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)

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

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)

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

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

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

Model I

Model II

+/+ 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

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

SSA

Creating a stoichiometry matrix K2 K3 K4 K5 K6 K7’ K8’ K9’*Pc K9’*(1-Pc) P1 -1 P2 1 P3 P4 P5 P6 P7 P8

Results

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>

Laplace Transform Replication of Fig. 8

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)

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

FSP Replication of Fig. 8

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

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

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: ODEFUN1=@(t,P)Am*P; ODE solver for the function handle: Data=ode45(ODEFUN1,tvec,Pi); Solution matrix: P=Data.y; time results: time=Data.x;

Inserting theoretical equations (one round of translocation) eFlu1=@(t)0.58*[0.22*exp(-11*t)+0.78*exp(-0.9*t)]+0.4; Eq. (64) eFRET1=@(t)0.75-0.58*[0.85*exp(-0.9*t)+0.15*exp(-0.2*t)];

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

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. 265-274, May 2015. [2] J. Chen, et al. “Dynamic pathways of -1 translational frameshifting.” Nature, vol. 512, pp. 328-332, 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. 408-424, Jan 2016.   [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. 1619-1631, Jun 2014.