particles in a box Everything with a Single Particle
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IMPORTANCE OF SINGLE-MOLECULE ANALYSIS IN CELLULAR BEHAVIOR
Biological systems often operate at the single-molecule level, where stochastic fluctuations play a critical role in function. In cellular environments, understanding how individual molecules transition between states is essential for interpreting key biochemical processes. For example, in patch-clamp experiments, researchers frequently analyze the behavior of a single ion channel, revealing insights into gating dynamics and membrane potential regulation. These single-molecule observations help bridge the gap between macroscopic kinetics and the intricate, probabilistic nature of molecular interactions within cells.
WHAT IS IT?
Analyzing Chemical Equilibrium and Kinetics Using a Single-Molecule Simulation
Chemical reactions occur through the stochastic movement of individual molecules. Unlike bulk kinetics, which averages molecular behavior over thousands of molecules, this simulation focuses on a single molecule to directly observe state occupancy, transition times, and their connection to equilibrium and reaction rates.
This model simulates a single molecule that can switch between two states. The states differ in standard enthalpy (H°) and entropy (S°). The height difference represents ΔH°, while the width of each sector represents ΔS°. The two sectors are separated by a barrer (Ea).
HOW IT WORKS
State Occupancy and Equilibrium
The time a molecule spends in each state reflects the probability of being in that state. Over long periods, this probability corresponds to the equilibrium constant (Keq):
Keq = (residence time in red) / (residence time in green)
Since equilibrium is governed by Gibbs free energy (ΔG), Keq can be used to estimate ΔG:
ΔG = ΔG° + RT ln(Keq)
Thus, tracking residence time in each state provides direct estimates of Keq and ΔG, offering a microscopic understanding of equilibrium.
WAITING TIME AND RATE CONSTANTS
Defining Waiting Time
The waiting time represents how long a molecule remains in a state before transitioning. This waiting time is related to the rate constant (k) for the reaction:
k₁ = 1 / (average waiting time in state 1)
k₋₁ = 1 / (average waiting time in state 2)
Using Rate Constants to Estimate Equilibrium
At equilibrium, forward and reverse transition rates balance:
k₁[A] = k₋₁[B]
Rearranging:
Keq = k₁ / k₋₁
This shows that equilibrium constants can be obtained both from state occupancy and from rate constants derived from waiting times.
ACTIVATION ENERGY (Ea) FROM RATE CONSTANTS
According to the Arrhenius equation, the rate constants follow:
k = A * exp(-Ea / (R * T))
Taking the natural logarithm:
ln(k) = ln(A) - (Ea / R) * (1 / T)
By measuring rate constants across different temperatures and plotting ln(k) vs. 1/T, the slope gives Ea/R, allowing estimation of Ea for both forward and reverse reactions:
Ea₁ = slope forward * R
Ea₋₁ = slope reverse * R
In this single molecule simulation, the estimation of the rate constants by the waiting time at different temperatures can be used to assess the Ea for the forward and reverse reaction
KEY TAKEAWAYS
- Time spent in each state determines Keq, which allows the estimation of ΔG.
- Waiting time provides rate constants k₁ and k₋₁.
- Rate constants can be used to estimate Keq, reinforcing equilibrium calculations.
- Arrhenius analysis on rate constants enables Ea estimation.
This single-molecule approach connects stochastic molecular behavior to fundamental kinetic and thermodynamic principles, providing insights into reaction equilibrium and activation energy.
COPYRIGHT AND LICENSE
Copyright 1997 Uri Wilensky.
This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.
Commercial licenses are also available. To inquire about commercial licenses, please contact Uri Wilensky at uri@northwestern.edu.
This model was created as part of the project: CONNECTED MATHEMATICS: MAKING SENSE OF COMPLEX PHENOMENA THROUGH BUILDING OBJECT-BASED PARALLEL MODELS (OBPML). The project gratefully acknowledges the support of the National Science Foundation (Applications of Advanced Technologies Program) -- grant numbers RED #9552950 and REC #9632612.
This model was converted to NetLogo as part of the projects: PARTICIPATORY SIMULATIONS: NETWORK-BASED DESIGN FOR SYSTEMS LEARNING IN CLASSROOMS and/or INTEGRATED SIMULATION AND MODELING ENVIRONMENT. The project gratefully acknowledges the support of the National Science Foundation (REPP & ROLE programs) -- grant numbers REC #9814682 and REC-0126227. Converted from StarLogoT to NetLogo, 2002.
Comments and Questions
globals [ box-edge1 box-edge2 leftbox-edge rightbox floor1 floor2 entropy1 ; entropy2 ; temperature R yboltz enthalpy1 ; enthalpy2 ; scale Eafloor initialstate ; particles in initially in state 1 ΔG ΔG0 ΔG0e -ΔG0e ΔS ΔH Kin1 Kin2 K tracer a stateLeft stateRight waitingLeft waitingRight previous-state waitingTimes1to2 waitingTimes2to1 logWaitingTimes1to2 logWaitingTimes2to1 waitingTime ] breed [ particles particle ] breed [ flashes flash ] flashes-own [birthday] particles-own [ mass energy ;; particle info last-collision state ] to setup clear-all set waitingTimes1to2 [] set waitingTimes2to1 [] set logWaitingTimes1to2 [] set logWaitingTimes2to1 [] set-default-shape particles "circle" set-default-shape flashes "plane" set R 1.9858775 / 1000 ;constante de gases en kcal/mol/K ; set entropy1 0.3 set entropy1 1 - entropy2 set enthalpy1 0 ifelse enthalpy1 <= enthalpy2 [set floor1 0 set floor2 enthalpy2 * scale] ; scale 1kcal= scale y coordenate [set floor2 0 set floor1 -1 * scale * enthalpy2] ; scale 1kcal=scale y coordinate set Eafloor max (list floor1 floor2) + Ea * scale ; set initialstate 0.1 set box-edge2 (max-pxcor) set box-edge1 min-pxcor + (entropy1 * max-pxcor * 2) set ΔH enthalpy2 - enthalpy1 set ΔS R * ln (entropy2 / entropy1) ;calculo entropía set ΔG0 ΔH - temperature * ΔS ;calculo Δ G cero teórica make-box set initialstate 1 - avance make-particles update-variables reset-ticks set a 1 end to go ask particles [ bounce ] ask particles [ timeInState ] update-variables update-plots display set a a + 1 if single = true [stop] ; if a > 1 [stop] end to update-variables set ΔG0 ΔH - temperature * ΔS ;calculo Δ G cero teórica set Kin1 stateLeft;count turtles with [state = 1] set Kin2 stateRight; count turtles with [state = 2] set K Kin2 / (Kin1 + 1E-10) set ΔG0e (- R * temperature * ln (K + 1E-10)) set -ΔG0e -1 * ΔG0e set ΔG R * temperature * ln (K + 1E-10) + ΔG0 ; set-current-plot "Waiting Time" ; set-plot-pen-mode 1 ; set-histogram-num-bars 10 ; histogram waitingTimes1to2 ;; Waiting time for state 1 → state 2 end to bounce ;; particle procedure ifelse state = 1 [ set yboltz floor1 - scale * ln (random-float 1) * R * temperature if yboltz >= Eafloor [set xcor min-pxcor + random (max-pxcor - min-pxcor)] ] [ set yboltz floor2 - scale * ln (random-float 1) * R * temperature if yboltz >= Eafloor [set xcor min-pxcor + random (max-pxcor - min-pxcor)] ] ifelse yboltz > max-pycor [set ycor max-pycor] [set ycor yboltz] ifelse (xcor >= min-pxcor and xcor < box-edge1) [set state 1 set color green] [set state 2 set color red] end ;; Particle Procedure - Track Waiting Time Before Transitions to timeInState ;; Increment time in the current state ifelse state = 1 [ set stateLeft stateLeft + 1 ] [ set stateRight stateRight + 1 ] ;; Increment waiting time in the current state set waitingTime waitingTime + 1 ;; Check if transition occurs and store waiting time if state != previous-state [ if previous-state = 1 [ set waitingTimes1to2 lput waitingTime waitingTimes1to2 set logWaitingTimes1to2 lput ln waitingTime logWaitingTimes1to2 set waitingTime 0 ;; Reset waiting time for next cycle ] if previous-state = 2 [ set waitingTimes2to1 lput waitingTime waitingTimes2to1 set logWaitingTimes2to1 lput ln waitingTime logWaitingTimes2to1 set waitingTime 0 ;; Reset waiting time for next cycle ] ] ;; Update previous-state for next tick comparison set previous-state state end ;;; ;;; drawing procedures to make-box ; white the part of the box that is inactive ifelse floor1 <= floor2 [ ask patches with [ (pxcor > box-edge1) and (pycor < floor2) ] [ set pcolor white ] ] [ ask patches with [ (pxcor < box-edge1) and (pycor < floor1) ] [ set pcolor white ] ] ; limite superior ask patches with [ pycor > max-pycor - 5 ] [ set pcolor blue ] ; limite inferior ask patches with [ (pycor < floor1 + 5 and pycor > floor1 ) and (pxcor <= box-edge1) ] [ set pcolor yellow ] ask patches with [ (pycor < floor2 + 5 and pycor > floor2) and (pxcor >= box-edge1) ] [ set pcolor yellow ] ; limite izquierdo ask patches with [ (pxcor < min-pxcor + 5) and (pycor >= floor1) ] [ set pcolor yellow ] ; limite derecho ask patches with [ (pxcor > max-pxcor - 5) and (pycor >= floor2) ] [ set pcolor yellow ] ; limite central ask patches with [ (pxcor < box-edge1 + 3 and pxcor > box-edge1 - 3) and (pycor <= Eafloor) ] [ set pcolor yellow ] end ;;; ;; creates initial particles to make-particles create-particles number-of-particles [ set size 15 ifelse random-float 1 < initialstate [ set xcor min-pxcor + random (box-edge1 - min-pxcor) set yboltz floor1 - scale * ln (random-float 1) * R * temperature ifelse yboltz > max-pycor [set ycor max-pycor] [set ycor yboltz] ] [ set xcor box-edge1 + random (box-edge2 - box-edge1) set yboltz floor2 - scale * ln (random-float 1) * R * temperature ifelse yboltz > max-pycor [set ycor max-pycor] [set ycor yboltz] ] ifelse (xcor >= min-pxcor and xcor < box-edge1) [set state 1 set color green] [set state 2 set color red] ;;random-position ] set K count turtles with [state = 2] / (count turtles with [state = 1] + 0.00000000000000000000001) set ΔG0e (- R * temperature * ln (K + 0.0000000000001)) set ΔG R * temperature * ln (K + 0.0000000000001) + ΔG0 end ; Copyright 1997 Uri Wilensky. ; See Info tab for full copyright and license.
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