
Na+, K+, Cl−Channels and
Transporters Due to Apoptosis by
Data on Cell Ion and Water Content
Alteration
Valentina E. Yurinskaya 1†, Igor A. Vereninov 2† and Alexey A. Vereninov 1
*
1Laboratory of Cell Physiology, Institute of Cytology, Russian Academy of Sciences, St-Petersburg, Russia, 2Peter the Great
St-Petersburg Polytechnic University, St-Petersburg, Russia
Monovalent ions are involved in a vast array of cellular processes. Their movement across
the cell membrane is regulated by numerous channels and transporters. Identification
of the pathways responsible for redistribution of ions and cell water in living cells
is hampered by their strong interdependence. This difficulty can be overcome by
computational analysis of the whole cell flux balance. Our previous computational
studies were concerned with monovalent ion fluxes in cells under the conditions of
balanced ion distribution or during transition processes after stopping the Na+/K+pump.
Here we analyze a more complex case—redistribution of ions during cell apoptosis
when the parameters keep changing during the process. New experimental data for
staurosporine-induced apoptosis of human lymphoma cells U937 have been obtained:
the time course of changes in cellular K+, Na+, Cl−, and water content, as well
as Rb+fluxes as a marker of the Na/K pump activity. Using a newly developed
computational tool, we found that alteration of ion and water balance was associated
with a 55% decrease in the Na+/K+-ATPase rate coefficient over a 4-h period, with a
time-dependent increase in potassium channel permeability, and a decrease in sodium
channel permeability. The early decrease in [Cl−]iand cell volume were associated with
an ∼5-fold increase in chloride channel permeability. The developed approach and
the presented executable file can be used to identify the channels and transporters
responsible for alterations of cell ion and water balance not only during apoptosis but
in other physiological scenarios.
Keywords: apoptosis, monovalent ions, channel parameters computation, ion transport, cell water, apoptotic
volume decrease
INTRODUCTION
A characteristic feature of apoptosis, one of the basic genetically encoded cell death mechanisms,
in contrast to accidental death, is that it is not associated with cell swelling or plasma membrane
rupture (Galluzzi et al., 2018). The apoptotic volume decrease (AVD) is a common but not
mandatory symptom of the cell apoptosis (Maeno et al., 2000, 2006; Okada et al., 2001; Yurinskaya
et al., 2005a,b; Bortner and Cidlowski, 2007). Cell swelling in apoptosis is prevented by the specific
alteration of the monovalent ion balance in apoptotic cells, as the monovalent ions are major cell

Yurinskaya et al. Monovalent Ions in Apoptosis
water regulators. It is believed that monovalent ions play
an important role in apoptosis (Lang et al., 2005; Lang and
Hoffmann, 2012; Bortner and Cidlowski, 2014; Hoffmann et al.,
2015; Kondratskyi et al., 2015; Jentsch, 2016; Wanitchakool
et al., 2016). However, this opinion is based mostly on the fact
that ion channels and transporters are altered somehow during
apoptosis and that their pharmacological or genetic modification
has an effect on apoptosis. The mechanism of specific apoptotic
alteration of cell ion and water balance has gotten much
less attention than the molecular identity of channels and
transporters involved in apoptosis. The mechanistic studies are
hampered by the interdependence between ion fluxes via the
numerous channels and transporters in the plasma membrane.
This difficulty can be overcome by the computational analysis of
whole-cell ion flux balance, which has been developed for normal
cells (Jakobsson, 1980; Lew and Bookchin, 1986; Lew et al., 1991;
Terashima et al., 2006; Vereninov et al., 2014, 2016). However,
no successful analyses have been done on apoptotic cells. We
have studied the relationships between alterations of the Na+/K+
pump or K+, Na+, and Cl−channels and transporters and the
apoptotic alteration of the entire cell water and ion balance
in U937 cells treated with staurosporine (STS) and etoposide
(Yurinskaya et al., 2005a,b, 2011). However, we lacked the
necessary experimental data and a proper programme code for
computation of transient processes in cell ion homeostasis and
analyzed only apoptotic cells at a single time point, 4 h. Here, we
studied ionic events during apoptosis development from 30 min
to 4 h. The background data included K+, Na+, Cl−, and water
contents and ouabain-sensitive and -resistant Rb+influx in U937
cells that were induced to undergo apoptosis by STS. An original
algorithm of the numerical solution of the cell monovalent ion
flux balance equations and the programme code were developed,
which allowed us to account for the continuous changes in the
Na+/K+pump activity. To our knowledge, this is the first attempt
to study the dynamics of the alteration of K+, Na+, Cl−, and
water contents during apoptosis. The approach developed to
study STS-induced apoptosis in U937 cells may be recommended
for identification of channels and transporters responsible for
alteration of cell ion and water balance in various situations.
METHODS
Reagents
RPMI 1640 medium and fetal bovine serum (FBS, HyClone
Standard) were purchased from Biolot (Russia). STS and
ouabain were from Sigma-Aldrich (Germany), Percoll was
purchased from Pharmacia (Sweden). The isotope 36Cl−was
from “Isotope” (Russia). Salts were of analytical grade and were
from Reachem (Russia).
Cell Cultures
U937 human histiocytic lymphoma cells were obtained from the
Russian Cell Culture Collection (Institute of Cytology, Russian
Academy of Sciences, cat. number 160B2). The cells were
cultured in RPMI 1640 medium supplemented with 10% FBS at
37◦C and 5% CO2overlay. For the induction of apoptosis, the
cells, at a density of 1 ×106cells per ml, were exposed to STS for
0.5–4 h. All the incubations were done at 37◦C.
Determination of Cell Ion and Water
Contents
The experimental methods used in this work have been described
in detail earlier (Yurinskaya et al., 2005a,b, 2011; Vereninov
et al., 2007, 2008). In summary, the cells were pelleted in
RPMI medium, washed five times with MgCl2solution (96 mM)
and treated with 5% trichloroacetic acid (TCA). TCA extracts
were analyzed for ion content. Intracellular K+, Na+, and Rb+
contents were determined by flame emission on a Perkin-Elmer
AA 306 spectrophotometer. To determine the intracellular Cl−,
the cells were cultured for 90 min or more at 37◦C in RPMI
medium containing 36Cl−(0.12 µCi ml−1). The radioactivity of
36Cl−in TCA extracts was measured using a liquid scintillation
counter (Beckman LS 6500). The intracellular Cl−content was
calculated, taking into account the specific activity of 36Cl−(∼2
counts min−1µmol−1). The TCA precipitates were dissolved in
0.1 N NaOH and analyzed for protein by the Lowry procedure,
with serum bovine albumin as a standard. The cell ion content
was calculated in micromoles per gram of protein.
Cell water content was determined by measurements of the
buoyant density of the cells in continuous Percoll gradient.
Percoll solution was prepared according to the manufacturer’s
instructions, and a thick cell suspension (0.1–0.2 ml, ∼3×106
cells) was placed on the solution surface and centrifuged for
10 min at 400 ×g (MPW-340 centrifuge, Poland). The buoyant
density of the cells was estimated using density marker beads
(Sigma-Aldrich, Germany). The water content per gram of
protein, vprot, was calculated as vprot. =(1-ρ/ρdry)/[0.72(ρ-1)],
where ρis the measured buoyant density of the cells and ρdry is
the cell dry mass density, which was 1.38 g ml−1. The proportion
of protein in dry mass was 72%.
Cellular ion concentration was calculated from the values
of ion and water content per gram of protein. In view that
intracellular water content in living cells is much more variable
than ion content, and because the ions and protein were assayed
in one sample whereas cell water content in parallel samples the
ion content per gram of protein should be considered as more
reliable than ion concentration in cellular water.
The Na+/K+Pump Rate Coefficient
Determination
The pump rate coefficient was determined based on the assay
of the ouabain-sensitive Rb+influx and cell Na+content. The
cells were incubated in medium with 2.5 mM RbCl and with or
without 0.1 mM ouabain for 10 min. The rate coefficient of the
Na+/K+pump (beta) was calculated as the ratio of the Na+
pump efflux to the cell Na+content given the assumption of the
simple linear dependence of Na+efflux on cell Na+in the studied
range of concentrations. The pump Na+efflux was calculated
from ouabain-sensitive (OS) Rb+influx assuming proportions of
[Rb]oand [K]oof 2.5 and 5.8 mM, respectively, and Na/K pump
flux stoichiometry of 3:2.
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Yurinskaya et al. Monovalent Ions in Apoptosis
Calculation of the Monovalent Ion Flux
Balance
The mathematical model of cell ion homeostasis and the
algorithm of the numerical solution of the flux balance were
described in detail earlier (Vereninov et al., 2014, 2016). The
reader can reproduce all presented computed data and perform
new calculations for various parameters by using the executable
file to programme code BEZ01B (How to use programme code
BEZ01B.zip in Supplementary Material). This software differs
from the previous BEZ01 by the additional parameter kb, which
characterizes a decrease in the pump rate coefficient βwith time.
Symbols and definitions used are shown in Table 1. The input
data used in calculation as file DATAB.txt in Supplementary
Material (see BEZ01B.zip in Supplementary Material) are the
following: extracellular and intracellular concentrations (na0,
k0,cl0, and B0;na,k, and cl); kv; the pump rate coefficient
(β); the pump Na/K stoichiometric coefficient (γ); parameter
kb; channel permeability coefficients (pna,pk,pcl); and the rate
coefficients for the Na+-Cl−(NC), K+-Cl−(KC), and Na+-
K+-2Cl−(NKCC) cotransporters (inc,ikc,inkcc). The results of
our computations appear in the file RESB.txt in Supplementary
Material (Table 2) after running the executable file.
The flux equations were:
dNai
dt =V{(pNau([Na]iexp(u)−[Na]o)/g−β[Na]i
+JNC +JNKCC}
dKi
dt =V{(pKu([K]iexp(u)−[K]o)/g+β[Na]i/γ
+JNKCC +JKC}
dCli
dt =V{(pClu([Cl]oexp(u)−[Cl]i)/g+JNC
+JKC +2JNKCC}
Here uis the dimensionless membrane potential (MP) related to
absolute values U(mV) as U=uRT/F =26.7ufor 37◦C and
g=1−exp(u). The left-hand sides of these three equations
represent the rates of change in the cell ion content, Nai=
[Na]iV, Ki=[K]iV, Cli=[Cl]iV. The right-hand sides express
fluxes via channels, the Na/K pump, and cotransporters. The
rate coefficients pNa,pK,pCl characterizing channel ion transfer
are similar to the Goldman’s coefficients. Fluxes JNC, JKC, JNKCC
depend on internal and external ion concentrations as
JNC =iNC([Na]o[Cl]o−[Na]i[Cl]i)
JKC =iKC([K]o[Cl]o−[K]i[Cl]i)
JNKCC =iNKCC([Na]o[K]o[Cl]o[Cl]o−[Na]i[K]i[Cl]i[Cl]i)
Here iNC, iKC , and iNKCC are the rate coefficients for
cotransporters (inc,ikc,inkcc in program symbols, Vereninov
et al., 2014). Transmembrane electrochemical potential
differences for Na+, K+, and Cl−were calculated as: ∆µNa
=26.7·ln([Na]i/[Na]o)+U,∆µK=26.7·ln([K]i/[K]o)+U,
and ∆µCl =26.7·ln([Cl]i/[Cl]o)-U, respectively. The values
of electrochemical potential differences for Na+, K+and Cl−,
denoted in program symbols as mun,muk, and mucl, are
important because they show the driving force and the direction
of ion movement via channels and transporters under the
indicated conditions. It is the changes in ∆µNa,∆µK, and ∆µCl
that are responsible for the possible fast effects of MP on ion
fluxes via “electroneutral” transporters.
Statistical Analysis
Experimental data are presented as the mean ±SEM. P<
0.05 (Student’s ttest) was considered statistically significant.
Reliability of the calculated data is discussed further.
RESULTS
Computational Approach to the Solution of
the Problem of How the Entire Cell Ion and
Water Balance Depends on the State of
Various Channels and Transporters
The first of the two main aims of the present study is the
demonstration of the computational approach to the solution of
the problem of how the entire cell ion and water balance depends
on the parameters of various channels and transporters. The
second aim is the analysis of the ion and water balance changes
during apoptosis in real U937 cells. This aim is an example of
using the developed approach. Some background points should
be considered first. The basic mathematical model used in our
approach is similar to the known model developed by pioneers
for analysis of ion homeostasis in normal cells (Jakobsson, 1980;
Lew and Bookchin, 1986; Lew et al., 1991). Our algorithm of
the numerical solution of the flux equations and basic software
was published earlier (Vereninov et al., 2014, 2016). Some minor
differences in mathematical models used by previous authors
consist in the number of transporters included in the calculations.
Only the Na+/K+pump and electroconductive channels were
considered in the early computational studies of cell ion
balance. Lew and colleagues were the first who found that the
Na+/K+pump and electroconductive channels cannot explain
monovalent ion flux balance in human reticulocytes because they
cannot explain the non-equilibrial Cl−distribution under the
balanced state without NC (Lew et al., 1991). Cotransporters NC
and KC were investigated by Hernández and Cristina (1998).
The NKCC cotransport was included in ion balance modeling in
cardiomyocytes (Terashima et al., 2006). Our software accounts
for Na+, K+, and Cl−channels, the Na+/K+pump and the NC,
KC and NKCC cotransporters. We found that NC is necessary
as a rule in the calculation of the resting monovalent ion flux
balance in U937 cells, while NKCC and KC are not. Nevertheless,
the parameters characterizing these two transporters are present
in our code, and fluxes via transporters can be accounted for if
these parameters differ from zero.
Two points may worry experimentalists. First, the Na+/K+
pump activity is characterized by a single rate coefficient.
However, a set of ion binding sites are known in the pump, and its
operation kinetics in biochemical studies is described commonly
by more than one parameter. The single rate coefficient is used
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Yurinskaya et al. Monovalent Ions in Apoptosis
TABLE 1 | Symbols and definitions.
Definitions In text and figures In files
DATAB.txt,
RESB.txt
Units
Ion species Na+, K+, Cl−, Rb+Na, K, Cl
Types of cotransport NC, NKCC, KC
Concentration of ions in cell water or external medium [Na]i, [K]i, [Cl]i, [Na]o, [K]o, [Cl]ona, k, cl, na0, k0,
cl0
mM
External concentrations of membrane-impermeant
non-electrolytes such as mannitol introduced in artificial media
B0 mM
Intracellular ion contents Nai,Ki,Climmol, may be related to g cell
protein or cell number, etc.
Intracellular content of membrane-impermeant osmolytes Ammol, may be related to g cell
protein or cell number, etc.
Cell water volume Vml, may be related to g cell
protein or cell number, etc.
Membrane-impermeant osmolyte concentration in cell water A/V*1,000 mM
Cell water content per unit of A V/A ml mmol−1
Mean valence of membrane-impermeant osmolytes, A z z Dimensionless
Permeability coefficients pNa, pK, pCl pna, pk, pcl min−1
Pump rate coefficient βbeta min−1
Na/K pump flux stoichiometry γgamma Dimensionless
Membrane potential, MP UmV
Dimensionless membrane potential u=UF/RT uDimensionless
Net fluxes mediated by cotransport JNC, JNKCC , JKC NC, KC, NKCC µmol min−1(ml cell water)−1
Na efflux via the pump –β[Na]iPUMP µmol min−1(ml cell water)−1
K influx via the pump β[Na]i/γPUMP µmol min−1(ml cell water)−1
Net fluxes mediated by channels Channel µmol min−1(ml cell water)−1
Unidirectional influxes of Na, K or Cl via channels or cotransport IChannel, INC,
IKC, INKCC
µmol min−1(ml cell water)−1
Unidirectional effluxes of Na, K, or Cl via channels, or
cotransport
EChannel, ENC,
EKC, ENKCC,
µmol min−1(ml cell water)−1
Time derivatives of concentrations prna, prk, prcl mM min−1
Cotransport rate coefficients iNC , iKC inc, ikc ml µmol−1min−1
iNKCC inkcc ml3µmol−3min−1
Ratio of “new” to “old” media osmolarity when the external
osmolarity is changed
kv Dimensionless
Number of time points between output of results hp Dimensionless
Transmembrane electrochemical potential difference for Na+,
K+, or Cl−
∆µNa,∆µK,∆µCl mun, muk, mucl mV
Ratio of ouabain-sensitive to ouabain-resistant Rb+(K+) influx OSOR OSOR Dimensionless
Parameter βdecreases linearly with time with coefficient kb min−1
because of the evaluation of the properties of all the ion binding
sites of the pump in experiments in whole cells is infeasible
and because it appears to be quite sufficient for the calculation
of entire-cell ion homeostasis. This idea was demonstrated by
the quantitative prediction of the dynamics of monovalent ion
redistribution after stopping the Na+/K+pump (Vereninov et al.,
2014, 2016). Single rate coefficients for characterizing the ion
carriage kinetics via transporters are commonly used for the
same reason. The second point causing disapproval might be that
an integral permeability coefficient is used in the calculation of
the flux balance for all Na+or K+or Cl−channels, whereas
a great variety of channels for each ion species is located in
the plasma membrane. The single permeability coefficients are
commonly used in the analysis of the entire-cell flux balance
because in an analysis of such a complex system with many
channels and transporters, the matter of primary importance is to
understand whether ion flux changes due to alteration of the force
driving the ions or by properties of the channels or transporters
per se.
Computation of Ion Flux Balance in Cells
Similar to U937 Cells
Parameters in absolute units are used in our calculations. Their
initial, “standard,” values are obtained from the calculation
based on the distribution of monovalent ions and ouabain-
sensitive Rb+(K+) influx measured in cells under normal
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Yurinskaya et al. Monovalent Ions in Apoptosis
TABLE 2 | Results of computation.
t U na k cl V/A mun muk mucl prna prk prcl
(A) TIME COURSE OF VARIABLES
0−44.3 33.0 152.0 45.0 12.50 −82.8 42.9 19.0 0.00000 0.00000 0.00000
24 −44.5 36.1 148.9 45.2 12.52 −80.7 42.2 19.3 0.07612 −0.07696 0.00308
**
216 −44.7 38.0 147.0 45.1 12.51 −79.5 41.7 19.4 0.00007 0.00004 −0.00040
240 −44.7 38.0 147.0 45.1 12.51 −79.5 41.7 19.4 0.00004 0.00005 −0.00032
(B) PARAMETER VALUES (COPY OF THE FILE DATAB.TXT)
na0 k0 cl0 B0 kv na k cl beta gamma
140 5.8 116 48.2 1.0 33 152 45 0.039 1.5
pna pk pcl inc ikc inkcc hp kb
0.00382 0.022 0.0091 3E-5 0 0 240 0
(C) FLUX BALANCE UNDER THE BALANCED STATE
Net flux PUMP Channel NC KC NKCC
Na −1.4811 1.0451 0.4359 0.0000 0.0000
K 0.9874 −0.9878 0.0000 0.0000 0.0000
Cl 0.0000 −0.4363 0.4359 0.0000 0.0000
Influx PUMP IChannel INC IKC INKCC
Na 0.0000 1.1012 0.4872 0.0000 0.0000
K 0.9874 0.2627 0.0000 0.0000 0.0000
Cl 0.0000 0.4082 0.4872 0.0000 0.0000
Efflux PUMP EChannel ENC EKC ENKCC
Na −1.4811 −0.0561 −0.0513 0.0000 0.0000
K 0.0000 −1.2505 0.0000 0.0000 0.0000
Cl 0.0000 −0.8445 −0.0513 0.0000 0.0000
zOSOR A/V*1000
−1.75 3.76 79.95
Transition of the system to the balanced state as displayed in the file RESB.txt in Supplementary Material. The values similar to those for U937 cells with a rather high U and ∆µCl were
chosen for this example of a transition to the balanced state. The displayed values of fluxes as well as OSOR correspond to the latest time point. The values of fluxes for other moments
can be obtained by setting the necessary time interval with the hp value. The presented flux data do not include the fluxes involved in one-for-one exchange because they have no effect
on cell ion or water content or MP and can be ignored here. The flux data clearly demonstrate how the net fluxes via different channels and transporters compensate for each other and
come finally, under appropriate conditions, to a fully balanced ion distribution when the balance of influx and efflux is achieved for all ion species and prna, prk, prcl tend to zero.
**Time points not shown.
physiological conditions and the cell balanced state. The
parameters varied until the tested values give a calculated entire
ion and water homeostasis similar to that in real cells. The
“standard” parameters can vary in real cells depending on the cell
physiological state, the age of the culture, the conditions of cell
cultivation, etc.
Nevertheless, these parameters can remain invariant under
a varying environment. We found that the kinetics of the
disturbance of cell ion and water balance caused by blocking the
Na+/K+pump when the intracellular K+/Na+ratio is highly
changed and even reversed can be predicted sufficiently well
by calculation with the invariant parameters (Vereninov et al.,
2014, 2016). A set of examples is presented in Figure 1 to
show how changes in a single channel or transporter species
(one permeability coefficient or rate constant) can alter the
intracellular concentrations of all major ions, cell water content
and the MP. Unlike pNa and iNC, changes in pK or pCl lead,
over the course of 60–100 min, to a new balanced state. Even
this small set of examples demonstrates that some effects seem
to be unexpected at first sight. Intracellular K+concentration
decreases monotonically with the pNa increase, while the
intracellular K+content decreases initially and increases further
due to the superposition of the initial drop MP and the slow
increase in cell water-volume. An increase in the coupled
equivalent transport Na+and Cl−(iNC,) causes a decrease in
cell K+concentration, [K+], and, in contrast, an increase in
cell K+content because of changes in cell water volume and
in MP. The [K+] and MP are shifted in this case in opposite
directions. It should be stressed that the effects of parameter
variation are highly dependent on the cell species. Our previous
paper presented the typical dependences for the cells with high
MP and high intracellular K/Na ratio, for the cells with low MP
and high K/Na ratio (high potassium erythrocytes) and for the
low-MP and low-K/Na-ratio cells (low-potassium erythrocytes
of some carnivores and ruminants) (Vereninov et al., 2014).
Cells such as U937 and their variant with relatively high ∆µCl
(19.4 mV) are chosen as an example in Figure 1 to make the pCl
effect more obvious.
The ion and water redistribution caused in U937 cells by
stopping the Na+/K+pump was studied earlier in silico and
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Yurinskaya et al. Monovalent Ions in Apoptosis
FIGURE 1 | The calculated effects of an abrupt increase in the permeability coefficients of K+, Na+, Cl−channels, or the NC cotransport rate coefficient on cell K+,
Na+, and Cl−content and concentrations, water-volume (V/A) and MP (U). The data were calculated by using the software BEZ01B. The initial parameters were na0
140, k0 5.8, cl0 116, B0 48.2, kv 1, beta 0.039, gamma 1.5, pna, 0.00382, pk 0.022, pcl 0.0091, inc 0.00003, ikc =inkcc =0, kb 0, and hp 300, i.e., much like
U937 cells; the changed parameters are shown on the plots.
in an experiment (Vereninov et al., 2014, 2016). Here, it is
interesting to demonstrate this case as an example of asynchrony
in changes of K+, Na+, and Cl−after blocking the pump
(Figure 2). In the earlier stage, the electroneutrality of the net ion
fluxes is achieved mainly by the balance of fluxes K+outward
and Na+inward via channels, whereas, later, the Cl−influx
becomes significant. There is no alteration of total intracellular
osmolytes during the equal K+/Na+exchange, and it is for
this reason that no swelling occurs after blocking the pump
for a rather long time. It should be stressed that no specific
carrier is responsible for the balanced K+/Na+exchange. This
result is realized via electroconductive channels only due to
the dependence of fluxes on MP. The long-term balanced state
in monovalent ions and water distribution after stopping the
pump is unattainable, and cell swelling will go on infinitely.
However, the kinetics of the entire process may be different
in dependence on the pCl level. It should be noted that cell
water content and intracellular concentration are changing
synchronously. It is the low Cl−channel permeability that
saves real cells from swelling for a long time after blocking the
Na+/K+pump.
Due to a large number of adjustable parameters and
experimental observations, the usual statistical verification of
reliability of calculated data is not applicable. In order to
determine the sensitivity of results to experimental errors and
the choice of parameters, one can simply repeat calculations
for slightly different input values. It should be noted that
sensitivity of results to such changes must be determined for
specific conditions. Figure 3 shows the effects of changes in the
permeability of Na+, K+, and Cl−channels on the time course of
ion redistribution caused by blocking of the Na+/K+pump.
Changes in K+, Na+, Cl−, and Water
Contents During Early Apoptosis in U937
Cells Induced by STS
Most data on the redistribution of monovalent ions during
apoptosis relates to the 4–5 h stage (see references in Arrebola
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Yurinskaya et al. Monovalent Ions in Apoptosis
FIGURE 2 | The effect of pCl on the time course of the ion and water balance disturbance caused by turning off the pump. The data were calculated by using the
software BEZ01B with the following parameters: na0 140, k0 5.8, cl0 116, B0 48.2, kv 1, na 52.6, k144.2, cl 11.9, beta 0 (at the initial balanced value of 0.039),
gamma 1.5, pna 0.006, pk 0.06, pcl 0.1 (triangles) or 0.001 (circles) or 0.0001 (solid lines), inc =ikc =inkcc =0, kb 0.
FIGURE 3 | Dependence of calculated K+, Na+, Cl−redistribution dynamics after blocking the pump on K+, Na+, and Cl−channel permeabilities. Experimental
data are shown by the large symbols. Solid lines—data calculated for the following parameters: na0 140, k0 5.8, cl0 116, B0 48.2, kv 1, na 35, k156, cl 70, beta
0.001, pna 0.00301, pk 0.023, pcl 0.00405, inc 3.4E-5, ikc =inkcc =0, (before blocking the pump beta was 0.039), gamma 1.5. Dotted lines with small
symbols—data calculated for parameter values indicated above the graphs.
et al., 2005a). Our simultaneous determination of K+, Na+, Cl−,
and water contents in U937 cells treated with STS for 4 h was
published earlier (Yurinskaya et al., 2011). The data related to
this apoptosis stage confirmed the osmotic mechanism of AVD,
i.e., they showed that the water loss was caused mostly by the loss
of the total monovalent ion content and much less by a decrease
in content of the “impermeant intracellular anions,” A−. The
initial changes in all major monovalent ions and water content
during apoptosis have been studied much less, although the early
cell shrinkage is supposed to be crucial for triggering apoptosis.
Our current data on the changes in ion and water content
during STS apoptosis in U937 cells with the earliest time point
30 min are presented in Table 3. The values of the independently
determined ion content and water content correspond to the
osmotic mechanism of AVD at the early stages as well as at the
4 h stage studied before. The data on water content in Table 3
were obtained by the best method, i.e., by cell buoyant density.
These data agree well with the data obtained by using a Coulter
counter and flow cytometer (Yurinskaya et al., 2017). Calculation
TABLE 3 | Changes in K+, Na+, Cl−, and water contents during the early stages
of STS apoptosis in U937 cells.
Time K+Na+Cl−A−Water
min µmol ·(g prot.)−1ml/g
0 712 ±22 192 ±8 246 ±11 658 6.08 ±0.08
30 615 ±12 175 ±10 133 ±4 657 5.37 ±0.21
120 595 ±13 179 ±4 109 ±5 665 4.70 ±0.05
240 493 ±21 261 ±5 117 ±4 637 4.85 ±0.08
Means ±SEM from three independent experiments with duplicate determinations.
of the changes in K+, Na+, and Cl−net fluxes underlying the
changes in cell ion and water content shows that for the first hour,
the K+loss is electrically balanced predominantly by the Cl−
loss, whereas later it is mostly balanced by the Na+gain (Table 4,
last columns).
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Yurinskaya et al. Monovalent Ions in Apoptosis
The apoptotic changes in ion content obtained in our study
by flame photometry and radiotracer assay are very close to
the data obtained by the X-ray microanalysis in U937 cells
during several types of apoptosis, including early STS apoptosis
(Arrebola et al., 2005b, 2006). Unlike that X-ray microanalysis
study, we could more easily validate changes in cell water content
during apoptosis and therefore better estimate ion concentrations
per cell water volume. This approach enabled us to calculate the
entire cell electrochemical system and, in this way, to identify
channels and transporters critical for AVD and underlying
monovalent ion redistribution.
Matching the Real and Calculated
Changes in Cell K+, Na+, and Cl−
Concentrations During Apoptosis
The real changes in Na+, K+, Cl−and water contents
during STS apoptosis in U937 cells differ from the calculated
example presented in Figure 1. Evidently, other changes in
rate parameters during the transient process can occur in real
cells. Indeed, a decrease in the Na+/K+pump activity is a
peculiar feature of apoptosis and has been revealed without
any computation, particularly in U937 cells treated with STS
(Arrebola et al., 2005a,b; Vereninov et al., 2008; Yurinskaya
et al., 2010, 2011). The rate coefficient of the Na+/K+pump can
be calculated by OS Rb+influx and intracellular Na+content
(Vereninov et al., 2014, 2016). We found that the OS Rb+influx
for the first 4 h of STS apoptosis in U937 cells decreased mostly
linearly (Yurinskaya et al., 2010).
The linear decrease in the pump rate coefficient with time
was accounted for in the current programme code BEZ01B.
Figure 3A shows the transient process during STS apoptosis
in U937 cells calculated under the assumption that the pump
rate coefficient decreases linearly due to the decrease in the
coefficient kb, as found by OS-Rb+influx assay in experiment.
The values calculated for this simplest model (lines) correspond
approximately to the real ion concentrations (symbols) for K+
(circles) and Na+(triangles) but differ significantly for Cl−
(squares). The additional assumption that triggering apoptosis is
accompanied by stepwise increases in pCl and pK and by a slight
decrease in pNa improves the agreement between calculated
and real values for Cl−in the first 30–40 min, but not later
(Figure 3B). A change in pCl alone becomes ineffective because
of the small ∆µCl. The agreement may be obtained for the
whole 4 h time interval by assuming that pK further decreases
(Figure 3C). How unique is the found fitting? By trial and error,
we found that a pNa decrease and a pK increase alone without a
pCl increase could be sufficient to get agreement between real and
calculated chloride concentrations for the first 30 min. However,
this case should be rejected because the value OSOR becomes
unacceptably low.
The joint effect of pK, pNa and pCl shift is interesting. The
cells shown in Figure 4 had initially a rather low U(−29.9 mV)
and a mucl of ∼1.5 mV under the normal state (Table 4). The
variation of pCl alone at so small a mucl has no significant effect.
A decrease in pNa hyperpolarizes cells promptly, and an increase
in pK alone hyperpolarizes cells as well (Table 5). As a result,
the pCl increase becomes effective and sufficient to get both
the necessary agreement between real and calculated chloride
concentrations for the initial 30–40 min and the necessary OSOR.
The inc and pCl parameters change cell water and [Cl]iin
opposite directions (Figure 1). However, we could not replace
the pCl increase with the inc decrease in our fitting procedure,
as the [Cl]idecrease in the latter case is too small. We come to
the conclusion that an increase in pCl is a critical factor for the
complex water and ion rearrangement at the initial stage of STS
apoptosis in U937 cells, whereas its role becomes less significant
or even non-significant later.
We conclude that the redistribution of K+, Na+, and Cl−
underlying AVD in the studied U937 cells treated with STS is
caused (1) by a progressive linear decrease in the pump rate
coefficient from the initial 0.029 to 0.013 at 4 h, (2) by a significant
increase in pCl (0.0125 to 0.068) and changes in pK (0.0115
to 0.03 and later 0.02), and (3) by a moderate decrease in pNa
(0.0041 to 0.003). The most critical factors for changes in cell K+
and Na+are suppression of the pump, an increase in pK and a
decrease in pNa, whereas the early decreases in Cl−and water
content (early AVD) are associated primarily with an increase in
pCl by about 5 times and an increase in pK by about 2.6 times.
Figure 5 shows sensitivity of the calculated results to “trial”
variations in the permeability coefficients of Na+, K+, and Cl−
channels in the case of apoptosis. Similar trials convinced us that
the values of channel permeabilities and transporter parameters
obtained by computation for apoptotic U937 cells are reliable
and unique.
The calculated MP in the considered model of apoptosis
was slightly hyperpolarizing (by 12 mV). Our preliminary results
from flow cytometry using DiBAC4(3) did not show a significant
change in MP under STS-induced apoptosis (unpublished data).
These results differ from previous reports of MP depolarization
during apoptosis, e.g., in the Fas-L induced apoptosis of Jurkat
cells (Franco et al., 2006). Further studies are required to
determine whether MP changes are highly dependent on the
apoptosis inducer and/or on the cell species or whether cell
depolarization occurs in more severe apoptosis.
DISCUSSION
Monovalent ion channels and transporters are involved in
apoptosis (Burg et al., 2006; Lang et al., 2007; Hoffmann et al.,
2009, 2015; Dezaki et al., 2012; Lang and Hoffmann, 2012; Orlov
et al., 2013; Kondratskyi et al., 2015; Jentsch, 2016; Pedersen et al.,
2016; Wanitchakool et al., 2016). However, this phenomenon
may be caused by two reasons: because the monovalent ions are
the major cell volume regulators and should be responsible for
AVD only for this reason, or because they also play important
roles in cell signaling by affecting MP. It is not easy to distinguish
these two causes at present. We aimed to answer the question
of how alteration of distinct channels and transporters affects
the balance of monovalent ion fluxes across the cell membrane,
cell water content and MP at apoptosis. We studied the time
course of the monovalent ion balance redistribution during the
first 4 h development of apoptosis induced in U937 cells treated
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Yurinskaya et al. Monovalent Ions in Apoptosis
TABLE 4 | Changes in cell [K+], [Na+], [Cl−], MP (U) and net fluxes calculated upon a linear decrease in the pump rate coefficient and stepwise changes in the K+, Na+,
and Cl−channel permeability corresponding to the experimental data on apoptotic alteration of K+, Na+, and Cl−concentrations and pump fluxes.
Time, min beta pk pna pcl U na k cl V/A mun muk mucl Net fluxes
Na+K+Cl−
Before 0.029 0.0115 0.0041 0.0125 −29.9 32 117 40 8.26 −69.3 50.3 1.5 0 0 0
0 0.029 0.03 0.003 0.068 −34.6 32.0 117 40.0 8.26 −74.0 45.6 6.2 0 0 0
10 0.028 −38.5 32.3 116.4 33.5 7.82 −77.7 41.6 5.3 −0.119 −0.622 −0.733
20 0.028 −42.0 32.7 115.7 28.4 7.51 −80.8 37.9 4.4 −0.065 −0.481 −0.540
30 0.027 −45.0 33.3 114.9 24.6 7.29 −83.3 34.8 3.5 −0.020 −0.366 −0.381
60 0.025 0.02 −44.8 33.8 114.3 22.3 7.16 −82.8 34.8 0.8 0.031 −0.079 −0.048
120 0.021 −45.1 37.7 110.3 21.7 7.13 −80.1 33.5 0.3 0.086 −0.078 0.008
** ........................................................
210 0.015 −43.2 47.2 100.9 23.2 7.21 −72.2 33.1 0.1 0.139 −0.110 0.029
240 0.013 −42.2 51.4 96.7 24.0 7.25 −68.9 33.0 0.1 0.163 −0.127 0.036
The initial parameters were na0 140, k0 5.8, cl0 116, B0 48.2, kv 1, na 32, k 117, cl 40, gamma 1.5, inc 0.000003, and kb 0.000068. The changed parameters, including beta, pna, pk,
and pcl, are shown in the Table. **Time points not shown. The data were obtained by using the code BEZ01B. The time of channel alteration is indicated by horizontal lines. Outward
net fluxes are defined as negative.
FIGURE 4 | Time course of [K+], [Na+] and [Cl−] in real U937 cells treated with 1 µM STS (symbols) and calculated (lines) for different parameter datasets.
Symbols—experimental data, means ±SEM from three independent experiments with duplicate determinations. Small SEM values are masked by symbols.
Lines—calculated data obtained for the parameters indicated on the graphs. The initial parameters were na0 140, k0 5.8, cl0 116, B0 48.2, kv 1, na 32, k117, cl 40,
beta 0.029, gamma 1.5, pna 0.0041, pk 0.0115, pcl 0.0125, inc 0.000003, ikc =inkcc =0, and kb 0.000068. The changed parameters are shown in the layers
head. (A) Linear decrease in beta only. (B) Decrease in beta and changes in pna, pk, and pcl.(C) Additional decrease in pk. Shaded regions show significant
disagreement of experimental and predicted values. The calculated data were obtained by using code BEZ01B.
with STS as the established model of apoptosis with significant
AVD. Apoptosis in U937 cells is accompanied by rapid changes
in light scattering and cell water (volume) balance, whereas the
positive annexin test and intensive generation of apoptotic bodies
are revealed, starting at 3–4 h (Yurinskaya et al., 2017). The
identification of channels and transporters responsible for the
observed changes in monovalent ion distribution, water balance
and the pump fluxes was based on the computational modeling
of these changes. Such an approach was applied here to study
apoptosis for the first time, although the monovalent flux balance
under the normal physiological state and during redistribution of
ions due to stopping the pump has been calculated successfully
before (see ref. in: Vereninov et al., 2014, 2016). Our previous
code was modified currently to account for a continuous decrease
in the pump rate coefficient.
One of the most detailed experimental studies of the kinetics
of the monovalent ion balance rearrangement during apoptosis
was performed by X-ray microanalysis and in U937 cells
treated with STS in particular (Arrebola et al., 2005a,b). The
experimental data obtained by flame emission and radiotracer
assays in our study agree very well with the data obtained by
this quite different method. Unfortunately, the accurate cell water
content evaluation is hard to combine with the X-ray elemental
microanalysis. Therefore, the complete mathematical model of
the monovalent ion flux balance could not be developed using
only those data.
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Yurinskaya et al. Monovalent Ions in Apoptosis
TABLE 5 | The effects of pK, pNa, and pCl shifts at the initial stage of apoptosis on U,mucl, and ion concentrations.
t pk pna pcl U na k cl V/A mun muk mucl
pk shift
Initial 0.0115 0.0041 0.0125 −29.9 32.0 117.0 40.0 8.26 −69.3 50.3 1.5
00.03 0.0041 0.0125 −41.1 32.0 117.0 40.0 8.26 −80.6 39.1 12.7
15 −42.1 35.1 113.7 36.6 8.03 −79.0 37.4 11.3
30 −42.9 37.2 111.5 33.8 7.83 −78.3 36.0 10.0
45 −43.6 38.6 110.0 31.5 7.69 −78.0 34.9 8.8
60 −44.2 39.2 109.0 29.5 7.57 −78.0 34.1 7.7
pna shift
Initial 0.0115 0.0041 0.0125 −29.9 32.0 117.0 40.0 8.26 −69.3 50.3 1.5
0 0.0115 0.003 0.0125 −36.6 32.0 117.0 40.0 8.26 −76.0 43.6 8.2
15 −37.2 30.6 118.3 38.0 8.12 −77.8 43.3 7.4
30 −37.8 29.7 119.1 36.2 8.00 −79.2 42.8 6.7
45 −38.4 29.2 119.5 34.6 7.90 −80.2 42.4 6.1
60 −38.9 28.9 119.7 33.3 7.81 −81.0 41.9 5.6
pk and pna shift
Initial 0.0115 0.0041 0.0125 −29.9 32.0 117.0 40.0 8.26 −69.3 50.3 1.5
00.03 0.003 0.0125 −44.8 32.0 117.0 40.0 8.26 −84.2 35.4 16.4
15 −46.3 32.5 116.3 35.6 7.96 −85.3 33.8 14.7
30 −47.6 32.8 115.8 31.9 7.72 −86.4 32.3 13.1
45 −48.8 33.1 115.4 28.8 7.53 −87.3 31.1 11.5
60 −49.7 33.3 115.0 26.2 7.38 −88.1 30.0 10.0
pk,pna and pcl shift
Initial 0.0115 0.0041 0.0125 −29.9 32.0 117.0 40.0 8.26 −69.3 50.3 1.5
00.03 0.003 0.068 −34.6 32.0 117.0 40.0 8.26 −74.0 45.6 6.2
15 −40.4 32.3 116.2 30.7 7.65 −79.6 39.7 4.9
30 −45.2 32.6 115.6 24.4 7.28 −84.2 34.7 3.6
45 −48.7 32.9 115.1 20.5 7.07 −87.4 31.1 2.5
60 −50.9 33.2 114.7 18.3 6.95 −89.4 28.8 1.6
The initial parameters were as follows: na0 140, k0 5.8, cl0 116, B0 48.2, kv 1, beta 0.029, na 32, k 117, cl 40, gamma 1.5, inc 0.000003, kb 0.
FIGURE 5 | Dependence of the calculated K+, Na+, and Cl−redistribution dynamics during STS-induced apoptosis on K+, Na+, and Cl−channel permeabilities.
Experimental data are shown by the large symbols. Solid lines—data calculated for the following parameters: na0 140, k0 5.8, cl0 116, B0 48.2, kv 1, na 32, k117, cl
40, beta 0.029, gamma 1.5, pna 0.003, pk 0.03, pcl 0.068, inc 3E-6, ikc =inkcc =0, kb 0.000068. Dotted lines with small symbols—data calculated for parameter
values indicated above the graphs. OSOR values are given for a 4 h time point.
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Yurinskaya et al. Monovalent Ions in Apoptosis
Earlier we tried to relate the changes in ion and water contents
to the monovalent fluxes in the Na+/K+pump, K+, Na+, and
Cl−channels and certain cotransporters in U937 cells after 4 h
of STS-induced apoptosis (Yurinskaya et al., 2011). We came to
the conclusion that the Na+/K+pump suppression accompanied
by a decrease in Na+channel permeability might be responsible
for AVD under the considered conditions. However, our current
computational tool had not been developed at that time, the
experimental data were limited to single time point 4 h, and
the assumption was used that the balanced monovalent ion
distribution is reached at 4 h of STS-induced apoptosis in U937
cells. More complete current data show that the cells at the 4 h
time point are far from the balanced state (Table 3). There is
significant Na+gain (0.163) that is by ¾ balanced by K+leak
(0.127) and ¼ by the gain of Cl−(0.036).
Currently, we substantially revised and developed our
previous conception of the participation of the major channels
and transporters in AVD during the STS-caused apoptosis
of U937 cells. It remains true that a slow decrease in the
Na+/K+pump activity is a primary factor responsible for
AVD at the late (4 h) stage of apoptosis. Recalculation of the
data published earlier (Yurinskaya et al., 2011) with use of
the current programme code and without intricate hypotheses
confirmed a decrease in pNa at 4 h. The current data show
that the pNa decrease at 4 h is significant indeed. The most
interesting and important phenomenon is a more than 5-fold
increase in the Cl−channel permeability, which is much more
important at the early stage. It is remarkable that the effect
of the pCl increase disappears further because of a decrease
in intracellular Cl−concentration and associated decrease in
chloride electrochemical potential difference, ∆µCl (mucl in
Table 4). The effects of the early increase in pK and a decrease in
pNa on Uare significant because they lead to an increase in ∆µCl
that drives chloride outward. A large body of electrophysiological
evidence published recently indicates that the state of the chloride
channels can change upon initiation apoptosis (Hoffmann et al.,
2015; Kondratskyi et al., 2015; Jentsch, 2016; Pedersen et al., 2016;
Wanitchakool et al., 2016). However, there were no attempts to
use these data for the quantitative description of early AVD.
The current computations show that changes in not a single
type of channel but in K+, Na+, and Cl−channels and
in the Na+/K+pump are responsible for the apoptotic ion
balance alteration and that the effect of various channels and
transporters on ion balance may be different at different stages
of apoptosis. Certainly, the question arises how many parameters
can provide accordance between the calculated and real data? The
computation enables us to answer this question, although certain
time may be needed. In the case of STS-induced apoptosis in
U937 cells in our experiments, we can exclude alternative variants
by taking into account additionally the value of OSOR, which
appeared to be different in different parameter setups, giving
sufficiently good accordance between the real and calculated data.
In other cases, the problem could be solved probably not by using
OSOR but by some other way. The computation shows also how
the real behavior of cells should depend on the initial state of cells.
Certainly, as soon as basic experimental data vary, the obtained
numerical values of parameters will vary also.
A skeptical view is spread among the experimentalists on the
using calculations in analysis of the ion flux balance in cells.
There is also a great deal of sometimes convoluted discussion
about the merits and validity of certain assumptions that need
to be made for the models and real data to be reconciled.
As believed it is very difficult to verify the models and their
conclusions. In this regard, we should note the following. No
hypothetical assumptions are used in our calculations as well as in
similar studies of previous authors. The calculations are based on
the “model-independent” theory. Two mandatory fundamental
principles are used: macroscopic electroneutrality of ion transfer
and osmotic balance between internal and external media in
animal cells. All known types of monovalent ion pathways are
accounted for in calculations of ion flux balance. These pathways
are identified by the forces driving ions across the cell membrane
which calculation is indisputable at present. The calculations do
not depend on any peculiarity of molecular structure of proteins
involved in ion transfer. Our study is not a modeling-imitation
but a tool for solution the problem which cannot be solved
currently without computation. The calculations include the
interrelationships in the movement of ions across the membrane,
due to their overall dependence on the membrane potential
and contribution to the osmotic balance. This allows identifying
changes in the pathways itself.
If a required set of experimental data is given a unique solution
appears independently on any hypotheses on the number and
types of channels and transporters which could present in the
cell membrane. Our system of the flux equations accounts
all currently known types of ion transfer across membrane
characterized only by the ion driving forces: electrochemical
potential difference for movement of single ion species
(electrodiffusion through electroconductive channels), the sum
of electrochemical potential differences for the linked movement
of several species of ions (cotransport, countertransport), and
a combination of the electrochemical and chemical potential
differences in case of the Na+,K+-ATPase pump. Computer
decides what number of transporting units of each type should
be for implementation of two physically mandatory demands and
which ionic pathways do not play a role under given conditions,
particularly, at existent electrochemical gradients of each ion
species. The mandatory demands are electroneutrality of the
any macroscopic ion redistribution and osmotic balance between
a distensible animal cell and the medium. Any hypothesis
on the mechanism regulating cell water and ion content
or membrane potential must be checked for these demands
implementation. This cannot be done without computation in
a system with a numerous species of ions and numerous ion
pathways. Experimentalists avoid calculation and prefer using
inhibitors and genetic cell modification simply because there is
no sufficiently suitable tool for computation. We attempted to
reduce computational tool deficiency.
As to general validation of our tool, the most strong argument
here is the well prediction of the time course of the complex
redistribution of ions caused by stopping the Na+/K+pump
using the parameters obtained from analysis of the balanced ion
distribution. A full reverse of the intracellular K+/Na+ratio and
a strong cell depolarization take place in this case. One could
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Yurinskaya et al. Monovalent Ions in Apoptosis
suppose that properties of channels and transporters will change.
However, calculations with invariant parameters of channels and
transporters showed well matching of calculated and real data.
One could suppose that the case of parameters alteration occurs
more often than the parameters constancy. Of course, it would
be nice to have another independent method for determining
parameters, but it still needs to be looked for. Current study
demonstrates using the computational tool for evaluation the
parameter changing.
Notes Added in Response to Some
Readers of Our Preprint Posted at
BioRchiv (Yurinskaya et al., 2018)
Some of our readers have expressed doubt that using our tool,
one can get a unique set of parameters that provide agreement
between experimental and calculated data. Therefore, a few
comments would be in order.
The problem studied and discussed in our manuscript means
mathematically a search for coefficients of a system of ordinary
differential equations with non-linear right-hand sides. There is
a vast area of applied mathematics dealing with similar problems
using numerical methods (see e.g., Kahaner et al., 1989; Bonnans
et al., 2006). It is known that the problem can be solved in
a practically meaningful sense only if certain constraints on
parameters and variables are accepted which are obtained from
the knowledge of properties of a real object. Studying of our
real object showed that the effects of the different parameters
on the state of the system may vary widely in different areas.
This indicates the poor “conditionality” of the so-called Hessian
matrix. It is impossible to predict theoretically in our case how
many sets of parameters will give one and the same results.
However, the use of our executable file allows us to solve
the problem by trial testing various parameters in the area of
interest. The procedure in practice is not too time consuming.
This is what we did in our previous (Vereninov et al., 2014,
2016) and current studies. Some obvious constraints are included
directly in our source code. These are: U<0, [Na]i,o >0,
[K]i,o >0, [Cl]i,o >0, V>0. The initial parameters in our
treatment are found for the “balanced state of cell” (this common
physiological term means that the inward and outward fluxes
for each of ion species capable of crossing the cell membrane
are equal). This means, mathematically, that only those systems
are considered for which a stationary state is possible. If explicit
constraints are not enough, it is the researcher himself who
knows the real object should find additional constraints that
allow finding a unique set of parameters or at least to limit the
number of possible sets. In some cases, searching for certain
parameters may be unnecessary because the executable file shows
that the driving forces (mun, muk, mucl) are low or zero in
the considered area. In other cases, using specific blockers of
channels or transporters might help. A bit of creativity and
inventiveness is required here. Our articles demonstrate how a
problem was solved in some real cases. In the present study
the ratio of ouabain-sensitive to ouabain-resistant components
of the rubidium influx (OSOR) helps to choose the right set
of parameters. OSOR is obtained from experimental data easily
and reliably. Its value is included in the output table. Using
our executable file helps to determine which parameter(s) most
strongly affect cell ion homeostasis under given conditions and
to find the optimal experimental protocol for studying its role in
the considered phenomena.
CONCLUSIONS
1. The experimental data on the time course of K+, Na+, and
Cl−concentrations and ouabain-sensitive and -resistant Rb+
influx in U937 cells treated with STS for 0.5–4 h enabled
us to evaluate the changes in the pump rate coefficient and
to compute alterations of the K+, Na+, and Cl−channel
permeability coefficients associated with the initial stages of
apoptosis and AVD.
2. The redistribution of K+, Na+, and Cl−underlying AVD
in U937 cells is caused (1) by a progressive decrease in the
Na+/K+pump rate coefficient from an initial 0.029 to 0.013
at 4 h, (2) by a significant increase in pCl (0.013–0.068) and
increases in pK (0.012–0.03, later 0.02), and (3) by a moderate
decrease in pNa (0.004–0.003). The most critical factors for
changes in cell K+and Na+are the suppression of the pump,
an increase in pK and a decrease in pNa, whereas the early
decrease in Cl−and water content (early AVD) are associated
primarily with an increase in pCl by ∼5 times and an increase
in pK by ∼2.6 times.
3. Our approach demonstrates how to calculate the dependence
of cell ion and water balance on the states of channels and
transporters in the plasma membrane and is recommended
for analyzing redistribution of monovalent ions and water not
only during apoptosis but in other cases as well.
AUTHOR CONTRIBUTIONS
All authors contributed to the design of the experiments,
performed the experiments, and analyzed the data. IV developed
algorithm of numerical solution of the problem and wrote
the programme code. AV wrote the manuscript with input
from all authors. All authors have approved the final version
of the manuscript and agree to be accountable for all
aspects of the work. All persons designated as authors qualify
for authorship, and all those who qualify for authorship
are listed.
ACKNOWLEDGMENTS
This manuscript has been released as a Pre-print at BioRxiv
online Dec. 4, 2018; doi: http://dx.doi.org/10.1101/486811
(Yurinskaya et al., 2018). We thank Dr. Tatyana Goryachaya for
excellent assistance in the experiments with cells. The research
was supported (VY, AV) by the Grant of Russian Federation
(No. 0124-2019-0002, No. 0124-2019-0003).
SUPPLEMENTARY MATERIAL
The Supplementary Material for this article can be found
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00058/full#supplementary-material
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Yurinskaya et al. Monovalent Ions in Apoptosis
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Conflict of Interest Statement: The authors declare that the research was
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Frontiers in Cell and Developmental Biology | www.frontiersin.org 14 April 2019 | Volume 7 | Article 58