Department of Computer Science, Federal University of Minas Gerais, Av. Antônio Carlos 6627, Belo Horizonte, MG, Brazil

Abstract

Background

Recently there has been a growing interest in the application of Probabilistic Model Checking (PMC) for the formal specification of biological systems. PMC is able to exhaustively explore all states of a stochastic model and can provide valuable insight into its behavior which are more difficult to see using only traditional methods for system analysis such as deterministic and stochastic simulation. In this work we propose a stochastic modeling for the description and analysis of sodium-potassium exchange pump. The sodium-potassium pump is a membrane transport system presents in all animal cell and capable of moving sodium and potassium ions against their concentration gradient.

Results

We present a quantitative formal specification of the pump mechanism in the PRISM language, taking into consideration a discrete chemistry approach and the Law of Mass Action aspects. We also present an analysis of the system using quantitative properties in order to verify the pump reversibility and understand the pump behavior using trend labels for the transition rates of the pump reactions.

Conclusions

Probabilistic model checking can be used along with other well established approaches such as simulation and differential equations to better understand pump behavior. Using PMC we can determine if specific events happen such as

Background

Computational modeling has been increasingly used in the field of systems biology to examine the dynamics of biological processes. Traditionally, the modeling of biochemical pathways is based on a set of non-linear ordinary differential equations (ODE) to describe the evolution of average molecular concentrations over time

The main alternative modeling paradigm, originally proposed by Gillespie

Recently there has been considerable interest in the application of model checking

We propose that PMC be used in addition to stochastic and deterministic simulation in order to amplify the understanding of the biological system. For example, PMC can give clues about the existence of some events that can be later checked with stochastic simulation through the recovery of traces where the specific event happens. It also can support biologists suggesting interesting but uncommon aspects that can be verified experimentally.

In this paper we will use PMC for the modeling and analysis of the sodium-potassium exchange pump (Na,K-pump) in a quantitative way. This pump is an important transport system present in all animal cell and responsible for keeping the potassium and sodium concentrations inside the cell, respectively, high and low. Low sodium concentrations and high potassium concentrations in the cell cytoplasm are essential for basic cellular functions such as excitability, secondary active transport and volume regulation. In the brain, about one-half of the Adenosine Tri-Phosphate (ATP) provided by oxidate metabolism is used to power the Na,K-pump

A formal specification of this system has already been developed using the

We will describe how the pump mechanism can be modeled using probabilistic model checking taking into consideration a discrete chemistry approach and the Law of Mass Action aspects. We also will present some significative properties about the pump reversibility that can be addressed directly with model checking, whereas with other traditional approaches, such as deterministic and stochastic simulation, they can not be easily covered. Finally, we will reason about the pump behavior in terms of trend labels for the transition rates of the pump reactions which compute if there is a greater probability that the system takes specific transitions. These trends allow us to identify, for example, why the Na,K-pump goes more slowly in the forward direction over time, justifying the long periods of time to exhibit its reversibility.

Methods

Sodium-potassium exchange pump

The sodium-potassium exchange pump is found in the plasma membrane of virtually all animal cells and is responsible for the active transport of sodium and potassium across the membrane. One important characteristic of this pump is that both sodium and potassium ions are moving from areas of low concentration to high concentration, i.e., each ion is moving against its concentration gradient. This type of movement can only be achieved using the energy from the hydrolysis of ATP molecules. Figure _{1} and _{2}, with inward-facing (_{1}) and outward-facing (_{2}) binding sites for sodium ions (Na^{+} ) and potassium ions (K^{+}), respectively. The intracellular and extracellular forms of Na^{+} and K^{+} ions are explicitly identified as _{i} is the inorganic phosphate group and _{i}_{i}^{th} step in the cycle. For example, _{1} is the forward rate for the first step reaction _{i} ~ P indicates that the phospharyl group is covalently bound to _{i}. The pump mechanism is decomposed into a set of six elementary and reversible reactions. The enzyme in its conformation _{1} and with ATP already bound binds to three sodium ions inside the cell (step 1). This reaction stimulates ATP hydrolysis and then the release of Adenosine D—Phosphate (ADP) inside the cell, forming a phosphorylated enzyme intermediate (step 2). Extrusion of Na^{+} ions is completed by a conformational change (_{2}) and dissociation of the resulting complex (step 3). In this new shape, the pump has high affinity with potassium ions. Then, two potassium ions outside the cell bind to the pump enzyme and because of this reaction the enzyme is dephosphorylated (step 4). A further conformational change in which the enzyme binds ATP (step 5) is followed by the release of the two potassium ions inside the cell (step 6). Finally, the pump enzyme restores its original form that is capable of reacting with

The sodium-potassium exchange pump mechanism.

**The sodium-potassium exchange pump mechanism.** The Na,K-pump that moves two potassium ions from outside the cell to inside and three sodium ions from inside the cell to outside by the breakdown of ATP molecules.

Reaction scheme for sodium-potassium pump.

**Reaction scheme for sodium-potassium pump.** Reaction scheme for sodium-potassium pump mechanism based on Albers-Post model. _{1} and _{2} refer to conformationally distinct form of the pump, ^{+} and ^{+}, respectively.

Experimental data associated with the sodium-potassium pump cycle

Parameter

Value

Units

[

0.02200

M

[

0.14000

M

[

0.12700

M

[

0.01000

M

[ATP]

0.00500

M

[_{i}

0.00495

M

[

0.00006

M

_{1}

2.5 × 10^{11}

M^{–3}s^{–1}

_{2}

10^{4}

s^{–1}

_{3}

172

s^{–1}

_{4}

1.5 × 10^{7}

M^{–2}s^{–1}

_{5}

2 × 10^{6}

M^{–1}s^{–1}

_{6}

1.15 × 10^{4}

s^{–1}

_{1}

10^{5}

s^{–1}

_{2}

10^{5}

M^{–1}s^{–1}

_{3}

1.72 × 10^{4}

M^{–3}s^{–1}

_{4}

2 × 10^{5}

M^{–1}s^{–1}

_{5}

30

s^{–1}

_{6}

6 × 10^{8}

M^{–2}s^{–1}

cell volume

10^{–12}

l

temperature

310

K

Normal physiological parameters associated with the scheme in Fig. _{i}_{i}_{i}

Probabilistic model checking

Suppose _{0} is a starting state, _{0}, _{0} |= _{≥θ}(_{≥0} denote the set of non-negative reals and

•

• _{≥0} is the transition rate matrix, which assigns rates to each pair of states;

• ^{AP} is a labelling function which associates each state with a set of atomic propositions.

The probability of a transition between states ^{–R(s,s′).t}. The time spent in state _{s′∈S }

where _{≥0}. There are two types of CSL properties: transient **X** (next) operator and the **U**^{I}**X**Φ is true if Φ is satisfied in the next state, whereas Φ_{1 }**U**^{I}_{2} is true if Φ_{2} is held at some time instant in the interval _{1} holds. Other operators can be derived from this minimal set of CSL operators. Two of them, which will be used in this work, are the **F**^{I}**G**^{I}**U**, **F** and **G** which means that the interval is [0, ∞].

Furthermore, PRISM lets a CTMC be augmented with **rewards,** which are structures that associate real values with states or transitions. The state-rewards are accumulated in proportion to the time spent in the state, whereas the transition-rewards are accumulated each time the transition is taken. In PRISM, these are described using the

**rewards** ”**endrewards**

construct and are specified using multiple reward items of the form

to describe state-rewards and transition-rewards, respectively. In the previous definition,

Given the definition of the reward items, some properties can be used to recover amounts related to them. For example, the property ”what is the expected number of reactions between species A and B before a reaction between species A and C happens?” can only be asked with the reward mechanism. Two of the property types related to rewards which will be used in this work are

PRISM algorithm

The techniques that are implemented in PRISM to solve the PMC problem for CTMC models with rewards include

Sodium-potassium pump specification

Discrete chemistry

The entities in our model are ion species (_{i}, ADP) and the Na,K-ATPase (the pump enzyme) which can interact through six elementary reactions (see Fig.

In order to convert the initial amount of molecules and ions given in molarity ([

where _{A} is the Avogadro constant.

Moreover, in order to convert the rates from continuous chemistry to discrete chemistry we have used the Gillespie’s conversion

where _{2}, the reagents are

Law of mass action

The law of mass action states that a reaction rate is proportional to the concentration of its reagents. Then, we have to take into account the reagent concentrations in our model. Regarding the discrete chemistry conversion discussed in last subsection and the fourth step in the Na,K-pump cycle (see Fig.

the final rate _{4} is given as follows:

Given a reagent species, we have to raise its concentration to its molecularity. The final rates for the other sodium-pump mechanism steps are obtained similarly, see

PRISM specification

We now illustrate how to specify our Na,K-pump model in the PRISM language. Part of the model is presented in **ctmc** and comprises a set of **na**, **k**, **atp**, **adp**, **p** and the **pump** enzyme. There are also two finite-ranging variables in k module: **pump** and each of them represents one possible enzyme state, according with the cycle in Fig.

The behavior of each module, i.e. the changes in states which it can undergo, is specified by a number of

Transitions in different modules labelled with the same action occur simultaneously. The rate of synchronized transitions is equal to the product of the individual rates of the commands of the different modules that synchronize. In our model, the required updates when the fourth pump reaction (given in (3)) happens are represented by the commands labelled with action **r4**. Then, there is a decrease in the number of potassium ions outside the cell and the pump changes its state. The final rate when this reaction happens is ^{2}. We can extend our existing model by allowing more than one pump to occur in the system using the **NP** constant, which can assume any integer value to represent the number of pumps.

Finally, we add reward structures to our model as shown in **kOut** and **time**. The former assigns the current number of potassium ions outside the cell to every state in the system. This can be used to compute the expected amount of potassium ions outside the cell in a specific time for example. The latter simply assigns a state-reward of 1 to all states in the model and it is useful to analyze the total expected time before an event occurs.

Results and discussion

In the following analysis, all properties have been obtained considering a model with only one pump. Moreover, Table ^{–20} liters for analysis in the following sections. This type of abstraction strategy is common for modeling biological systems as discussed in

Model size ranging the cell volume.

Cell Volume(l)

#States

#Transitions

Time to Build

Time to Check

10^{–22}

9

16

0.03 sec

2 min 45.54 sec

10^{–21}

32

62

0.29 sec

51.94 sec

10^{–20}

194

386

47.45 sec

4 min 45.35 sec

10^{–19}

1 838

3 674

1 h 48 min 29.03 sec

1 h 2 min 18.98 sec

10^{–18}

?

?

> 7 days

?

Model size (

Discovering rare events

Uncommon events can have a significant impact in any system and particularly in biological systems. For example, if a particular combination of reagents can cause a pump to block permanently, it can cause cell death. No matter how unlikely this event is, if it happens the consequences are critical. Traditional analysis methods such as stochastic simulations can miss uncommon or rare events, because they simulate random paths in the evolution of the system, and if the event is rare, it is not likely that it will be simulated in a viable amount of time. PMC, however, can identify these events by looking for them. By stating a property that is true if such an event occurs, PMC can identify the conditions for its occurrence, and as a consequence, uncover hidden but potentially important behaviors in the system.

Our first analysis shows how model checking can be used to identify rare events in the Na,K-pump. Figure

Traditional solutions for potassium outside the cell in the sodium-potassium pump model.

**Traditional solutions for potassium outside the cell in the sodium-potassium pump model.** Potassium concentration in M outside the cell over time for the ODE approach (dashed line and y axis on the right) and count of potassium ions outside the cell given by a simulation trace (solid line and y axis on the left).

As can be seen in the ODE approach, the potassium outside the cell is decreasing until around 2 seconds, and its concentration converges on about 0.0018 M. However, this average behavior hides some important traces, as it is shown in the same figure, where the potassium count outside the cell, after the fast decrease until about 2 time units, will oscillate around 12 and might end, i.e. it can eventually reach value 0. However, the probability of this event ^{–3}) during the first 10 seconds. This probability value was determined using the CSL property

Thus, whereas a deterministic simulation

However PMC can provide stochastic simulation with some hints in this sense. As it lets us know in advance that the rare event happens with probability equal to 6.33 × 10^{–3} in the first 10 seconds, if the stochastic simulation time being considered is 10, in a sample of 1000 traces, for example, about 6 or 7 of them will probably show the rare event.

It is also important to note that in PMC the time is continuous, while in stochastic simulation it is discrete. Hence, if the duration of an event of interest is smaller than the time step being considered in the simulation, it will not be captured, whereas it will be considered in the PMC model. As will be shown, PMC can give some clues for stochastic simulation in order to address these issues. The CSL property shown below (followed by the model checker answer)

ensures that in all model traces the potassium outside the cell, in fact, will eventually end. Additionally, in order to know about the expected time for this event to happen, properties (6) and (7) can be used:

Property (6) means

Finally, the following properties are used to reason about the

There is more than one state where potassium outside the cell is over and, therefore, **min** and **max** are used to return the minimum and maximum expected time to reach a state where

where ”kOutOver” is a label to

Thus, two significant events in the system

Reversibility of the sodium-potassium pump

Due to the fact that there are backwards and forwards transitions for all reactions involved in the Na,K-pump mechanism, as is shown in Fig. _{i}.

Without loss of generality, we will study this reversible pump behavior in terms of the potassium amount outside the cell, which will be the species under observation. We can see this pump reversibility as an infinite oscillation between two values, the initial amount of potassium outside the cell,

where

Understanding the pump cycle

In this section we present a study of the Na,K-pump mechanism in terms of the rates in the cycle shown in Fig.

We now introduce some definitions and extensions in the previous PRISM pump model that will be used later. First, we compute the positive or ascending trend _{i}

where _{s′∈S} r_{i}_{i}_{j}_{j}

Thus, we add trend formulas to the previous PRISM model for all transition rates using PRISM resources (_{1}.

The rate transition _{1} is computed by the formula **rate_r1** and it is different to 0 when the current pump state is _{1}. ATP and there is enough sodium inside the cell _{1} is determined in the same way as described in Sect. _{1} and the formula exit rate represents their summation. The probability that _{1} is taken in the current state is given by formula **rate_r1_d**, whereas the label **trend_r1_up** represents if _{1} really has an ascending trend, i.e. ↑ _{1} is 1. Now, we can use the CSL property (13) to identify the rates that never have a positive trend during the system evolution and, consequently those rates that always have an ascending trend:

In our pump model, [

The results for property (13) are summarized in Fig. _{1}.ATP) are labelled with the rates for the transition between the current state and the next state, which is given by the direction of the arrow. Associated with each arrow, there is also a sign that indicates if the transition rate has always a positive trend (+) or a negative trend (-), and, finally, if the trend can be negative and positive during the system evolution (+/-).

Summarization of the rate trends in sodium-potassium pump.

**Summarization of the rate trends in sodium-potassium pump.** Summarization of the trend for all transition rates in the pump cycle presented in Fig.

We can see that the forward rates _{1} and _{2} always have a negative trend, while _{6} always has a positive trend during the system evolution. Moreover, the trends for the forward rates _{3}, _{4} and _{5} can be positive or negative, depending on the changes in the amount of substrates during the pump evolution.

In order to identify the moment when these forward transition rates which don’t have only a positive or negative trend during the system evolution change their trends, we have again extended our PRISM model with the following transition-rewards

In the **plusKout** is a reward that assigns 1 to each transition from the state K_{2}._{2} to state _{2} ~ P, which results in the releasing of two potassium ions outside the cell. On the other hand **minusKout** is a reward that assigns 1 to each transition from the state _{2} ~ P to the state K_{2}._{2}, which results in the consumption of two potassium ions outside the cell. CSL property (14) determines _{i} starts to have a positive trend

Using property (14), we can see that _{3}, _{4} and _{5} (forward rates) start to have a negative trend only when the potassium outside the cell is, respectively, 21, 7 and 7 (the initial amount of potassium outside the cell in our model is 61).

Thus, we can divide the pump operation into three main steps, as is shown in Fig. _{1} and _{6}, once _{3} is taken, the system might complete easily the cycle in the forward direction, because the forward rates _{3}, _{4}, _{5} and _{6} have a positive trend in the most of the time. The backward rate _{2} needs that the pump goes in the forward direction awhile, increasing the amount of ADP inside the cell, in order to exhibit a positive trend. When potassium outside the cell reaches the value 21, the rates _{3} and _{2} changes their trends, starting the intermediate step (B). In this step, the pump can still move in forward direction. The last step (C), starts when the potassium outside the cell reaches the value 7, causing changes in the trends of the forward rates _{4} and _{5}. First, rate _{4} no longer has a positive trend, while the negative trend of the backward rate _{3} is replaced by a positive one. This happens due to the increase of sodium outside the cell, which gives strength to _{3}, and the decrease of potassium outside the cell, which weakens _{4}. Second, the forward rate _{5} also stops exhibiting a positive trend, whereas the trend of the backward rate _{4} starts to be ascending. This change is caused by the accumulation of P_{i} inside the cell and the reduction of ATP due to the pump movement in the forward direction. In step (C) there is a low probability, although is not impossible, that the pump continues its operation in the forward direction, given that the only forward rate with positive trend is _{6}, delaying the depletion of potassium outside the cell. In fact, there is a strong general trend for the pump to move backwards, returning to the intermediate step, where the system stays most of the time. Additionally, the pump can move backwards from the intermediate step, returning to the initial configuration. However, this takes long periods of time, given that it is necessary to move against the positive trends of the forward rates _{3}, _{4}, _{5} and _{6} in the initial step.

The main steps of the sodium-potassium pump system in terms of rate trends.

**The main steps of the sodium-potassium pump system in terms of rate trends.** Steps of the sodium-potassium pump determined by the transition rate trends. There are arrows only for the rates that exhibit positive trends. Pump states in gray are those whose the rate in the forward direction has positive trend, whereas those in white have the backward rate exhibiting a positive trend. The thickness of the central arrow in the forward direction indicates the strength to the general trend of the pump in this direction. Thus, the thicker the arrow, the greater the tendency of the pump to run in the forward direction. (A) Initial step. The double circle represents the initial state of the system. (B) Intermediate step. (C) Final step.

As shown in the previous sections, the depletion of potassium outside the cell and the pump reversibility are events that can happen in the pump model. However, they can take longs periods of time to be completed. So the study of this section is important to indicate the reasons for this delay. For example, it is possible to see that the first obstacle in the normal operation of the pump is the accumulation of ADP inside the cell which causes the reversion of the _{3} trend. This may indicate a specific aspect of the system that merits further studies. This result may lead to a more precise study because it tells us in detail what has happened (accumulation of ADP inside the cell) and not simply that the pump has reversed its behavior. Results such as these can uncover important hidden behaviors that can speed up further experiments and increase their accuracy.

Validation of the PRISM model

In this section we will show that our PRISM model can produce similar results when compared to the stochastic and deterministic simulations. Property (15) allows us to know the expected amount of potassium outside the cell in time

The label ”kOut” is a reward name defined as shown in

PRISM curves for the sodium-potassium pump model.

**PRISM curves for the sodium-potassium pump model.** Expected amount of potassium outside the cell over time (dashed line) given by property (15) and count of potassium ions outside the cell given by a PRISM simulation trace (solid line).

Conclusions

In this work we use a stochastic modeling approach and

We have presented a quantitative formal specification of the Na,K-pump, based on a set of elementary reactions. All the process to build the model in the PRISM tool, taking into account a discrete chemistry and the Law of Mass Action has been described. Moreover, we have also checked some rare quantitative properties such as the depletion of sodium potassium outside the cell and the pump reversibility that can be addressed easily using model checking, whereas with the other traditional approaches, such as simulation and ODE methodology, it can be difficult.

Furthermore, using model checking we have shown that these events happen infinitely often. These properties cannot be addressed using simulations, given that they are, by definition, time-finite approaches and, additionally, do not construct the mathematical model which represents all possible states that a system can be.

Moreover, we have used transition rate trends, in order to understand the pump behavior and why it takes a long period of time to express completely the reversibility property.

Finally, we have shown that probabilistic model checking can be used along with other well established approaches to extend the pump behavior knowledge. Then, after we know that the event

In practice, the main objective of this work is to provide biologists with hints related to important and interesting events that should be checked in more detail using biological experiments. Thus, biological experiments could be preceded by model checking analysis, which can be used very efficiently, for example, for rejecting impossible hypothesis or for orienting biologists toward logical possible situations. In this way, instead of performing many experiments, the biologists will focus on those that are as pointed out as possible by the mathematical model.

Future works include making our Na,K-pump model more dynamic, adding other actual cell membrane aspects and systems. In order to deal with the large state space, given the big number of ions and molecules, an abstraction of CTMCs based on discrete levels of concentrations, namely CTMC with levels

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

MAC and SVAC carried out the model checking studies, analysis and validation of the results. MAC studied the biological system, found the experimental data related to it, built the model in the PRISM tool and created the CLS properties about the biological system. SVAC and ACF participated in the study design and coordination. All authors helped to draft the manuscript and were involved in the read, review and approval of the final manuscript.

Acknowledgements

This study received financial support from CAPES, FAPEMIG and CNPQ. The authors wish to thank to PRISM team to the tool support and Jader Cruz for his assistance in the validation process of the results.

This article has been published as part of