Team:Edinburgh/Phage Replication

Phage Replication

A basic activity in our biorefinery is the degradation of cellulose, due to the presence of enzymes. For our phage system, we are not only concerned with the activities and amount of enzymes, but also with the metabolism and activities of bacteriophage.

M13 Replication

M13 is a filamentous bacteriophage: a worm-like virus approximately 1 um long with a 10 nm diameter that infects only E. coli.

• The viral particle consists of a single-stranded, closed circular DNA core surround by a protein coat.
• Prior to virus assembly, the coat proteins are fixed in the bacterial membrane by transmembrane domains.
• During assembly, viral DNA is extruded through the membrane and concomitantly enveloped by coat proteins.
• The ends of the assembled virus are capped by four minor coat proteins, and the length of the filament is covered by several thousand copies of the major coat protein(P8).

From Slonczewski and Foster (2010).

• The M13 phage attacks E. coli (host), multiplies in the host cell cytoplasm, and is released without causing the bacteria’s death (non-lytic).

Model construction

At first thought, the rate of change of population = production rate of population - loss rate of population

Model for non lytic M13 phage

Equations

• dx/dt=a*k1*x-b*v*x

Rate of change of quantity of uninfected E. coli equals to the uninfected E. coli replicate itself minus the E. coli infected by M13 phage.

• dy/dt=a*k2*y+b*v*x

Rate of change of quantity of infected E. coli equals to the quantity of infected E. coli replicate itself plus the E. coli infected by M13 phage.

• dv/dt=c*y-b*v*x-m*v

Rate of change of quantity of free phage equals to the phage released by infected E. coli minus the phage which is to infect an E. coli and the decayed phage.

X(t) — uninfected E. coli

Y(t) — infected E. coli

V(t) — free phage

a — replication coefficient of E. coli

b — transmission coefficient of phage

c — replication coefficient of phage

m — decay rate of phage

K1, K2 — account for the difference of the rate of replication between infected E.coli and uninfected E.coli

Simulations

The MATLAB code uses a Runge-Kutta method of order four to solve the system.

Figure1 simulation value: x0=2.00E3 y0=v0=2.00E5

The simulation runs under the condition that the amount of uninfected E.coli is significantly smaller than others.The quantity of uninfected E.coli keeps at a low level, which may have economical meaning in practical, since our goal is to get free displayed phage. Besides, the figure also shows the infected E. coli population dominates the population of free phage.

Figure2 simulation value: x0=v0=2.00E5 y0=2.00E3

The same as Figure1, in this case, the population of infected E.coli also dominates the population of free phage. However, the excess amount of uninfected E.coli results in large amount of free phage infecting E.coli. Therefore, over the simulation time of 15 hours, we will get least free phage among this three cases. And the increasing rate of free phage rise significantly in this case, this is probably because large amount of free phage infected E.coli therefore leads to a significant rise in the amount of infected phage, which can release free phage later.

Figure3 simulation value: x0=y0=v0=2.00E5

A decrease of the slope of rise of uninfected E.coli is observed during the first 8 simulation hours. And even an increase of the slope of the fall of uninfected E.coli is observed later. This means probably that as the population of free phage increasing, more e.colis are infected by free phage. After 15 hours, we can get the most free phage among the three cases.

From these results it is evident that the population of the bacteriophages M13 primarily depends on the the population of infected E.coli, which is the host of bacteriaphages. And the slowing down of bacterial metabolism have little effect on the reproduction of phage.

References

• Gregory A. Weiss， Sachdev S. Sidhu(2000)[http://www.utoronto.ca/sidhulab/pdf/08.pdf: Design and Evolution of Artificial M13 coat Proteins]
• Slonczewski JL, Foster JW (2010) [http://www.wwnorton.com/college/biology/microbiology2/ch/11/etopics.aspx Microbiology: An Evolving Science], 2nd edition. W. W. Norton & Company
• Robert J.H Payne, Vincent A. A. Jansen(2011)[http://personal.rhul.ac.uk/ujba/115/jtb01.pdf Understanding Bacteriaphage therapy as a density-dependent kinetic process]
• Cattoen C (2003) [http://msor.victoria.ac.nz/twiki/pub/Groups/GravityGroup/PreviousProjectsInAppliedMathematics/bacteria-phage_REPORT.pdf Bacteriaphage mathematical model applied to the cheese industry]
• A.O. Converse, J.D. Optekar(1993) [http://onlinelibrary.wiley.com/doi/10.1002/bit.260420120/pdf: A synergistic Kinetics Model for Enzymatic Cellulose Hydrolysis Compared to degree-of-synergism Experimental Results]