1. What is frameshifting?
Frame-shifting used to synthesize multiple
proteins from a single mRNA with overlapping
ORFs [1]
Schematic of frame-shifting in modified dnaX gene in E. coli [2]
Viral frameshifting has been recognized as a method of condensing viral RNA by utilizing the same mRNA to encode for multiple proteins and increase the overall replication efficiency of the virus
It consists of overlapping ORFs, as well as a slippery sequence and the presence of a downstream secondary structure (possibly a stem loop (hairpin) or pseudo-knot structure)

2. What is frameshifting?
Can occur in (+) and (-) direction relative to the current ORF
The mechanisms for inducing a (+) frameshift and a (-) frameshift can vary
The slippery sequence ensures correct pairing between the mRNA and tRNA before and after the frameshift and varies for genes and organism
The location of the Shine-Dalgarno, slippery sequence, and stem loop structure in relation to one another may determine the timing of the -1 frameshift
The process is not quantitatively well understood and very few papers exist with both quantitative modeling and the required detail to allow for replication

3. The Model We Considered
“Model of the pathway of -1 frameshifting: Kinetics” Xie (2016)
Since various mechanisms and factors that affect FS have been proposed, Xie (2016) considered mechanisms that had available kinetic and biochemical data
The factors that were evaluated include
aa-tRNA kinetics inside the ribosome (via change in fluorescence of fluorescein (Flu))
Change in fluorescence resonance energy transfer (FRET) between tRNA (Flu) and a nonfluorescent acceptor (AttoQ) attached to protein S13
Elongation factor G (EF-G) binding and dissociation using a FRET donor placed on a ribosomal protein (L12)

4. Biochemical Data From Caliskan (2014)
Xie (2016) developed model to replicate the experimental fluorescence data of Caliskan (2014) and an additional model with modifications
Figure 4 in Caliskan (2014):Movement of tRNALeu

5. Kinetic Model of -1 PFR from Caliskan (2014)
Figure 5 in Caliskan (2014):Movement of tRNALeu

6. The Three Primary Models
Xie (2016) developed a slightly modified model to replicate the experimental fluorescence data of Caliskan (2014) and an additional model, which was modified from another paper they published on long pausing in frameshifting, Xie (2016)
The three models are:
(1) -/- mRNA
(2) +/+ mRNA Model I
(3) +/+ mRNA Model II
-/- mRNA model lacks both a slippery sequence and pseudoknot
+/+ mRNA models include both frameshifting elements as well as intersubunit rotations involved in translocation of tRNA-mRNA complex (since -1 frameshifting occurs during this step)
Model I and Model II only consider the frameshifting pathway since over 75% of translating ribosomes frameshift

7. Model I

8. Model II

9. +/+ mRNA Models Both +/+ mRNA models have two sub-models
Case I: One round of translocation
Case 2: Two rounds of translocation
Model II also takes into consideration intrasubunit rotation

10. Stochastic Simulation Algorithm
The Direct Method was used for the stochastic simulations
Each model required different stoichiometry matrices and propensity functions

11. SSA

12. Creating a stoichiometry matrixK2 K3 K4 K5 K6
K7’
K8’
K9’*Pc
K9’*(1-Pc) P1 -1 P2 1 P3 P4 P5 P6 P7 P8

13. Results

15. Laplace Transform & Moment Generating Function Analysis
P(a) = C(sI-A)-1P0
P(t) = ℒ-1(s)
M(s) = 𝑑 𝑑𝑠 𝑛 (𝐶(𝑠𝐼−𝐴) −1 𝑃 0 )
M(s)|s=0 = <tn>

16. Laplace Transform Replication of Fig. 8

17. Laplace Transform and Initial and Final Value Theorem
lim 𝑠→0 𝑠𝐹(𝑠) = lim 𝑡→∞ 𝐹(𝑡)
lim 𝑠→∞ 𝑠𝐹(𝑠) = lim 𝑡→0 𝐹(𝑡)

18. FSP Solution After solving for A Matirx
Solve for expm(A*(t-to)) at various points in time.
Uses an error term, but since probability can bleed out form the system, it is physically meaningful to our problem

19. FSP Replication of Fig. 8

20. Setup the matrix of coefficients (one round of translocation)
Define rate constants
Matrix of coefficients: Am = [ -k(2) ; k(2) -k(3) ; k(3) -k(4) ; k(4) -(k(5)+kd(1)) ; k(5) -(k(6)+kd(1)) ; k(6) -(k(7)+kd(1)) 0 0 k(10) ; k(7) -(k(8)+kd(1)) ; k(8) -(k(9)+kd(1)) ; (1-Pu)*k(9) -(k(10)+kd(1)) ; Pu*k(9) 0 -(k(11)+kd(2)) 0 0 0; k(11) -(k(12)+kd(2)) 0 0; k(12) -(k(13)+kd(3)) 0; k(13) -kd(3) ];

21. Define Flu1 & FRET1 Eq.s (49-50).
CFlu1=[1 1 1 A1 A2 A2 A2 A2 A2 A2 A2 A3 A3];
CFRET1=[1 B1 B1 B2 B3 B3 B3 B3 B3 B1*B3 B1*B3 B4 B4];
Flu1=CFlu1*P Eq.(49)
FRET1=1-CFRET1*P Eq.(50)
P is calculated after running ODE

22. ODE Initial conditions for probability:
Pi=zeros(1,13)';
Setting the initial value for P1 according to the article:
Pi(1)=1;
Function handle for the system of equations:
ODE solver for the function handle:
Data=ode45(ODEFUN1,tvec,Pi);
Solution matrix:
P=Data.y;
time results:
time=Data.x;

23. Inserting theoretical equations (one round of translocation)
Eq. (64)

24. Example results
Fig. 8 (e) & (f)
+/+ mRNA, Model II

25. References
[1] N. Caliskan, F. Peske, and M. V. Rodnina. “Changed in translation: mRNA recoding by -1 programmed
ribosomal frameshifting.” Trends Biochem Sci, vol. 40, no. 5, pp , May 2015.
[2] J. Chen, et al. “Dynamic pathways of -1 translational frameshifting.” Nature, vol. 512, pp , Aug.
2014.
[3] B. L. Bailey, K. Visscher, and J. Watkins. “A stochastic model of translation with -1 programmed
ribosomal frameshifting.” Phys Biol, vol. 11, no. 1 (016009), Feb 2014.
[4] P. Xie. “Model of the pathway of -1 frameshifting: Long pausing.” Biochem Biophys Rep, vol. 5, pp.
, Jan  
[5] P. Xie. “Model of the pathway of -1 frameshifting: Kinetics.” Biochem Biophys Rep, vol. 5, pp. 453-
467, Feb 2016.
[6] N. Caliskan, et al. “Programmed -1 Frameshifting by Kinetic Partitioning during Impeded Translocation.” Cell, vol. 157, pp , Jun 2014.