# Theory Notes¶

## Computation of cell dry mass¶

The concept of cell dry mass computation was first introduced by Barer [Bar52]. The dry mass $$m$$ of a biological cell is defined by its non-aqueous fraction $$f(x,y,z)$$ (concentration or density in g/L), i.e. the number of grams of protein and DNA within the cell volume (excluding salts).

$m = \iiint f(x,y,z) \, dx dy dz$

The assumption of dry mass computation in QPI is that $$f(x,y,z)$$ is proportional to the RI of the cell $$n(x,y,z)$$ with a proportionality constant called the refraction increment $$\alpha$$ (units [mL/g])

$n(x,y,z) = n_\text{intra} + \alpha f(x,y,z)$

with the RI of the intracellular fluid $$n_\text{intra}$$, a dilute salt solution. These two equations can be combined to

(1)$m = \frac{1}{\alpha} \cdot \iiint (n(x,y,z) - n_\text{intra}) \, dx dy dz.$

In QPI, the RI is measured indirectly as a projected quantitative phase retardation image $$\phi(x,y)$$.

$\phi(x,y) = \frac{2 \pi}{\lambda} \int (n(x,y,z) - n_\text{med}) \, dz$

with the vacuum wavelength $$\lambda$$ of the imaging light and the refractive index of the cell-embedding medium $$n_\text{med}$$. Integrating the above equation over the detector area $$(x,y)$$ yields

(2)$\iint \phi(x,y) \, dx dy = \frac{2 \pi}{\lambda} \iiint (n(x,y,z) - n_\text{med}) \, dx dy dz$

If the embedding medium has the same refractive index as the intracellular solute ($$n_\text{med} = n_\text{intra}$$), then equations (1) and (2) can be combined to

$m_\text{med=intra} = \frac{\lambda}{2 \pi \alpha} \cdot \iint \phi(x,y) \, dx dy.$

For a discrete image, this formula simplifies to

(3)$m_\text{med=intra} = \frac{\lambda}{2 \pi \alpha} \cdot \Delta A \cdot \sum_{i,j} \phi(x_i,y_j)$

with the pixel area $$\Delta A$$ and a pixel-wise summation of the phase data.

## Relative and absolute dry mass¶

If however the medium surrounding the cell has a different refractive index ($$n_\text{med} \neq n_\text{intra}$$), then the phase $$\phi$$ is measured relative to the RI of the medium $$n_\text{med}$$ which causes an underestimation of the dry mass if $$n_\text{med} > n_\text{intra}$$. For instance, a cell could be immersed in a protein solution or embedded in a hydrogel with a refractive index of $$n_\text{med}$$ = $$n_\text{intra}$$ + 0.002. For a spherical cell with a radius of 10µm, the resulting dry mass is underestimated by 46pg. Therefore, it is called “relative dry mass” $$m_\text{rel}$$.

$m_\text{rel} = \frac{\lambda}{2 \pi \alpha} \cdot \iint \phi(x,y) \, dx dy,$

If the imaged phase object is spherical with the radius $$R$$, then the “absolute dry mass” $$m_\text{abs}$$ can be computed by splitting equation (1) into relative mass and suppressed spherical mass.

$\begin{split}m_\text{abs} &= \frac{1}{\alpha} \cdot \iiint (n(x,y,z) - n_\text{med} + n_\text{med} - n_\text{intra}) \, dx dy dz \\ &= m_\text{rel} + \frac{4\pi}{3\alpha} R^3 (n_\text{med} - n_\text{intra})\end{split}$

For a visualization of the deviation of the relative dry mass from the actual dry mass for spherical objects, please have a look at the relative vs. absolute dry mass example.

## Range of validity¶

Variations in the refraction increment may occur and thus the above considerations are not always valid. For a detailed discussion of the variables that affect the refraction increment, please see [BJ54].

### Dependency on imaging wavelength¶

Barer and Joseph measured the refraction increment of several proteins in dependence of wavelength. In general, short wavelengths (366nm) yield values close to 0.200mL/g while long wavelengths (656nm) yield smaller values close to 0.180mL/g (table 3 in [BJ54]).

### Dependency on protein concentration¶

The refraction increment has been reported to be linear for a wide range of protein concentrations. Barer and Joseph found that bovine serum albumin exhibits a linear refraction increment up to its limit of solubility (figure 2 in [BJ54]). They additionally received a personal communication stating that this is also the case for gelatin.

### Dependency on pH, temperature, and salts¶

The refraction increment is little dependent on pH, temperature, and salts [BJ54].

### Refraction increment and the mass of cells¶

Dry mass and actual mass of a cell differ by the weight of the intracellular fluid. This weight difference is defined by the volume of the cell minus the volume of the protein and DNA content. While it seems to be difficult to define a partial specific volume (PSV) for DNA, there appears to be a consensus regarding the PSV of proteins, yielding approximately 0.73mL/g (see e.g. reference [Bar57] as well as [HGC94] and question 843 of the O-manual referring to it). For example, the protein and DNA of a cell with a radius of 10µm and a dry mass of 350pg (cell volume 4.19pL, average refractive index 1.35) occupy approximately 0.73mL/g · 350pg = 0.256pL (assuming the PSV of protein and DNA are similar). Therefore, the actual volume of the intracellular fluid is 3.93pL (94% of the cell volume) which is equivalent to a mass of 3.93ng resulting in a total (actual) cell mass of 4.28ng. Thus, the dry mass of this cell makes up approximately 10% of its actual mass which leads to a total mass that is about 2% heavier than the equivalent volume of pure water (4.19ng).

## Default parameters in DryMass¶

• The default refraction increment is $$\alpha$$ = 0.18mL/g, as suggested for cells based on the refraction increment of cellular constituents by references [BJ54] and [Bar53]. The refraction increment can be manually set using the configuration key “refraction increment” in the “sphere” section.
• The default refractive index of the intracellular fluid in DryMass is assumed to be $$n_\text{intra}$$ = 1.335, an educated guess based on the refractive index of phosphate buffered saline (PBS), whose osmolarity and ion concentrations match those of the human body.