ecoli.processes.spatiality.diffusion_network

Diffusion Network

DiffusionNetwork models Brownian diffusion using a network of nodes and edges. Each node acts as a cellular compartment and each edge connects those compartments, indicating where diffusion can occur.

This process class models diffusion based off of Fick’s law. The following equation is used:

  • Diffusion: \(\frac{dc}{dt} = \frac{DA}{V} * \frac{dc}{dx}\)

  • \(D\): Diffusion constant

  • \(A\): Cross-sectional area of edge

  • \(V\): Volume of node

This diffusion equation is solved using implicit Euler to derive a matrix, M, that updates the concentration, c(t), on each timestep.

  • Concentration update: \(c_{t+1} = M^{-1} * c_{t}\)

This process takes in molecular weights in order to solve for molecule hydrodynamic radii, which is it turn used to solve for diffusion constants. These calculations assume that all molecules are spherical proteins. For more information on how this is done, see calculate_rp_from_mw and compute_diffusion_constants_from_rp. However, if a molecule radius is known, it can be passed in as radii. Similarly, if the diffusion constant is known, it can be passed in as a property of an edge as diffusion_constants.

Note

This model treats all molecule classes as concentrations and is a deterministic solution. This model should only be used when there is a sufficient number of molecules such that they can be treated deterministically.

class ecoli.processes.spatiality.diffusion_network.DiffusionNetwork(parameters=None)[source]

Bases: Process

Models diffusion between a network of connected nodes.

Ports:

  • nodes: Expects a store which is a dict of node names (the keys of the dict) to a dict, which has the key value pairs for length, volume, and molecules.

Parameters:

parameters –

A dictionary of configuration options. The following configuration options may be provided:

  • nodes (list): A list of node names.

  • edges (dict): Maps edge names (the keys of the dict) to a dict (the values of the dict), which must include the key-value pairs of nodes to a list of nodes each edge connects, and cross_sectional_area to the area of that edge. Additionally, known diffusion constants can be included as diffusion_constants in units of um^2/s, and edge-specific scaling of the diffusion constants can be included as, diffusion_scaling_constant.

  • mw (dict): Maps from names of molecules (the keys of the dict) to their molecular weights in units of fg (the values of the dict).

  • mesh_size (float): Mesh size in units of nm.

  • time_step (float): The time step used in units of s.

  • radii (dict): Maps from molecule names of molecules (the keys of the dict) to their known hydrodynamic radii in units of nm (the values of the dict). This is an optional parameter.

  • temp (float): Temperature of experiment in units of K. This is an optional parameter.

calculate_request(timestep, states)[source]
defaults: Dict[str, Any]
{   'edges': {},
    'mesh_size': 50,
    'mw': {},
    'nodes': [],
    'radii': {},
    'temp': 310.15,
    'time_step': 0.1}
name = 'diffusion_network'
next_update(timestep, state)[source]
ports_schema()[source]

Dynamically constructs ports – one port for each node in DiffusionNetwork.parameters['nodes']. Each node has variables volume, length, and a molecules dict with molecule names mapped to their counts.

ecoli.processes.spatiality.diffusion_network.array_from(d)[source]
ecoli.processes.spatiality.diffusion_network.array_to(keys, array)[source]
ecoli.processes.spatiality.diffusion_network.calculate_rp_from_mw(molecule_ids, mw)[source]

This function compute the hydrodynamic radius of a macromolecules from its molecular weight. It is important to note that the hydrodynamic diameter is mainly used for computation of diffusion constant, and can be different from the observed diameter under microscopes or the radius of gyration, especially for loose polymers such as RNAs. This function is not E coli specific.

References: Bioinformatics (2012). doi:10.1093/bioinformatics/bts537

Parameters:

  • molecule_ids – List of molecule ids.

  • mw – molecular weight of the macromolecules, units: fg.

Returns:

the hydrodynamic radius (in unit of nm) of the macromolecules using the following formula:

  • rp = 0.0515*MW^(0.392) nm (Hong & Lei 2008) (protein)

These parameters are also possible for other macromolecule types, however all molecules are currently assumed to be proteins.

  • rp = 0.0566*MW^(0.38) nm (Werner 2011) (RNA)

  • rp = 0.024*MW^(0.57) nm (Robertson et al 2006) (linear DNA)

  • rp = 0.0125*MW^(0.59) nm (Robertson et al 2006) (circular DNA)

  • rp = 0.0145*MW^(0.57) nm (Robertson et al 2006) (supercoiled DNA)

ecoli.processes.spatiality.diffusion_network.compute_diffusion_constants_from_rp(molecule_ids, rp, mesh_size, edges, temp)[source]

Warning

The default values of the parameters are E coli specific.

This function computes the hypothesized diffusion constant of macromolecules within the nucleoid and the cytoplasm region. In literature, there is no known differentiation between the diffusion constant of a molecule in the nucleoid and in the cytoplasm up to the best of our knowledge in 2020. However, there is a good reason why we can assume that previously reported diffusion constant are in fact the diffusion constant of a protein in the nucleoid region:

  1. The image traces of a protein within a bacteria usually cross the nucleoid regions.

  2. The nucleoid region, compared to the cytoplasm, should be the main limiting factor restricting the magnitude of diffusion constant.

  3. The same theory of diffusion constant has been implemented to mammalian cells, and the term ‘rh’, the average hydrodynamic radius of the biggest crowders, are different in mammalian cytoplasm, and it seems to reflect the hydrodynamic radius of the actin filament (note: the hydrodynamic radius of actin filament should be computed based on the average length of actin fiber, and is not equal to the radius of the actin filament itself.) (ref: Nano Lett. 2011, 11, 2157-2163). As for E coli, the ‘rh’ term = 40nm, which may correspond to the 80nm DNA fiber. On the other hand, for the diffusion constant of E coli in the true cytoplasm, we will expect the value of ‘rh’ term to be approximately 10 nm, which correspond to the radius of active ribosomes.

Using these terms for scaling a baseline diffusion constant (calculated from Enstein-Stokes equation), a cytosol-specific diffusion calculation can be obtained).

Ref: Kalwarczyk, T., Tabaka, M. & Holyst, R. Bioinformatics (2012). doi:10.1093/bioinformatics/bts537

This function computes the hypothesized diffusion constant of macromolecules within the nucleoid region by scaling the hypothesized cytoplasm diffusion constant. There is evidence that as molecules grow in size, they become more excluded from E. coli’s nucleoid because the DNA polymers form a meshgrid with an estimated mesh size of 50 nm (ref: Xiang et al., bioRxiv (2020)). The equation implemented here assumes that molecules move through the network by encountering openings between the DNA meshgrid greater than the hydrodynamic radius.

Ref: Brian Amsden Macromolecules (1999). doi:10.1021/ma980922a

D_0 = K_B*T/(6*pi*eta_0*rp)
ln(D_0/D_cyto) = ln(eta/eta_0) = (xi^2/Rh^2 + xi^2/rp^2)^(-a/2)
D_0 = the diffusion constant of a macromolecule in pure solvent
eta_0 = the viscosity of pure solvent, in this case, water
eta = the size-dependent viscosity experienced by the macromolecule.
xi = average distance between the surface of proteins
rh = average hydrodynamic radius of the biggest crowders
a = some constant of the order of 1
rp = hydrodynamic radius of probed molecule

In this formula, since we allow the changes in temperature, we also consider the viscosity changes of water under different temperature:

eta_0 = A*10^(B/(T-C))
A = 2.414*10^(-5) Pa*sec
B = 247.8 K
C = 140 K

Ref: Dortmund Data Bank

Parameters:

  • molecule_ids – List of molecule ids.

  • rp – List of radii corresponding to each molecule id. unit: nm

  • mesh_size – Size of meshgrid openings. unit: nm

  • edges – Dictionary of edges.

  • temp – The temperature of interest. unit: K.

Returns:

the diffusion constant of the macromolecule in units um**2/sec

ecoli.processes.spatiality.diffusion_network.plot_diff_range(diffusion_constants, rp, out_dir='out')[source]
ecoli.processes.spatiality.diffusion_network.plot_output(output, nodes, out_dir='out')[source]