^{1}, Christian M. Lastoskie

^{2}, Carol A. Fierke

^{3}and Ann Marie Sastry

^{4,a)}

### Abstract

We present a mobile trap algorithm to sense zinc ions using protein-basedsensors such as carbonic anhydrase (CA). Zinc is an essential biometal required for mammalian cellular functions although its intracellular concentration is reported to be very low. Protein-basedsensors like CA molecules are employed to sense rare species like zinc ions. In this study, the zinc ions are mobile targets, which are sought by the mobile traps in the form of sensors. Particle motions are modeled using random walk along with the first passage technique for efficient simulations. The association reaction between sensors and ions is incorporated using a probability upon an ion-sensor collision. The dissociationreaction of an ion-bound CA molecule is modeled using a second, independent probability . The results of the algorithm are verified against the traditional simulation techniques (e.g., Gillespie’s algorithm). This study demonstrates that individual sensor molecules can be characterized using the probability pair , which, in turn, is linked to the system level chemical kinetic constants, and . Further investigations of CA-Zn reaction using the mobile trap algorithm show that when the diffusivity of zinc ions approaches that of sensor molecules, the reaction data obtained using the static trap assumption differ from the reaction data obtained using the mobile trap formulation. This study also reveals similar behavior when the sensor molecule has higher dissociation constant. In both the cases, the reaction data obtained using the static trap formulation reach equilibrium at a higher number of complex molecules (ion-bound sensor molecules) compared to the reaction data from the mobile trap formulation. With practical limitations on the number sensors that can be inserted/expressed in a cell and stochastic nature of the intracellular ionic concentrations, fluorescence from the number of complex sensor molecules at equilibrium will be the measure of the intracellular ion concentration. For reliable detection of zinc ions, it is desirable that the sensors must not bind all the zinc ions tightly, but should rather bind and unbind. Thus for a given fluorescence and with association-dissociation reactions between ions and sensors, the static trap approach will underestimate the number of zinc ions present in the system.

Support for this work provided by the W. M. Keck Foundation to three of the authors (A.M.S., C.M.L. and C.A.F.) and the National Science Foundation through an NSF PECASE to the last author (A.M.S.) is gratefully acknowledged. Computations were performed on a Sun Fire Cluster V480, a gift of Sun Microsystems, Incorporated to two of the authors (C.M.L. and A.M.S.). The authors also gratefully acknowledge this equipment support.

I. INTRODUCTION

II. METHODS

A. Assumptions and implementation of the mobile trap algorithm

B. Verification of implementation: Satisfaction of the law of mass action

C. Correlations between and and between and

D. Comparative study between static and mobile traps

III. RESULTS AND DISCUSSION

A. Verification of implementation

B. Correlation between and

C. Correlation between and

D. Comparison with Gillespie’s algorithm

E. Static versus the mobile traps

F. Contrast with the well-stirred hypotheses and general implications of work

G. Implications of findings for intracellular modeling of zinc transport and comparisons with other approaches

IV. CONCLUSIONS AND FUTURE WORK

## Figures

(a) Movement of a zinc ion (smaller circle) through randomly distributed CA molecules (larger circles) and binding to a CA molecule. The paths of CA molecules are not shown. (b) Various variables used in simulations that are obtained from the current particle positions.

(a) Movement of a zinc ion (smaller circle) through randomly distributed CA molecules (larger circles) and binding to a CA molecule. The paths of CA molecules are not shown. (b) Various variables used in simulations that are obtained from the current particle positions.

Reaction data obtained from the mobile trap algorithm with a of 0.9. The numbers and diffusion coefficients of particles of types and are 6 and 12 and and , respectively. Reaction data were fitted with an analytical solution with of and uncertainty of 54%.

Reaction data obtained from the mobile trap algorithm with a of 0.9. The numbers and diffusion coefficients of particles of types and are 6 and 12 and and , respectively. Reaction data were fitted with an analytical solution with of and uncertainty of 54%.

Survival time vs concentration of CA with a rectangular hyperbola curve fit [per Eq. (19)] for three different values, 0.05, 0.01, and 0.005. Simulation times shown are average values for 100 realizations of each case.

Survival time vs concentration of CA with a rectangular hyperbola curve fit [per Eq. (19)] for three different values, 0.05, 0.01, and 0.005. Simulation times shown are average values for 100 realizations of each case.

Average reaction rate (1/survival time) vs concentration of CA with a straight-line curve fit [per Eq. (19)] for three different values, 0.05, 0.01, and 0.005. Simulation times shown are average values for 100 realizations of each case.

Average reaction rate (1/survival time) vs concentration of CA with a straight-line curve fit [per Eq. (19)] for three different values, 0.05, 0.01, and 0.005. Simulation times shown are average values for 100 realizations of each case.

A straight line passing through vs data allows interpolation for a particular value corresponding to a particular value.

A straight line passing through vs data allows interpolation for a particular value corresponding to a particular value.

A typical plot of the number of free CA molecules as a function of time, obtained from selective CA dissociation simulations, with 12 complex CA molecules, and and of 0.0055 and , respectively. The time duration between two successive CA selections for dissociation was ; the total simulated duration was . For this case, the value is , which is very close to the experimental value of .

A typical plot of the number of free CA molecules as a function of time, obtained from selective CA dissociation simulations, with 12 complex CA molecules, and and of 0.0055 and , respectively. The time duration between two successive CA selections for dissociation was ; the total simulated duration was . For this case, the value is , which is very close to the experimental value of .

Comparison between reaction data obtained from our implementation of Gillespie’s algorithm and the mobile trap simulations. The numbers of CA and zinc particles were 12 each, and the simulated duration was . Data represent average values for ten realizations for each case.

Comparison between reaction data obtained from our implementation of Gillespie’s algorithm and the mobile trap simulations. The numbers of CA and zinc particles were 12 each, and the simulated duration was . Data represent average values for ten realizations for each case.

Number of complex CA-Zn molecules as a function of time when 20 zinc molecules react with a hypothetical variant of CA having a of 0.055 via the forward CA-Zn reaction. The number of CA molecules is 200. Reaction curves are shown for stationary (solid) and mobile (dotted) CA molecules. Cases shown are for fast zinc ions reacting with CA molecules (a) and slow zinc ions reacting with CA molecules (b).

Number of complex CA-Zn molecules as a function of time when 20 zinc molecules react with a hypothetical variant of CA having a of 0.055 via the forward CA-Zn reaction. The number of CA molecules is 200. Reaction curves are shown for stationary (solid) and mobile (dotted) CA molecules. Cases shown are for fast zinc ions reacting with CA molecules (a) and slow zinc ions reacting with CA molecules (b).

Number of complex CA-Zn molecules as a function of time when 200 zinc ions react with 20 CA molecules in the reversible reaction. The reaction curves are for stationary CA (solid curve) as well as mobile CA (dotted curve) molecules. The value was 0.055 and the value was . Cases shown are for fast zinc ions reacting with CA molecules (a) and slow zinc ions reacting with CA molecules (b).

Number of complex CA-Zn molecules as a function of time when 200 zinc ions react with 20 CA molecules in the reversible reaction. The reaction curves are for stationary CA (solid curve) as well as mobile CA (dotted curve) molecules. The value was 0.055 and the value was . Cases shown are for fast zinc ions reacting with CA molecules (a) and slow zinc ions reacting with CA molecules (b).

The point of time (deviation time) at which the reaction data from the mobile trap and the static trap methodology deviate from each other, plotted as a function of relative zinc mobility, the probability of association and dissociation; (a) normalized time of deviation as a function of relative zinc mobility, (b) reciprocal of deviation time as a function of the probability of association, and (c) deviation time as a function of the probability of dissociation.

The point of time (deviation time) at which the reaction data from the mobile trap and the static trap methodology deviate from each other, plotted as a function of relative zinc mobility, the probability of association and dissociation; (a) normalized time of deviation as a function of relative zinc mobility, (b) reciprocal of deviation time as a function of the probability of association, and (c) deviation time as a function of the probability of dissociation.

## Tables

Prior work in measurement of intracellular zinc concentration.

Prior work in measurement of intracellular zinc concentration.

Experimentally determined and values for various variants of human carbonic anhydrase. Asterisks indicate unreported values.

Experimentally determined and values for various variants of human carbonic anhydrase. Asterisks indicate unreported values.

values of CA-Zn reaction obtained using the straight lines as well the rectangular hyperbola method. Asterisks indicate unreported values.

values of CA-Zn reaction obtained using the straight lines as well the rectangular hyperbola method. Asterisks indicate unreported values.

values for various initial numbers of CA and zinc particles. These data were obtained from the mobile trap implementation and the selective CA dissociation simulations.

values for various initial numbers of CA and zinc particles. These data were obtained from the mobile trap implementation and the selective CA dissociation simulations.

Average numbers of CA-Zn association events and complex CA dissociation events from ten mobile trap and Gillespie-type simulations. For the mobile trap simulations, and values were 0.0055 and , respectively. For Gillespie’s algorithm simulations, and values were and , respectively.

Average numbers of CA-Zn association events and complex CA dissociation events from ten mobile trap and Gillespie-type simulations. For the mobile trap simulations, and values were 0.0055 and , respectively. For Gillespie’s algorithm simulations, and values were and , respectively.

The on rate constant values for CA-Zn reactions involving (a) fast zinc ions reacting with mobile and stationary CA molecules and (b) slow zinc ions reacting with mobile and stationary CA molecules. The probability of association is 0.055 in each case.

The on rate constant values for CA-Zn reactions involving (a) fast zinc ions reacting with mobile and stationary CA molecules and (b) slow zinc ions reacting with mobile and stationary CA molecules. The probability of association is 0.055 in each case.

On rate and off rate constants for CA-Zn reactions involving fast and slow zinc ions reacting with mobile and stationary CA molecules.

On rate and off rate constants for CA-Zn reactions involving fast and slow zinc ions reacting with mobile and stationary CA molecules.

The values of constants obtained from curve fitting Eq. (22) in Figs. 8(b), 9(a), and 9(b) data. These values show that the reaction data from the static trap and the mobile trap methodologies are characterized by different exponents and the saturation points.

The values of constants obtained from curve fitting Eq. (22) in Figs. 8(b), 9(a), and 9(b) data. These values show that the reaction data from the static trap and the mobile trap methodologies are characterized by different exponents and the saturation points.

The values of parameters used to simulate Zn-CA association-dissociation reactions using the static trap and the mobile trap approaches. The values of the deviation times were obtained using simulations.

The values of parameters used to simulate Zn-CA association-dissociation reactions using the static trap and the mobile trap approaches. The values of the deviation times were obtained using simulations.

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