wholecell.utils.spatial_tool

Functions that may be useful for future investigation in spatial model. Some of the functions contain default values specific to E coli. Please read before you use. References for the numbers are included in each function.

wholecell.utils.spatial_tool.compute_diffusion_constant_from_mw(mw, mtype=None, loc=None, temp=None, parameters=None)

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 2019. 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.

However, all the above statements are just hypothesis. If you want to compute the diffusion constant of a macromolecule in the whole E coli cell, you should set loc = ‘nucleoid’: the formula for this is obtained from actual experimental data set. When you set loc = ‘cytoplasm’, the entire results are merely hypothesis.

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

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

Parameters:

• mw – molecular weight(unit: Da) of the macromolecule

• mtype – There are 5 possible mtype options: protein, RNA, linear_DNA,

• circular_DNA –

• supercoiled_DNA. (and) –

• loc – The location of the molecule: ‘nucleoid’ or ‘cytoplasm’.

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

• parameters – The 4 parameters required to compute the diffusion constant: xi, a, rh_nuc, rh_cyto. The default values are E coli specific.

Returns:

the diffusion constant of the macromolecule, units: um**2/sec

Return type:

dc

wholecell.utils.spatial_tool.compute_diffusion_constant_from_rp(rp, loc=None, temp=None, parameters=None)

Warning: The default values of the ‘parameters’ are E coli specific.

This is the same function as ‘compute_diffusion_constant_from_mw’ except that it accepts the hydrodynamic radius of the macromolecules as the input. All hypothesis and statements in ‘compute_diffusion_constant_from_mw’ apply here as well.

Parameters:

• rp – the hydrodynamic radius of the macromolecule, units: nm.

• loc – The location of the molecule: ‘nucleoid’ or ‘cytoplasm’.

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

• parameters – The 4 parameters required to compute the diffusion constant: xi, a, rh_nuc, rh_cyto. The default values are E coli specific.

Returns:

the diffusion constant of the macromolecule, units: um**2/sec

Return type:

dc

wholecell.utils.spatial_tool.compute_free_volume_ratio(q, eta)

This is a function that computes the free volume ratio of existing macromolecules with respect to a new molecule based on the scaled particle theory. This is not E coli specific. It is important to note that the results may not be accurate for large q or eta > 0.50.

References: Malloggi, F. Soft Matter at Aqueous Interfaces. Lecture Notes in Physics (2016). doi:10.1007/978-3-319-24502-7. Chapter 3-4-2.

Parameters:

• q – the size ratio between the new particle and existing particles. For example, if the radius of the new particle = delta, and the radius of the existing particle = R, then q = delta/R.

• eta – the compaction ratio, or the volume occupancy of the existing particle. eta = n*v_particle/v_total

Returns:

the free volume ratio of the space = v_free/v_total.

Return type:

alpha

wholecell.utils.spatial_tool.compute_hydrodynamic_radius(mw, mtype=None)

This function compute the hydrodynamic diameter 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:

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

• mtype – There are 5 possible mtype options: protein, RNA, linear_DNA, circular_DNA, and supercoiled_DNA.

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) - 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)

wholecell.utils.spatial_tool.compute_n_blob(choice_model, choice_unit, bp_dna=None)

This function computes the number of DNA blobs within a bacteria. This functions contain multiple assumptions that demand careful examination.

Previous research (Ref: Skoko D, Wong B, Johnson R, Marko J (2004) Biochemistry 43: 13867-13874.) directly observed the structure of chromosomal DNA of E coli under AFM. They observed that the DNA are composed of 40nm & 80nm fibers. The 80nm fibers are the main type within the cells, but it is hypothesized to be folded from 2 threads of 40nm fibers. Therefore, we hypothesized that chromosomal DNA are composed of DNA blobs with diameter of 40nm.

For the model choice, there are 2 possible options. In the Flory chain option, the DNA within a blob is assumed to be self-avoidant. In the ideal chain option, the DNA within a blob is assumed to be in a melted state.

For the unit choice, there are 3 possible options. You can choose single base pair of DNA as the smallest unit in the formation of DNA blob. In conventional physics, 2*persistence length is regarded as the standard choice of the smallest unit in a polymer. It is important to note that, the persistence length of DNA can be drastically different under in vitro & in vivo conditions. The persistence length of DNA in vitro is ~ 50nm, while the persistence length of DNA in vivo can be down to 20nm. This is because the binding of DNA binding proteins & DNA supercoiling can decrease the energy of DNA and makes it softer. Another possible choice of the smallest unit of DNA is 10bp, which corresponds to the average binding distance between HU protein, a structural DNA binding protein.

Choosing different model setting and different smallest unit of DNA can result in drastically different results.

The default value of genomic size is set to be E coli specific.

Parameters:

• choice_model – the physical model of DNA within a blob. The 2 possible options are ‘ideal’ & ‘flory’.

• choice_unit – the choice of the smallest unit within a blob. The 3 possible options are ‘single’, ‘lp’, or an integer number indicating the number of base pairs within a single unit.

• bp_dna – the number of base pairs in the chromosome. The default value is set to be the chromosome size of E coli.

Returns:

the number of DNA blobs in a bacteria.

Return type:

n_blob

wholecell.utils.spatial_tool.compute_nucleoid_size(l_cell, d_cell, length_scaling_parameters=None, nucleoid_area_ratio=None)

Warning 1: This function contains default values that are specific to E coli grew on M9 media under aerobic condition only. The shape and size of nucleoid of a bacteria can be very different across species. Warning 2: This function is not suitable to compute the nucleoid size of E coli when its shape turn filamentaous. This is because the scaling formula of the length of the nucleoid is obtained from a dnaC mutant E coli strain. According to our reference, when the cells are allowed to replicate their DNA normally, a constant N/C area ratio is maintained even for filamentous variants (treated with cephalexin). However, if the DNA is not allowed to replicate, the N/C area ratio will decrease as the cell elongate. It is therefore recommended to carefully examine the condition when the length of the cells grow beyond 3 um. Warning 3: the default values of length_scaling_parameters and nucleoid_area_ratio are set to be E coli specific, grew on M9 media under aerobic condition. The N/C area ratio for E coli grew on LB under aerobic condition is close to 0.4. For E coli grew under anaerobic condition it is close to 0.5. Warning 4: It is also important to note that a single cell can contain more than 1 nucleoid, and this formula may not be suitable for these cases.

Reference on nucleoid length: Wu, F. et al. Curr. Biol. (2019). doi:10.1016/j.cub.2019.05.015 Reference on nucleoid/cytoplasm area ratio: Gray, W. T. et al. Cell (2019). doi:10.1016/j.cell.2019.05.017

Parameters:

• l_cell – the length of the cell, in units of um

• d_cell – the width of the cell, in units of um

• length_scaling_parameters – the parameters used for the scaling formula of the length of the nucleoid with respect to the length of the whole cell.

• nucleoid_area_ratio – the nucleoid/cytoplasm area ratio measured under microscope.

Returns:

the length of the nucleoid d_nuc: the diameter of the nucleoid

Return type:

l_nuc

wholecell.utils.spatial_tool.compute_rg_rna(n_nt)

This function computes the radius of gyration (in units of nm) of an RNA with length n_nt(in units of nt). This function is not E coli specific. It is important to note that radius of gyration can be very different from the hydrodynamic radius or the radius you can observe under microscopes, especially for RNAs.

References: Nucleic Acids Research, 2017, vol 45, 5: 2919-2934.

doi: 10.1093/nar/gkx023

Formula: Rg = a*N**nu. a = 0.366 +/- 0.146 nm, nu = 0.50 +/- 0.05

Parameters:

n_nt – the number of nucleotide in an RNA.

Returns:

The radius of gyration of an RNA in the unit of nm.