infection. Viral DNA degradation. CRISPR array. Protospacer recognition. Mutated protospacers. CRISPR e ector complexes. Initial protospacers. No recognition.
Optimal length of CRISPR arrays Alexander Martynov¹ Konstantin Severinov
1,2,3
Jaroslav Ispolatov
1,4
¹ Skolkovo Institute of Science and Technology, Moscow, Russia 2 Waksman Institute of Microbiology, Rutgers, The State University of New Jersey, Piscataway, NJ, USA 3 Institute of Molecular Genetics, Russian Academy of Sciences, Moscow, Russia 4 University of Santiago de Chile, Santiago, Chile
Model protospacer probability to mutate
Virus encounter
Number of spacers S
Poisson process
bound CRIPSR complexes interference efficiency
Interference probability
Cell survives
CRISPR array Mutated protospacers CRISPR effector complexes
Bacterial survival probability
Virus encounter
Evaluation of array composition that optimizes cell survival
Unequal crRNA production
Average encounter number with j-th virus over time t
E (t ) = ∏ exp[−rN j t (1 − I j )]
Viral DNA degradation
j
Interference probability Virus infection
Estimation of cell survival as a function of time
Cell dies
Time since spacer acquisition
No recognition
Spacer array length S
spacer-protospacer binding efficiency
Our goal was to evaluate the optimal CRISPR array composition given the different CRISPR-Cas system characteristics and virus environments. We designed an analytical model and studied its behavior in different regimes.
Initial protospacers
Cell survival as a fuction of system parameters
Ratio of i+1-th spacer concentration to i-th spacer
Protospacer recognition
efficiency
Binding efficiency
S
−1
Fraction of i-th spacer
Conclusions -
0.5
0.3
0.1
0.01 0.1
1− δ ) ] I = 1 − exp[− χ ∑ (1 + i −1 β δ δ (1 − ) i Interference 1
0.9
Virus mutation probability 1-µ
CRISPR-Cas system is an adaptive immune system in bacteria or archaea used to defend against the viruses. CRIPSR array functions as a virus recognition key that together with Cas proteins form CRISPR complexes can cleave virus DNA. Viruses tend to mutate protospacers - the DNA pieces that match spacers while cell pick new spacers to maintain immunity.
Cell modeled as fixed spacer array
Results
crRNA decay coefficient δ
Introduction
0.3
1
3
10
CRISPR effector binding efficeincy β
Within a given model there is an optimal CRISPR array length. High CRISPR efficiency leads to longer optimal CRISPR arrays while high virus mutation rate leads to shorter ones.
Optimal array length and maximal cell susvival as a function of protospacer mutation probability and spacer binding efficiency
We designed a new framework of thinking about the CRISPR array composition issue. Higher CRISPR efficiency leads to longer arrays and higher virus mutation rate lead to shorter arrays. Short CRISPR arrays can not be optimal and efficient unchanged while long ones can Unequal distribution of spacers in CRISPR complexes is beneficial to cell survival.
References [1] K. S. Makarova, N. V. Grishin, et al., A putative RNA-interference-based immune system in prokaryotes: computational analysis of the predicted enzymatic machinery, functional analogies with eukaryotic RNAi, and hypothetical mechanisms of action., Biol Direct. 2006 Mar 16;1:7 [2] E. V. Koonin and Y. I. Wolf, Evolution of the CRISPR-Cas adaptive immunity systems in prokaryotes: models and observations on virus-host coevolution, Mol. BioSyst., 2015,11, 20-2 [3] L. M. Childs, N. L. Held, et al. MULTISCALE MODEL OF CRISPR-INDUCED COEVOLUTIONARY DYNAMICS: DIVERSIFICATION AT THE INTERFACE OF LAMARCK AND DARWIN, Evolution. 2012 Jul;66(7):2015-29 [4] J. Iranzo, A. E. Lobkovsky, et al., Evolutionary dynamics of the prokaryotic adaptive immunity system CRISPR-Cas in an explicit ecological context, J Bacteriol. 2013 Sep;195(17):3834-44
For a large set of parameters, CRISPR system remains inefficient. Moreover, the length of the array is correlated with survival potential of the cell. Thus, short arrays could not be optimal and efficient while unchanged.
