# CellularMetabolismLabyrinth

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## RUNNING DOWNHILL A FREE ENERGY LANDSCAPE THROUGH A DYNAMIC LABYRINTH.

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The model was built on top the GasLab Gravity Box from the Model Library

Wilensky, U. (2002). NetLogo GasLab Gravity Box model. http://ccl.northwestern.edu/netlogo/models/GasLabGravityBox. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

Balls are isolated in a fixed condition until an enzyme decreases the activation energy and allows the jumping to a different state, always downhill the free energy gradient. Individual balls can be followed activating Trace

This model is a representation of the metabolic network inside a cell. The analogy is a generalization of the model MolecularThermodynamicsInBoxes (http://modelingcommons.org/browse/one*model/6895#model*tabs*browse*info).

Suppose that all possible chemical reactions among all metabolites in the cell are represented by boxes as described in MolecularThermodynamicsInBoxes. The cell cannot change the shape of the boxes, which are determined by the physicochemical properties of reactants and products; the only parameter amenable to manipulation is the activation energy (Ea). It can only decrease the Ea by providing an enzyme that catalyzes the reaction. It is important to notice that the Ea for the great majority of the chemical reaction in the cell is high enough to prevent them to proceed at room temperature at a speed compatible with the cellular metabolism. Glucose is a stable compound at room temperature and will not be oxidized by atmospheric oxygen. In contrast, inside de cell, glucose is rapidly oxidized to CO2 and H2O. Therefore, the cellular strategy to guide the energy gradient available in the environment to fuel its activity is quite simple: to dig the appropriate channels lowering the necessary Ea to direct the energy flux. The general principle is then straightforward. The box shapes cannot be modified, only the walls separating the compartments can be manipulated. However, the task for these nano-machines is formidable: to direct the energy flux in such a way to assure its homeostasis in a self-regulated manner. Which enzymatic path will be active at the necessary level, according to the signals sensed from the environment, needs to be determined in an automatic way to guarantee the cell survival and proper function. The cell metabolism is represented by an intricate labyrinth lying downhill a energy landscape. Walls (Eas) prevent the flux of energy, which can only follows a few opened pathways. Like a deceiving maze, the walls change over time in a coordinated way that allows the building of more wall-regulation devices, ultimately preserving the maze function.

## Comments and Questions

globals [ tick-delta ;; how much we advance the tick counter this time through max-tick-delta ;; the largest tick-delta is allowed to be box-edge ;; distance of box edge from axes init-avg-speed init-avg-energy ;; initial averages avg-speed avg-energy ;; current averages fast medium slow ;; current counts percent-slow percent-medium percent-fast ;; percentage of current counts aggregate-list ;; list of the sums of the temperature at each height i z yboltz R temp ] breed [ particles particle ] breed [ flashes flash ] flashes-own [birthday] patches-own [d] particles-own [ speed mass energy ;; particle info last-collision ] to setup clear-all set-default-shape particles "circle" set-default-shape flashes "plane" set max-tick-delta 0.1073 set R 1.9858775 / 1000 set temp 310 make-particles ;; box has constant size... set box-edge (max-pxcor) ;; make floor ; ask patches with [ pycor = min-pycor ] ; [ set pcolor brown ] ask n-of 10 patches with [pxcor mod 5 = 0 and pycor mod 5 = 0 and pxcor > min-pxcor and pxcor < max-pxcor and pycor > min-pycor and pycor < max-pycor ] [set pcolor blue] ask patches with [pcolor = blue] [ build-boxes ] make-box ;update-variables ;set init-avg-speed avg-speed ;set init-avg-energy avg-energy reset-ticks end to update-variables ; set medium count particles with [color = green] ; set slow count particles with [color = blue] ; set fast count particles with [color = red] ; set percent-medium (medium / (count particles)) * 100 ; set percent-slow (slow / (count particles)) * 100 ; set percent-fast (fast / (count particles)) * 100 set avg-speed mean [speed] of particles set avg-energy mean [energy] of particles end to go if random 10 < 1 [pathway] make-box if(count particles < 10) [ make-particles] ask particles [ bounce ] ask particles [ move ] ask particles [ if collide? [check-for-collision] ] ifelse (trace?) [ ask particle 0 [ pen-down ] ] [ ask particle 0 [ pen-up ] ] tick-advance tick-delta if floor ticks > floor (ticks - tick-delta) [ update-variables update-plots ] calculate-tick-delta ask flashes with [ticks - birthday > 1] [ die ] display end to calculate-tick-delta ;; tick-delta is calculated in such way that even the fastest ;; particle will jump at most 1 patch length in a tick. As ;; particles jump (speed * tick-delta) at every tick, making ;; tick length the inverse of the speed of the fastest particle ;; (1/max speed) assures that. Having each particle advance at most ;; one patch-length is necessary for it not to "jump over" a wall ;; or another particle. ifelse any? particles with [speed > 0] [ set tick-delta min list (1 / (ceiling max [speed] of particles)) max-tick-delta ] [ set tick-delta max-tick-delta ] end to bounce ;; particle procedure ;; get the coordinates of the patch we will be on if we go forward 1 let new-patch patch-ahead 1 let new-px [pxcor] of new-patch let new-py [pycor] of new-patch if [pycor] of new-patch <= min-pycor [ask new-patch [ sprout-flashes 1 [ set color brown; white set birthday ticks set heading 0 ] ] ] if new-py <= min-pycor[ ifelse ([who] of self = 0) [ pen-up setxy 0 max-pycor - 2; random-xcor max-pycor - 2 set color (random-float 4.9) + 5 set heading (random 180) + 90 stop ] [die] ] ;; if we're not about to hit a wall, we don't need to do any further checks if [pcolor] of new-patch != brown and [pcolor] of new-patch != blue [ stop ] if [pcolor] of new-patch = blue [ set heading heading + 180 boltzmann stop ] ;; if hitting the top or bottom, reflect heading around y axis if ([pcolor] of new-patch = brown and [d] of new-patch = 1) [ set heading (180 - heading) boltzmann stop ] if ([pcolor] of new-patch = brown and [d] of new-patch = 0) [ set heading (- heading)] ; ask patches with [[pycor] of new-patch <= min-pycor] ; [ sprout-flashes 1 [ ; set color white ; set birthday ticks ; set heading 0 ; ] ; ] end to move ;; particle procedure ;; In other GasLab models, we use "jump speed * tick-delta" to move the ;; turtle the right distance along its current heading. In this ;; model, though, the particles are affected by gravity as well, so we ;; need to offset the turtle vertically by an additional amount. The ;; easiest way to do this is to use "setxy" instead of "jump". ;; Trigonometry tells us that "jump speed * tick-delta" is equivalent to: ;; setxy (xcor + dx * speed * tick-delta) ;; (ycor + dy * speed * tick-delta) ;; so to take gravity into account we just need to alter ycor ;; by an additional amount given by the classical physics equation: ;; y(t) = 0.5*a*t^2 + v*t + y(t-1) ;; but taking tick-delta into account, since tick-delta is a multiplier of t. setxy (xcor + dx * speed * tick-delta) (ycor + dy * speed * tick-delta - gravity-acceleration * (0.5 * tick-delta * tick-delta)) factor-gravity end to build-boxes set i 1 while [[pcolor] of patch-at 0 i != blue] [ ask patch-at 0 i [set pcolor brown set d 0] set i i + 1 ] set i 1 while [[pcolor] of patch-at i 0 != blue] [ ask patch-at i 0 [set pcolor brown set d 1] set i i + 1 ] end to pathway ask up-to-n-of random 2 patches with [pcolor = blue] [ set i random 40 + 1 set z random 4 ; show z ; if z = 1 or z = 3 [if random 2 < 2 [stop]] para que hayan más agujeros verticales while [([pcolor] of patch-at-heading-and-distance (z * 90) i) = brown and i < random 200][ ask patch-at-heading-and-distance (z * 90) i [set pcolor black] set i i + 1 ; show i ] ; ask self [set pcolor blue] ] ask up-to-n-of random 10 patches with [pcolor = blue] [ set i random 40 + 1 set z random 4 while [([pcolor] of patch-at-heading-and-distance (z * 90) i) = black and i < random 200][ ask patch-at-heading-and-distance (z * 90) i [set pcolor brown] set i i + 1 ; show i ] ; ask self [set pcolor blue] ] end to boltzmann set yboltz -1 * (ln (random-float 1)) * R * temp; from the potencial energy that should have a boltzmann distribution; ; ifelse state = 1 [set yboltz yboltz + floor2][set yboltz yboltz + floor1] set speed 1 * (2 * gravity-acceleration * yboltz) ^ 0.5 ; the speed is calculated from the free fall formula. ; v^2 = 2gh is the speed of an object falling from an heigth h with a gravity g ;The 1.15 factor is a correction factor for ??????????????? ; set heading (180 - heading) end to factor-gravity ;; turtle procedure let vx (dx * speed) let vy (dy * speed) - (gravity-acceleration * tick-delta) set speed sqrt ((vy ^ 2) + (vx ^ 2)) ; recolor set heading atan vx vy end to check-for-collision ;; particle procedure ;; Here we impose a rule that collisions only take place when there ;; are exactly two particles per patch. We do this because when the ;; student introduces new particles from the side, we want them to ;; form a uniform wavefront. ;; ;; Why do we want a uniform wavefront? Because it is actually more ;; realistic. (And also because the curriculum uses the uniform ;; wavefront to help teach the relationship between particle collisions, ;; wall hits, and pressure.) ;; ;; Why is it realistic to assume a uniform wavefront? Because in reality, ;; whether a collision takes place would depend on the actual headings ;; of the particles, not merely on their proximity. Since the particles ;; in the wavefront have identical speeds and near-identical headings, ;; in reality they would not collide. So even though the two-particles ;; rule is not itself realistic, it produces a realistic result. Also, ;; unless the number of particles is extremely large, it is very rare ;; for three or more particles to land on the same patch (for example, ;; with 400 particles it happens less than 1% of the time). So imposing ;; this additional rule should have only a negligible effect on the ;; aggregate behavior of the system. ;; ;; Why does this rule produce a uniform wavefront? The particles all ;; start out on the same patch, which means that without the only-two ;; rule, they would all start colliding with each other immediately, ;; resulting in much random variation of speeds and headings. With ;; the only-two rule, they are prevented from colliding with each other ;; until they have spread out a lot. (And in fact, if you observe ;; the wavefront closely, you will see that it is not completely smooth, ;; because some collisions eventually do start occurring when it thins out while fanning.) if count other particles-here = 1 [ ;; the following conditions are imposed on collision candidates: ;; 1. they must have a lower who number than my own, because collision ;; code is asymmetrical: it must always happen from the point of view ;; of just one particle. ;; 2. they must not be the same particle that we last collided with on ;; this patch, so that we have a chance to leave the patch after we've ;; collided with someone. let candidate one-of other particles-here with [who < [who] of myself and myself != last-collision] ;; we also only collide if one of us has non-zero speed. It's useless ;; (and incorrect, actually) for two particles with zero speed to collide. if (candidate != nobody) and (speed > 0 or [speed] of candidate > 0) [ collide-with candidate set last-collision candidate ask candidate [ set last-collision myself ] ] ] end ;; implements a collision with another particle. ;; ;; THIS IS THE HEART OF THE PARTICLE SIMULATION, AND YOU ARE STRONGLY ADVISED ;; NOT TO CHANGE IT UNLESS YOU REALLY UNDERSTAND WHAT YOU'RE DOING! ;; ;; The two particles colliding are self and other-particle, and while the ;; collision is performed from the point of view of self, both particles are ;; modified to reflect its effects. This is somewhat complicated, so I'll ;; give a general outline here: ;; 1. Do initial setup, and determine the heading between particle centers ;; (call it theta). ;; 2. Convert the representation of the velocity of each particle from ;; speed/heading to a theta-based vector whose first component is the ;; particle's speed along theta, and whose second component is the speed ;; perpendicular to theta. ;; 3. Modify the velocity vectors to reflect the effects of the collision. ;; This involves: ;; a. computing the velocity of the center of mass of the whole system ;; along direction theta ;; b. updating the along-theta components of the two velocity vectors. ;; 4. Convert from the theta-based vector representation of velocity back to ;; the usual speed/heading representation for each particle. ;; 5. Perform final cleanup and update derived quantities. to collide-with [ other-particle ] ;; particle procedure ;;; PHASE 1: initial setup ;; for convenience, grab some quantities from other-particle let mass2 [mass] of other-particle let speed2 [speed] of other-particle let heading2 [heading] of other-particle ;; since particles are modeled as zero-size points, theta isn't meaningfully ;; defined. we can assign it randomly without affecting the model's outcome. let theta (random-float 360) ;;; PHASE 2: convert velocities to theta-based vector representation ;; now convert my velocity from speed/heading representation to components ;; along theta and perpendicular to theta let v1t (speed * cos (theta - heading)) let v1l (speed * sin (theta - heading)) ;; do the same for other-particle let v2t (speed2 * cos (theta - heading2)) let v2l (speed2 * sin (theta - heading2)) ;;; PHASE 3: manipulate vectors to implement collision ;; compute the velocity of the system's center of mass along theta let vcm (((mass * v1t) + (mass2 * v2t)) / (mass + mass2) ) ;; now compute the new velocity for each particle along direction theta. ;; velocity perpendicular to theta is unaffected by a collision along theta, ;; so the next two lines actually implement the collision itself, in the ;; sense that the effects of the collision are exactly the following changes ;; in particle velocity. set v1t (2 * vcm - v1t) set v2t (2 * vcm - v2t) ;;; PHASE 4: convert back to normal speed/heading ;; now convert my velocity vector into my new speed and heading set speed sqrt ((v1t ^ 2) + (v1l ^ 2)) set energy (0.5 * mass * speed * speed) ;; if the magnitude of the velocity vector is 0, atan is undefined. but ;; speed will be 0, so heading is irrelevant anyway. therefore, in that ;; case we'll just leave it unmodified. if v1l != 0 or v1t != 0 [ set heading (theta - (atan v1l v1t)) ] ;; and do the same for other-particle ask other-particle [ set speed sqrt ((v2t ^ 2) + (v2l ^ 2)) set energy (0.5 * mass * (speed ^ 2)) if v2l != 0 or v2t != 0 [ set heading (theta - (atan v2l v2t)) ] ] ;; PHASE 5: final updates ;; now recolor, since color is based on quantities that may have changed ; recolor ; ask other-particle ; [ recolor ] end to recolor ;; particle procedure ifelse speed < (0.5 * init-particle-speed) [ set color blue ] [ ifelse speed > (1.5 * init-particle-speed) [ set color red ] [ set color green ] ] end ;;; ;;; drawing procedures ;;; ;; draws the box to make-box ask patches with [ pxcor = min-pxcor or pxcor = max-pxcor or pycor = min-pycor or pycor = max-pycor] [ set pcolor brown] end ;; creates initial particles to make-particles create-particles number-of-particles [ setup-particle random-position ; recolor ] calculate-tick-delta end to setup-particle ;; particle procedure set speed init-particle-speed set mass particle-mass set energy (0.5 * mass * speed * speed) set last-collision nobody set size 2 set color red if ([who] of self = 0) [set color white set pen-size 2] end ;; place particle at random location inside the box. to random-position ;; particle procedure setxy 0 max-pycor - 2; random-float (world-height - 3) + 1 set heading random-float 360 end to draw-energy-graph let n min-pycor repeat (max-pycor / 2) [ let x particles with [ (pycor >= n) and (pycor < (n + 4)) ] ifelse count x = 0 [ plot 0 ] [ let temperature mean [ energy ] of x plot temperature let list-pos (n + max-pycor) / 4 set aggregate-list (replace-item list-pos aggregate-list ((item list-pos aggregate-list) + temperature)) ] set n n + 4 ] end to draw-aggregate-graph [lst] foreach lst plot end ;; histogram procedure to draw-vert-line [ xval ] plotxy xval plot-y-min plot-pen-down plotxy xval plot-y-max plot-pen-up end to-report last-n [n the-list] ifelse n >= length the-list [ report the-list ] [ report last-n n butfirst the-list ] end ; Copyright 2002 Uri Wilensky. ; See Info tab for full copyright and license.

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## Attached files

File | Type | Description | Last updated | |
---|---|---|---|---|

CellularMetabolismLabyrinth.nlogo | data | To download a file that works | about 2 years ago, by Luis Mayorga | Download |

CellularMetabolismLabyrinth.png | preview | Preview for 'CellularMetabolismLabyrinth' | over 2 years ago, by Luis Mayorga | Download |

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