## in vitro COsmid packaging efficiency

The in vitro packaging efficiency is modelled using two decay functions as a function of the packaging substrate length x relative to the wild-type lambda DNA length x. The relative packaging efficiency y is defined such that the wild-type lambda phage DNA has an efficiency of 100%, while an expansion of the substrate size by 10% reduces the efficiency to 0.001 and a reduction of the substrate size by 30% reduces the efficiency to 0.01.

I. For the linear decay model, the relative packaging efficiency is expressed as:

Ln(y) = −30⋅ln(10)⋅(x/x0−1), when x > x0,

Ln(y) = −6.7⋅ln(10)⋅(1-x/x0), when x < x0,

II. For the quadratic decay model, it is given by:

Ln(y) = −300⋅ln(10)⋅(x/x0−1)^2, when x >x0,

Ln(y) = −22.2⋅ln(10)⋅(1-x/x0)^2, when x <x0,

These coefficients were specifically determined to accommodate the boundary conditions where the packaging efficiency is 1 at the wild-type length , and drops to 0.001 when the length is expanded by 10% (x =1.1⋅x) or 0.01 when the length is reduced by 30%(x=0.7*x0), respectively.

Although the packaging process for lambda DNA is highly complex—encompassing factors such as charge repulsion between the densely packed DNA, internal head pressure from the confined DNA, the strength of the packaging motor (terminase), the availability of ATP for energy, the role of ions like magnesium in stabilising the DNA structure within the capsid, and potential termination sequences on the substrate that can halt replication, preventing the synthesis of full-length packagable substrates.—this simplified evaluation is based solely on experimental boundary conditions. It uses the log-linear and log-quadratic decay functions to model packaging efficiency as a function of substrate length, providing a practical approach without delving into the full molecular details.