Automatic Control Laboratory, ETH Zurich, Physikstrasse 3 8092 Zurich, Switzerland

Center Systems Biology, Universität Stuttgart, Nobelstrasse 15 70569 Stuttgart, Germany

Abstract

Background

In this paper we apply a novel agent-based simulation method in order to model intracellular reactions in detail. The simulations are performed within a virtual cytoskeleton enriched with further crowding elements, which allows the analysis of molecular crowding effects on intracellular diffusion and reaction rates. The cytoskeleton network leads to a reduction in the mobility of molecules. Molecules can also unspecifically bind to membranes or the cytoskeleton affecting (i) the fraction of unbound molecules in the cytosol and (ii) furthermore reducing the mobility. Binding of molecules to intracellular structures or scaffolds can in turn lead to a microcompartmentalization of the cell. Especially the formation of enzyme complexes promoting metabolic channeling, e.g. in glycolysis, depends on the co-localization of the proteins.

Results

While the co-localization of enzymes leads to faster reaction rates, the reduced mobility decreases the collision rate of reactants, hence reducing the reaction rate, as expected. This effect is most prominent in diffusion limited reactions. Furthermore, anomalous diffusion can occur due to molecular crowding in the cell. In the context of diffusion controlled reactions, anomalous diffusion leads to fractal reaction kinetics. The simulation framework is used to quantify and separate the effects originating from molecular crowding or the reduced mobility of the reactants. We were able to define three factors which describe the effective reaction rate, namely ^{diff }
^{volume }
^{access }

Conclusions

Molecule distributions, reaction rate constants and structural parameters can be adjusted separately in the simulation allowing a comprehensive study of individual effects in the context of a realistic cell environment. As such, the present simulation can help to bridge the gap between

Background

The complex structured and crowded intracellular conditions

In order to investigate the effects of a given intracellular state on the reaction rate, we have developed an agent-based simulation which tracks individual molecules through a virtual cell containing a model cytoskeleton (see Figure _{0 }if sampled on short distances/times and a reduced _{eff }
_{eff }

Intracellular structure

**Intracellular structure**. Comparison of the 3D intracellular structures:

The mobility of the reactants is not the only factor determining the effective

While the adsorption (for instance to the cytoskeleton) and subsequent immobilization hampers the reactions

• **Metabolic channeling **

Metabolic channeling and scaffolds in signal transduction

**Metabolic channeling and scaffolds in signal transduction**. (a) Metabolic channeling in contrast to unconnected metabolic reactions with unbound enzymes

• **Regulation of signal transduction: **Cells are subject to many and sometimes contradictory signals. The information is carried from the receptors in the plasma membrane towards the nucleus by signaling molecules. Especially in multi-stage cascades, for example in MAPK (mitogen activated protein kinase) signaling, this transfer can be regulated by scaffolds. The scaffolds integrate several stages of the cascade in one place (see Figure

• **Pharmacokinetics and drug detoxification: **If drug molecules bind to proteins or membranes, they are likewise sequestered from the cytoplasm. This reduces both their action and their degradation, for example by enzymes of the CYP family in the liver

All these details are omitted in models based on differential equations in which only the number/concentration of the species is tracked. The general compartmentalization of the organism/cell can be included in the model if the respective compartments and transport rates are defined

A particle or agent based simulation allows exploring the effects introduced by the spatial organization of the cell including (transient) binding to the cytoskeleton. The

Results and discussion

The simulations are performed in a small model cell with a diameter of 7

**Contains further simulation results and details of the model setup**. **Simulation**. The simulation is available upon request from Michael Klann,

Click here for file

Effective diffusion

The mean squared displacement (MSD) of diffusing molecules should increase linearly with time according to

where _{eff }
_{0 }= 0.77 ± 0.01. This slowdown is in agreement with previous studies of the impact of the cytoskeleton structure on the diffusion of inert (i.e. molecules that do not interact with other molecules) tracer molecules for the given excluded volume fraction

The slowdown of the diffusion can also be explained by the local confinement of the obstacles _{0 }in the beginning, and later on proportional to the average _{eff}

If tracers bind transiently to the cytoskeleton and are therefore temporarily immobilized, the effective diffusion coefficient is further reduced (see Figure

Transient binding

**Transient binding**. (a) Transient binding of molecules to the cytoskeleton. (b) Diffusion rises proportionally to the fraction of unbound molecules. (c) Nonlinearity in 〈^{2}〉 induced by the dynamics of the reversible binding process. _{diss }_{binding }

Figure

Effective reaction rates

Diffusion-controlled reactions

The theory of diffusion controlled reactions requires to take into account the following points

• **Diffusion Limit: **The maximal reaction rate constant for a bimolecular reaction of two spherical molecules _{i }
_{j }
_{D }
_{i }
_{j}
_{i }
_{j}

• **Microscopic Reaction Rate Constant: **If not every encounter between two reactants leads to a reaction, the microscopic reaction rate constant _{micro }

• **Effective Macroscopic/Bulk Reaction Rate Constant: **The resulting reaction rate constant which is observed on the macroscopic level, corresponding to the rate constant of ODE models is determined as

Test setup

The test molecules in the simulation are enzyme _{E }
_{S }
^{2}/s. This leads to a diffusion limit of the reaction rate of _{D }
^{7 }l/(mol·s). The chosen macroscopic reaction rate for a test simulation can be given as a fraction of _{D }
_{macro }
^{5 }l/(mol·s) (1% of _{D}
_{macro }
^{6 }l/(mol·s) (10% of _{D}
_{macro }
^{7 }l/(mol·s) (30% of _{D}

The resulting reaction rate can be accessed from the change in the number of molecules. The noise in a stochastic simulation, however, hampers the identification of the current reaction rate. If the considered species is, in turn, created and destroyed by two reactions, it will accumulate to a dynamic equilibrium steady state. Averaging the steady state number over time reduces the stochastic noise in the result. This situation can be found

Model setup

**Model setup**. Description of an enzymatic reaction in a metabolic pathway based on mass action kinetics.

In order to reduce the complexity, the considered substrate species _{1}. It is consumed in the enzymatic reaction _{2}. The number of enzymes _{E}
_{E }
^{-7 }mol/l (20600 molecules). The macroscopic balance equation for the substrate concentration is in this model

which leads to the equilibrium steady state

Detailed simulation vs. ODE-model

This section compares the outcome of the detailed stochastic simulation with the result of the macroscopic ODE-model of Figure

The steady state of the

Results for the 'in vitro' setup

_{2}/_{D}

_{1 }[mol/(L · s)]

_{2 }[L/(mol · s)]

**
**

**
**

**rel. Error**

0.01

3.78 × 10^{-9}

7.57 × 10^{5}

2575

2558 ± 51

0.7%

0.1

3.78 × 10^{-8}

7.57 × 10^{6}

2575

2532 ± 51

1.7%

0.3

1.14 × 10^{-7}

2.27 × 10^{7}

2575

2619 ± 51

1.7%

Comparison of the steady state molecule numbers _{S }

The situation is quite different in the '_{1 }= _{1 }is held constant in the simulation, the bimolecular reaction is affected by the crowded intracellular conditions. The rate for the second reaction becomes

The steady state shifts accordingly to

In order to understand the corresponding change in the reaction rate, the overall effect (^{eff}

1. The first factor arises from the reduced free volume fraction _{0 }is calculated as number of molecules per cell/total cell volume). This factor has to be added only once (for _{E}
_{S}
_{2}
_{E}c_{S }
_{S }

2. The reduced effective diffusion has an influence on the reaction rate because it reduces the collision rate. For the present analysis it is assumed that the molecules react in the same way _{2}/_{D }

3. The hindered accessibility of the molecules due to steric effects of nearby obstacles contributes a further reduction ^{access }
^{access }

Inaccessible volume fraction

**Inaccessible volume fraction**. The restricted volume close to all structures in the cell reduces the interaction volume (green) for the reaction. In order to estimate the effect of the reduced interaction volume on the reaction rate, the fraction of the accessible reaction volume has to be averaged over all possible molecule positions in the given cell. In the complex environment of the given random intracellular architecture the calculation of the corresponding ^{access}

In combination the effective macroscopic bimolecular reaction rate is accordingly:

Table _{2}/_{D }
_{2 }is faster. For comparison also a simulation in a homogenized cell was conducted. This cell does not contain any hindering obstacles but the size is reduced by a factor of 0.695 so that the effective concentration of molecules matches the effective concentration in the detailed virtual cell. Also the diffusion is reset in order to match the respective effective diffusion - but only after the microscopic reaction rate was set based on the

Results for the 'in vivo' setup

_{2}/_{D}

**
f **

**
f **

**
f **

**
f **

**
**

**
**

**
**

Detailed model

0.01

1.44

1.00

0.97

1.39

1845 ± 37

1853 ± 40

1.00

cell

0.1

1.44

0.97

0.97

1.35

1879 ± 38

1839 ± 40

0.98

0.3

1.44

0.91

0.97

1.27

2058 ± 40

1996 ± 40

0.97

Homogenized/

0.01

1.44

1.00

1.00

1.43

1783 ± 36

1757 ± 40

0.99

averaged cell

0.1

1.44

0.97

1.00

1.39

1815 ± 37

1802 ± 40

0.99

0.3

1.44

0.91

1.00

1.32

1989 ± 39

2003 ± 40

1.01

Results of the effective reaction rate in the virtual cell. The free volume fraction for the test spheres with a radius of 2.5 nm is ^{volume }^{access }_{S }

In order to understand this result, it is necessary to recall the transient anomalous diffusion in the crowded environment. At short distances, the molecules still move with their original (fast) diffusion coefficient. Only on longer distances the tortuous way around the obstacles leads to a reduced mobility. The results indicate that the diffusion limited bimolecular reaction senses an intermediary effective diffusion coefficient which is slower than _{0 }but faster than the long term effective _{eff }
_{2}/_{D }

Future work could investigate this effect with respect to the local confinement, and also include the influence of unspecific and transient binding on the reaction rate, i.e. the nullified mobility of one or both of the reactants due to binding to cellular structures. In addition, also the combined action of individual reaction rate constants for different sub-states of a molecule (free, bound, phosphorylated at site x, etc.) could be analyzed in a more complex model.

Enzyme co-localization and metabolic channeling

This section considers more than one reaction of the enzymatic conversion of the metabolites in the cell. The pathway is simplified to a sequence of 4 reactions as shown in Figure

Metabolic channeling model

**Metabolic channeling model**. (a) Metabolic pathway and (b) possible enzyme distributions. Note, that the enzymes are immobilized (_{E }^{2}/s; right: ^{2}/s)).

The aim of the present simulations is to elucidate the effect of the localization of the enzymes. The most optimal setup promises to be the **enzyme channel (A) **of Figure **enzyme layer (B) **is modeled for comparison. These structured setups are furthermore compared to a uniform **random distribution (C) **of the enzymes in the cell and a **well mixed (D) **model based on ODE (in order to keep the models comparable, the corresponding stochastic solution of the ODE model is evaluated based on the Gillespie method

All enzymes are immobilized in the simulation, i.e. they stay at their fixed initial position with an artificial _{E }

The metabolites are moving with a diffusion coefficient of (i) ^{2}/s and (ii) ^{2}/s for a comparison of the mobility effects. The macroscopic reaction rate constant is set to ^{6 }l/(mol·s), which is fairly fast but not extremely diffusion limited (_{D }
^{7 }l/(mol·s) for ^{2}/s, and _{D }
^{8 }l/(mol·s) for ^{2}/s) - see Additional file

Since all setups are conducted with the same number of enzymes and the same reaction rates, they produce similar results (see Figure ^{2}/s. The overall development of the individual metabolite pools is shown in the Additional file

Likewise the setups where the enzymes are close to the plasma membrane (i.e. channel (A) or layer (B)) lead to a faster formation of the product because the substrate enters the cell through the plasma membrane (note, that this setup also promotes a faster export of the metabolite because it is produced next to the plasma membrane). This is clearly visible in the excerpt of the initial phase shown in Figure

The setup where the respective enzymes are co-localized in an enzyme channel indeed shows the fastest (initial) product formation rate. It can be expected that the differences between the enzyme distributions tested in this paper will increase if the Michaelis-Menten enzyme kinetics is used. The high local enzyme concentration close to the surface leads to a locally higher _{max }

The improved reaction rate in the co-localized channel structure is in agreement with the findings of Bauler et al.

The present study shows that the location of the reactants (here enzymes and metabolites) can play an important role. This work focused on the influence of spatial aspects and the possible enzyme co-localization, which allows a more realistic study of enzyme channeling properties. Future work could include more advanced reaction kinetics in order to verify channeling, as well as a study of the resulting control properties on the metabolic flux

Conclusions

The present simulation allows a detailed analysis of the effects of the intracellular properties on the reaction rates. The results have shown that the

In addition, the influence of the subcellular localization of the reactants was tested. The results show that the co-localization of enzymes in a metabolic-channeling framework can improve the product formation. It is worth noting that the advantage of a specific location of the enzymes is accompanied by the disadvantage of the reduced enzyme mobility. Hence the reaction rate will be reduced in the diffusion limited case. This reduction could even outbalance the superior product formation rate of enzyme channeling. Since this depends on the actual diffusion and reaction rate constants, further simulations are required in order to quantify the advantage of the channel configuration - especially in the context of a more advanced kinetics within the multi-enzyme-complex.

Thus the present simulation framework is a promising tool to investigate intracellular reactions and signal transduction processes in the detailed spatial organization of the cell

On the way to a model which is in agreement with living cells, several parameters like the correct cytoskeleton structure, molecular crowding, and additional unspecific interactions which can for example transiently immobilize the molecules have to be adjusted, giving a deeper insight into the cell

Methods

Description of the agent-based simulation

Only the molecules of interest are tracked in the simulation in order to reduce the complexity, which allows modeling the whole cell

where _{0 }is the diffusion coefficient.

The simulation only requires defining the particle radius and diffusion coefficient for each species as well as the initial number and distribution of the molecules. The particles are initially placed in the virtual cell at positions which are not restricted by the cytoskeleton or crowding molecules. Likewise all reactions have to be defined (educts, products, rate constants).

Reactions between molecules

A reaction between two molecules can only occur, if the reactants are close enough together. The reaction probability between two molecules is therefore given by the probability of the collision and the probability that a reaction occurs given that a collision is occurring. The claim of the simulation is that it can reproduce the macroscopic (mass action kinetics) rate constant _{ij }

The discrete time simulation framework complicates the estimation of the reaction probability. The position of the molecules is only known at _{n }
_{
n+1 }= _{n }

The gap between

with the initial probability density function is

The flux across the boundary within [_{0}. By this approach the requested true number of reactions can be estimated

Derivation of the simulation method

As shown by Pogson et al.

For simplicity, the units of the concentrations should be [molecules/^{3}] (note: the units of _{ij }
^{3}/molecules/s]. If _{i }
_{A}
^{15 }[liter/(^{3}mol)], where _{A }

and converted according to _{i }
_{cell }
_{i }

Δ

of the _{ij}

The corresponding 'reaction' volume

can be introduced _{ij }
_{ij}
**reaction**volume/molecule]. Replacing Δ_{ij }

So in the completely homogeneous framework the fraction of reacting molecules corresponds to a fraction of the volume in which all molecules react, while they do not react in the remaining volume.

From the perspective of the

This reaction radius is used by Pogson et. al. _{j }
_{i}

A molecule of species ^{1/3 }according to Equation (20), compensating the Δ

From macroscopic theory to one microscopic reaction: impact of diffusion

Initially, a uniform random distribution of molecules is assumed. On average, Δ_{i }
_{i }
_{j }

• The collision rate constant between the reactants is given by _{D }
_{i }
_{j}
_{i }
_{j}

• If not all collisions lead to a reaction but only a fraction which is determined by a microscopic reaction rate _{micro}

(cf. the results section on effective reaction rates). Obviously the microscopic reaction rate constant _{micro }
_{ij }

• The collision radius (_{i }
_{j}
_{D }
_{micro}

• **Solution: reaction probability in the interaction volume: **Both concepts, the flux/surface-based description and the macroscopic, volume-based framework can be brought into agreement in the following way:

1. The true collision radius

2. The microscopic reaction rate constant _{micro }

- the corresponding reaction volume should be in analogy to Equation(18)

but this will (most likely) not match the collision volume 4_{i }
_{j}
^{3}

- The mismatch is adjusted by introducing the reaction probability

which effectively reduces the reaction volume determined by the collision radius to the reaction volume given by Equation (23) while it retains the correct interaction surface.

This approach also reflects the nature of reactions in a probabilistic framework: the overall, macroscopic reaction probability is now determined by the probability to collide and the probability to react, given that a collision has occurred.

• **Resulting reaction algorithm: **Two particles

Adsorption to cellular structures

The association with the cytoskeleton can be described in the same way, and also the adsorption to surfaces like the plasma membrane. Since these objects are impenetrable, the reaction volume has to be outside of the cellular structures - leading to a reaction layer around them. The height of this layer is given by _{binding }
_{binding }
_{structure}
_{diss }
_{diss }

Remarks on reversible reactions

The binding and unbinding process to the cytoskeleton is a diffusion controlled process by itself. The details of the effective rates in diffusion controlled reversible processes have been studied for instance in

Parallelization

Parallelization of the simulation is possible and benefits from the fact that all agents are updated simultaneously with a global Δ

Review and comparison with other simulation methods

For the given purpose of the simulation environment, i.e. modeling of the cell including a realistic intracellular environment such as a detailed cytoskeleton structure, only agent-based off-lattice methods can be used. Therefore we leave out the spatial Gillespie method as well as derivatives thereof, and also the E-Cell plug-in of Arjunan and Tomita

We also require that the treatment of bimolecular reactions is implemented efficiently and as correctly as possible. As such, the Greens-function reaction dynamics allows to do large steps if reactants are far apart from each other which would suit this need

The distance-dependent reaction probability derived from the Fokker-Planck Equation as outlined above based on Equations (12) and (13) (cf.

Quantifying the influence of the reduced diffusion on a bimolecular reaction

The macroscopic reaction rate _{macro }
_{ij }
_{D }

_{D }

For a given macroscopic reaction rate, the microscopic reaction rate constant is accordingly given by Equation (3):

(given that the user does not try to exceed the diffusion limit with the macroscopic reaction rate, i.e. _{macro }
_{D}

and the effective macroscopic reaction rate can now be calculated based on Equation (3)

Inserting Equation (26) into Equation (28) leads to

If also the definition of _{D }
_{D, eff }

The initial (unperturbed) macroscopic reaction rate can be set into relation with the diffusion limit, defining

which leads to a simplification of Equation (30)

From this equation it can be deduced that the effective macroscopic reaction rate constant is reduced by the factor

Authors' contributions

MK developed, designed, and performed the simulations and drafted the manuscript. AL calculated the reaction probability based on the Fokker-Planck equation and revised the core functionality of the simulation. MR is the group leader, has initiated the program and revised the manuscript. All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank Martin Falk and Thomas Ertl for the valuable discussions about the visualization method as well as on the possible parallelization of the simulation and acknowledge the funding by the State of Baden-Württemberg/Center Systems Biology in Stuttgart.