The size of the immune repertoire of bacteria

TL;DR

This paper investigates the limitations on the size of bacterial CRISPR immune cassettes, revealing a fundamental trade-off between cassette size and immune response effectiveness, which constrains the evolution of bacterial adaptive immunity.

Contribution

It introduces a theoretical framework explaining why bacterial CRISPR cassettes are limited in size due to a trade-off between repertoire size and immune response efficiency.

Findings

Identifies a trade-off limiting CRISPR cassette size

Shows how this trade-off constrains immune repertoire evolution

Provides a quantitative model for immune response optimization

Abstract

Some bacteria and archaea possess an immune system, based on the CRISPR-Cas mechanism, that confers adaptive immunity against phage. In such species, individual bacteria maintain a "cassette" of viral DNA elements called spacers as a memory of past infections. The typical cassette contains a few dozen spacers. Given that bacteria can have very large genomes, and since having more spacers should confer a better memory, it is puzzling that so little genetic space would be devoted by bacteria to their adaptive immune system. Here, we identify a fundamental trade-off between the size of the bacterial immune repertoire and effectiveness of response to a given threat, and show how this tradeoff imposes a limit on the optimal size of the CRISPR cassette.

Equations

$$\displaystyle P_{\text{survival}}=1-\Big{(}$$

$$\displaystyle\times\sum_{d_{1}+d_{2}+\cdots=N_{p}}\big{[}1-\alpha({\bf v},{\bf d})\big{]}q({\bf d}|{\bf s})\Big{)}\,.$$

$$P_{\text{survival}}=1-\left(p_0+\sum _d{p}_1(1-\alpha (d))q(d)\right)$$

$$\displaystyle{P_{\text{survival}}=\left(1-e^{-\frac{\lambda}{\kappa}}\right)\times\left(1-\sum_{d<N_{p}}\big{(}1-\alpha(d)\big{)}q(d)\right)}\,,$$

$$0=1-{\lambda }^{\ast }\sum _{d=0}^{d_c}\mathcal{N}\left(\frac{1}{{\lambda }^{\ast }},\frac{1}{{\lambda }^{\ast }}\right)\left(\frac{3}{2{\lambda }^{\ast }}-\frac{d^2-({\lambda }^{\ast }{)}^{-2}}{2}\right),$$

$$1=$$

$$+\frac{1}{2}\left(\text{Erf}\left[\frac{1}{\sqrt{2{\lambda }^{\ast }}}\right]+\text{Erf}\left[\frac{d_c{\lambda }^{\ast }-1}{\sqrt{2{\lambda }^{\ast }}}\right]\right)$$

$$P_{\text{survival}}=\left(1-e^{-\frac{\lambda }{\kappa }}\right)\left(1-e^{-\frac{1}{\lambda }}\right).$$

$$\displaystyle 0=\frac{\partial}{\partial\lambda}P_{\text{survival}}\Big{|}_{\lambda^{*}}=\left(1-e^{-\frac{1}{\lambda^{*}}}\right)\frac{e^{-\frac{\lambda^{*}}{\kappa}}}{\kappa}-e^{-\frac{1}{\lambda^{*}}}\left(1-e^{-\frac{\lambda^{*}}{\kappa}}\right)\frac{1}{\left(\lambda^{*}\right)^{2}}.$$

$$\frac{1}{\kappa }f(a\sqrt{\kappa })=\frac{1}{a^2\kappa }f\left(\frac{\sqrt{\kappa }}{a}\right)$$

$$L^{\ast }=N_p{\lambda }^{\ast }=\sqrt{N_{p}K}.$$

Full text

I Introduction

All living things from bacteria to whales are under constant threat from viruses. To defend against these threats many organisms have developed innate mechanisms that make it harder for infections to occur (e.g., by impeding entry of viruses through the cell membrane), or that respond universally to the presence of infections (e.g. through inflammatory responses) Labrie2010 . While such innate defenses are very important, they likely cannot be as effective as defenses that specifically target particular infections. The challenge for developing specific responses is that they require a mechanism for learning to identify each incident virus, and a mechanism for remembering the defended targets. Vertebrates implement such a learning-and-memory approach in their adaptive immune system, which produces novel antibodies through random genomic recombination, selects effective immune elements when they bind to invaders, and then maintains a memory pool to guard against future invasions. Recently, we have learned that some bacteria also enjoy adaptive immunity Barrangou2014 – they maintain a cassette of “spacers” (snippets of DNA from previously encountered phage), and use these spacers via the CRISPR-Cas mechanism to identify and clear recurring infections by similar viruses.

The CRISPR-Cas mechanism for immunity has three stages Barrangou2014 ; Bonomo2018 . Following a first encounter with a virus, after a successful defense through another mechanism or if the virus is ineffective for some reason Hynes:2014aa , some of the Cas proteins recruit pieces of viral DNA and integrate these “spacers” into an array separated by palindromic repeated sequences in the CRIPSR locus of the bacterial genome. This array defines the CRISPR “cassette” which represents the memory of past infections. In the second stage, the entire CRIPSR locus is transcribed into RNA through a mechanism that depends on the CRISPR type of the bacterium. Finally, invading sequences are recognized by base-pairing with the CRISPR RNA. A successful match triggers cleavage of the virus through the action of complexes formed by other Cas proteins.

In lab experiments that expose naive bacteria, which start without spacers, to carefully controlled environments, the CRISPR cassette is often small, consisting of a few spacers acquired during interactions with phage. Wild-type bacteria can have larger cassettes, but even these contain only a few tens to at most a few hundreds of spacers Devashish_2015 ; Martynov2017 ; grissa2007crisprdb ; Barrangou2014 . Metagenomic analysis of the human gut microbiome has revealed CRISPR cassettes with an average size of twelve spacers Mangericao2016 . A broader analysis of all sequenced bacteria and archeae found cassette lengths clustered in the twenty to forty range Grissa:2007 . Similarly, a study of 124 strains of Streptococcus thermophilus revealed an average cassette size of 33 Horvath:2008 .

These findings lead to a puzzle. Why do bacteria maintain such small memories in their adaptive immune systems given that they have probably been exposed over generations to thousands of species of phage? Certainly, the size of the genome is not a constraint since bacterial genome sizes lie in the range of millions of base pairs. One idea could be that viruses evolve so quickly or are encountered so rarely that a deep memory is not useful Martynov2017 . Here we propose an alternative hypothesis: given limited resources for Cas-complex formation Vale2015a , a deep memory imposes an opportunity cost by reducing the chance for the spacer specific to an invader to be activated in time to cleave the virus before it reproduces. Thus, there is a tradeoff between effectiveness of the defense and the depth of memory.

We analyze this tradeoff quantitatively and demonstrate that it predicts an optimal size for the CRISPR cassette. This optimum is controlled by the number of Cas complexes that are available for cleaving viruses, and by the diversity of the phage landscape. In the limit that the phage landscape is very diverse, the size of the cassette is largely limited by the number of Cas proteins that the bacterium can produce while carrying on its other functions Vale2015a . Using the biologically relevant range for Cas protein concentration in a bacterium, we show that the optimal cassette size should typically lie in the range of 10-100 spacers, consistent with genomic observations Devashish_2015 ; Martynov2017 ; Mangericao2016 ; Grissa:2007 ; Horvath:2008 .

II Results

II.1 CRISPR as a probabilistic memory of phage

We consider a model of infection where bacteria encounter $j=1,\ldots,K$ types of phage, each with probability $f_{j}$. For simplicity, all types of phage are taken to be equally infectious and to have similar growth rates, conditions that are easily relaxed. In the CRISPR mechanism for adaptive immunity, bacteria incorporate snippets of phage DNA (“spacers”) into the CRISPR cassette. Upon later infection, the bacteria recruit CRISPR-Cas complexes with spacers that match the invading phage to cleave the viral DNA (Fig. 1).

Suppose that the cassette contains $L$ spacers, and that an individual bacterium maintains a population of $N_{p}$ Cas protein complexes that can be recruited to cleave invaders. The cassette configuration can be characterized in terms of a vector ${\bf s}=\{s_{1}\cdots s_{K}\}$ with entries counting the number of spacers specific to each phage type. The cassette size is the sum of $s_{j}$, i.e. $\sum_{j}s_{j}=L$, and quantifies the amount of immune memory stored by an individual bacterium. We describe the phage configuration in a given infection event as a vector ${\bf v}$ of length $K$ with entries indicating presence (1) or absence (0) of each viral type. Finally, we define the configuration of complexes ${\bf d}=\{d_{1}\cdots d_{K}\}$ as a vector with entries counting the number complexes specific to each phage during the CRISPR response. The total number of complexes is the sum of $d_{j}$, i.e. $\sum_{j}d_{j}=N_{p}$. In terms of these variables, the probability of surviving a phage infection using the CRISPR-Cas defense mechanism is

[TABLE]

Here, $p_{V}({\bf v})$ is the probability of encountering the phage configuration ${\bf v}$, $p_{S}({\bf s}|L)$ is the probability of having a cassette configuration ${\bf s}$ of length $L$, $\alpha({\bf v},{\bf d})$ is the probability of detecting all the viral types present in ${\bf v}$ given the configuration ${\bf d}$ of complexes, and $q({\bf d}|{\bf s})$ is the probability of producing the CRISPR-Cas configuration ${\bf d}$ given the set of spacers ${\bf s}$ and $N_{p}$ Cas protein complexes.

The form of the detection probability function $\alpha({\bf v},{\bf d})$ depends on the specific mechanism used by the CRISPR machinery to bind and degrade a phage. However, a critical number of specific complexes, $d_{c}$, is required for the CRISPR machinery to achieve targeting at the speeds measured in experiments Jones1420 . Thus, we consider two functional forms for the probability that a particular phage type can be detected with $d$ specific complexes: (i) a hard constraint $\alpha(d)=\theta(d-d_{c})$, where $\theta(x)$ is step function and is 0 if $x<0$ and 1 if $x>0$, and (ii) a soft constraint $\alpha(d)={d^{h}\over d^{h}+d_{c}^{h}}$. In both cases $d_{c}$ is an efficacy parameter that depends on biochemical rates, and determines the threshold on $d$ below which detection is rare and above which detection is common. The first functional form (step function) describes switch-like behavior where complexes bind to phage DNA if they exceed a certain concentration $d>d_{c}$. The second form mimics a Hill-like response, where the chance of binding increases gradually with the number of complexes. Here, we allow the binding of Cas-complexes with phage DNA binding to be cooperative. As the cooperativity $h$ increases, the binding behavior becomes increasingly switch-like.

II.2 There is an optimal amount of memory

In realistic settings we can make simplifying assumptions about the general model in eq. 1. For example, we can assume that successful infections of a bacterium by different phages occur with low probability and are independent. Of course, a given bacterium can be infected by multiple phages over its lifetime. Since the probability that a bacterium simultaneously encounters multiple phages is small, we assume that encounters are sequential (i.e., the viral configuration vector ${\bf v}$ has a single nonzero entry).

Second, we assume that a bacterium’s lineage encounters many and diverse phage types over multiple generations, i.e., $K\gg 1$. Because phages mutate readily, there is subtlety about what defines a “type”. We use a functional definition – a “type” of phage is defined by its specific recognition by a given spacer. Sometimes, after a bacterium becomes immune to a phage, single point mutations in the virus can produces “escapers” that evade recognition. By our definition these escapers are effectively a new type of virus that the bacterial population must deal with sequentially in future infections Deveau2008 ; Heidelberg2009 ; Sun2013 ; Houte2016 .

Third, we assume that spacers are uniformly sampled from the phage distribution over time. This distribution is not known experimentally, so we make a minimal assumption that all phage types are equally likely and occur with probability $1/K$. This is a conservative assumption, because bacterial immune memory confers the least advantage when faced with an unbiased (i.e., a minimally informative) phage environment. In effect, we are not interested here in the short-time co-evolutionary arms race between bacterium and phage, and are instead considering the long-term statistical features.

Finally, we assume that phage encounters from which spacers are acquired occur randomly. Thus each spacer in the CRISPR cassette has a probability $1/K$ of being specific for a given phage. Since the cassette size is much smaller than the number of viral types ($L\ll K$), it follows that the cassette will either have one or no spacer that can target a particular phage type – in other words $s_{i}=0,1$. We also assume that all spacers along the cassette are equally expressed, in the absence of alternative experimental evidence. A biased expression of spacers may explain finite cassette sizes Martynov2017 , but our model does not require such additional assumptions.

We derive an expression for the probability to survive a phage infection, given the cassette size $L$, the number of Cas complexes $N_{p}$ and the diversity of the phage population $K$ (Methods, Eq. 3). Across a wide range of parameters and for various choices of detection probability functions $\alpha(\bf{v},\bf{d})$, we find that there is an optimal amount of memory, consisting of a few tens of spacers in the CRISPR cassette, to maximize survival probability (Fig. 2 and Fig. S1). The optimum occurs because there is a trade-off between the amount of stored memory in CRISPR cassettes and the efficacy with which a bacterium can utilize its limited resources (i.e., Cas proteins) to turn memory into a functional response. If the CRISPR cassette is too small, bacteria do remember past phage encounters well enough to defend against future infections. On the other hand, if the cassette is too large, Cas complexes bind too infrequently to the correct spacer to provide effective immunity against a particular invading virus. The optimal amount of immune memory (cassette size) should lie in between these two extremes, with details that depend on the phage diversity, the number of Cas complexes, and the detection probability function $\alpha(\bf{v},\bf{d})$ (Fig. 2 and Methods).

II.3 Optimal memory depends on phage diversity

How does the optimal amount of CRISPR memory depend on the diversity of viral threats? If there are relatively few types of phage, an optimal strategy for a bacterium would be to maintain an effective memory for most threats and to match the viral variants with a cassette whose size grows with viral diversity. This matching strategy will eventually fail as the diversity of the phage population increases if the number of Cas proteins ($N_{p}$) is limited. We examined this tradeoff by measuring the optimal cassette size as we varied the viral diversity ($K$) while keeping the CRISPR machinery ($N_{p}$ and the detection probability $\alpha(\bf{v},\bf{d})$) fixed.

To characterize viral diversity, we defined a parameter $\kappa=K/N_{p}$ as the ratio of the number of phage types ($K$) and the number of Cas complexes ($N_{p}$). When phage diversity is low ($\kappa\leq 1$) the optimal amount of memory (cassette size) increases sub-linearly with viral heterogeneity (Fig. 3), approximately as a power law. When the detection probability $\alpha(\bf{v},\bf{d})$ is nearly switch-like, the optimal cassette size scales approximately as $L\sim\sqrt{K}$ (Fig. 3A,B). This implies that when viral diversity is low the amount of memory should increases with the diversity, but it is actually beneficial not to retain a memory of all prior phage encounters. Forgetting some encounters will allow the bacterium to mount a stronger response against future threats by engaging a larger number of Cas complexes for the threats that are remembered. This sub-linearity in the optimal amount of memory becomes stronger as the the number of phage-specific CRISPR-Cas complexes necessary for an effective response, $d_{c}$, increases.

When phage diversity is high ($\kappa>1$), the optimal amount of memory depends on the CRISPR mechanism via the response threshold $d_{c}$, but is independent of viral heterogeneity so long as $d_{c}\geq 2$ (Fig. 3; see SI for discussion of the special case $d_{c}=1$). In nature, phage are expected to be very diverse ($K\gg N_{p}$). Thus, our model predicts that the cassette size of a bacterium is determined by the expression level of Cas complexes $N_{p}$ and the detection threshold $d_{c}$ of the particular CRISPR mechanism that is used by the species.

II.4 Memory increases with detection efficacy

Detection efficacy depends on two key parameters: (i) the detection threshold $d_{c}$ and (ii) the number of available Cas proteins $N_{p}$. In Fig. 4, we show that the optimal cassette size for defending against a diverse phage population ($\kappa=K/N_{p}\gg 1$) decays as a power law of the detection threshold $\sim(d_{c}/N_{p})^{-\beta}$ with an exponent $\beta\simeq 1$ (Methods). This decay occurs because the spacers in a cassette compete to form complexes with Cas proteins; thus, having more distinct spacers effectively decreases the average number of complexes that would be specific to each infection. Thus, the smaller the cassette size, the more likely that the $d_{c}$ specific complexes required for an effective CRISPR response will be produced. The optimal cassette size is a compromise between this drive towards having less memory, and the drive to have a defense that spans the pathogenic landscape.

If the detection probability depends sharply on the number of bound complexes with a threshold $d_{c}$ (Fig. 4a), to a first approximation, fewer than $d_{c}$ complexes bound to a specific spacer are useless, as detection remains unlikely, and larger than this number is a waste, as it would not improve detection. In this case, if the expression of the Cas protein was a deterministic process, it would be optimal to have a cassette with $N_{p}/d_{c}$ spacers, each of which could be expressed and bind to exactly $d_{c}$ complexes, predicting that $L=(d_{c}/N_{p})^{-1}$. However, since gene expression is intrinsically stochastic, there would sometimes be more than $d_{c}$ bound complexes for a given spacer, and sometimes less. This stochastic spreading weakens the dependence of the optimal cassette size on the threshold $d_{c}$, resulting in the exponent $\beta$ and coefficient $C$ to be slightly less than one in the optimal cassette size scaling $L\sim C(d_{c}/N_{p})^{-\beta}$ (Fig. 4 insets); see Methods for a more detailed derivation. If the detection threshold is soft, the CRISPR mechanism effectiveness is less dependent on having at least $d_{c}$ complexes, specific to a phage. In addition, having a slightly higher number of complexes than the detection threshold can increase the detection probability. These effects combine to produce a scaling between the optimal cassette size and the detection efficacy, $L\sim(d_{c}/N_{p})^{-1}$ (Fig. 4b,c).

In summary, our model predicts that a more effective CRISPR mechanism (i.e., having lower detection threshold $d_{c}$ or a larger number of complexes $N_{p}$) should be associated with a greater amount of immune memory (i.e., larger cassette size $L$).

Our model also provides an estimate for typical cassette size in bacterial populations countering a diverse set of pathogens (regime of $K\gg N_{p}$). Assume that the typical number of Cas complexes is $N_{p}\sim 1000$, comparable to the copy number of other proteins in a bacterial cell Milo2013 ; Price2016 , and that rapid detection of an infecting phage requires a modest number of activated CRISPR-Cas complexes, with a detection threshold in range of $d_{c}\sim 10-100$ Jones1420 . Our model then predicts that the optimal CRISPR cassette size should lie in the range of $L\sim 10-100$, consistent with empirical observations Devashish_2015 ; Martynov2017 ; grissa2007crisprdb ; Barrangou2014 ; Mangericao2016 ; Horvath:2008 .

III Discussion

A bacterium’s ability to neutralize phage attacks depends on the number of spacers stored in its CRISPR cassette. In laboratory experiments, where viral diversity is limited, a few new acquired spacers per bacterium are enough to stabilize a bacterial population. A Streptococcus thermophilus population defending against continual phage attack acquired at most four new spacers over $\sim 80$ generations, with over $50\%$ of the population having only one new spacer. A different experimental design where individual bacteria with different spacers were mixed found that an overall spacer diversity of $\geq 20$ across the starting bacterial innoculum was enough to stabilize the population.

Surprisingly, in natural populations, where viral diversity may be high, cassette lengths are still small. Metagenomic analysis of the human gut microbiome revealed CRISPR cassettes with twelve spacers on average Mangericao2016 . A much broader analysis over all sequenced bacteria and archeae found cassette lengths clustered in twenty to forty range Grissa:2007 . In agreement with this global analysis, a study targeted at 124 strains of Streptococcus thermophilus revealed an average cassette size of 33 Horvath:2008 . One explanation for these observations may be that a bacterium in the wild encounters only a small set of phages (e.g. due to spatial constraints in habitats). Alternatively, it may be that the small cassette size results from a functional tradeoff in the bacterial immune system. Our theory examines both how these considerations – phage diversity and tradeoffs in the CRISPR mechanism – affect the optimal amount of immune memory that a bacterium should store.

We identified two qualitatively distinct regimes for the statistics of optimal CRISPR immune memory. When phage diversity ($K$) is much lower than the number of available Cas proteins ($N_{p}$), the CRISPR cassette length $L$ should increase sublinearly in $K/N_{p}$, approximately as the square root (Fig. 3). This resembles the previously identified optimal surveillance strategy of “square-root biased sampling” to catch rare harmful events while minimizing the wasteful resources spent on profiling innocents Press:2009 ; Mayer2015b .

When phage diversity is high ($K\gg N_{p}$), the number of available Cas proteins $N_{p}$ limits effective use of memory to mount an immune response. In this regime, the optimal cassette size increases linearly with $N_{p}$, $L\sim N_{p}/d_{c}$. Here $d_{c}$ quantifies the average number of Cas complexes specific to a a given phage that is necessary to mount an effective defense. Why would the number of Cas proteins be limited in the first place? Perhaps because there is in general a physiological cost to maintaining high levels of any protein within a cell. High sustained expression of the Cas system in particular may lead to auto-immune like phenotypes, where a bacterium’s lifespan is reduced because of the acquisition of spacers from its own genome Jiang2013 . This may explain why, in some bacteria, the expression of Cas proteins is controlled by a quorum sensing pathway that is triggered when the density of bacteria is high, making them more susceptible to infections Patterson2016 ; Kroghsboe_2012 ; Hoyland-Kroghsbo2017 .

To understand the tradeoffs that apply to CRISPR cassette size in the wild, we require a better understanding of both bacterial physiology and phage diversity in local environments. Unfortunately, quantifying phage diversity from metagenomic data is challenging. Phage genomes are constantly changing as they undergo error-prone replications due to sub-optimal use of hijacked bacterial replication machinery. Additionally, phages often swap pieces of DNA with their hosts as they are assembled and packaged inside an infected bacterium, leading to drastic changes in their gene synteny. These large-scale genomic changes challenge the reliability of genome assembly and alignment techniques in quantifying phage diversity in a given community. At the same time, the expression levels of Cas proteins are also not well-quantified across broad bacterial families. However, we can expect that these questions will be addressed by ongoing advances in metagenomics and in high-throughput expression measurements in bacterial communities.

The authors of Martynov2017 proposed a different theory for the small size of CRISPR cassettes. They suggested that the CRISPR mechanism reads out more recently stored memories first – i.e., CRISPR is a Last-In-First-Out (LIFO) memory system. In this picture, as old memories become less useful, they are harder to access and will degrade by genetic drift, restricting the size of the “functional” CRISPR cassette. Our model proposes an alternative explanation — i.e., CRISPR cassettes are small simply because they are adapted for optimizing the immune response in a world where threats are diverse while the CRISPR machinery for detecting and cleaving phage is subject to molecular bottlenecks. The alternative theories can be experimentally distinguished by better understanding the readout mechanism of the CRISPR cassette. Of course it could also be that both mechanisms are at play in limiting the utility of deep immune memory.

Vertebrates also possess an adaptive immune system that learns from past infections to defend against future threats. It has been suggested that pathogen detection in vertebrates is optimized by biasing the immune repertoire towards sensing rare infections with a higher chance than warranted by their frequency Mayer:2015 . This sort of optimization is unlikely to be relevant to individual bacteria given the small cassette size and high cost of CRISPR proteins. However, it is known that across a bacterial population spacer abundances are highly variable but distributed in a stereotypic way Bonsma-Fisher:2018 . It is possible that this stereotyped distribution represents an optimally adaptive immune strategy for the population as a whole. To study this question, our framework could be extended to analyze optimal strategies for distributed adaptive immunity in microbial communities.

For both vertebrates and bacteria, a major challenge is to characterize dynamics of the immune system as it chases a diverse pathogenic population that is itself evolving to evade detection by its hosts. This out-of-equilibrium process can last over an extended evolutionary period Nourmohammad:2016 . Most models, including the one presented here, ignore these dynamical aspects of the co-evolving CRISPR-phage system, which may partly influence the statistics of stored memory, including the optimal cassette length. Accounting for these dynamics will shed additional light on the organization of immune memory and the de-novo response to evolving pathogens in organisms with well-adapting immune systems.

IV Methods

IV.1 Probability of successful immune response

Applying the assumptions described in the Result section, the complete model (eq. 1) reduces to,

[TABLE]

where $p_{0}$ and $p_{1}$ are respectively the probabilities for a bacterium to have [math] or $1$ spacers specific to the infecting phage, $d$ is the number of specific complexes, $q(d)$ is the probability of producing $d$ complexes, and $\alpha(d)$ is the probability that a cassette producing $d$ specific complexes recognizes the phage. We assume that it is unlikely for a bacterium to carry more than 1 spacer against a given phage ($p_{1}\approx 1-p_{0}$), and take infections to be random events drawn from a pool of $K$ distinct viruses. Hence, the probability that none of the spacers in a cassette of size $L$ recognizes an invading phage is $p_{0}=(1-1/K)^{L}\approx e^{-L/K}$. The survival probability is

[TABLE]

where $\kappa=K/N_{p}$ and $\lambda=L/N_{p}$ denote a normalized viral diversity and immune capacity, respectively. We assume that transcription events occur independently and are equally likely. Thus, $q(d)$ in eq. 3 is given by a binomial density describing the probability of having $d$ complexes specific to a given phage, given $N_{p}$ Cas proteins and an equal probability of selecting any one of the $L$ spacers to produce each complex. A successful detection typically requires activation of multiple complexes with a minimum (critical) number, $d_{c}$. Accordingly, we choose the detection probability function $\alpha(d)$ to be a threshold function that saturates to $1$ at $d\geq d_{c}$ (SI).

IV.2 Optimal cassette size

The optimal cassette size relative to the number of Cas proteins, $\lambda=L/N_{p}$, can be evaluated by optimizing the survival probability \frac{\partial}{\partial\lambda}P_{\text{survival}}\Big{|}_{\lambda^{*}}=0. In the biologically realistic regime, where the number of complexes is both large $N_{p}\gg 1$, and large compared to the cassette size $N_{p}/L\gg 1$, the binomial probability density for the number of specific complexes $d$ in eq. 3 can be approximated by a Gaussian $\mathcal{N}\left(\lambda^{-1},{\lambda}^{-1}(1-\frac{\lambda^{-1}}{N_{p}})\right)\simeq\mathcal{N}\left(\lambda^{-1},{\lambda}^{-1}\right)$, up to quantities of order $\mathcal{O}(1/\sqrt{N_{p}})$. In this limit, the optimization criterion gives

[TABLE]

where we assumed a diverse pool of viruses $\kappa\gg 1$ and a sharp recognition function $\alpha(d)=\theta(d-d_{c})$. Approximating the sum in eq. (3) with an integral with strict boundaries $[0,d_{c}]$, we arrive at an equation for the optimal cassette size $\lambda^{*}$,

[TABLE]

In the limit that the cassette size is much smaller than the number of available complexes $L/N_{p}=\lambda\ll 1$, with $d_{c}\lambda$ finite, the optimal cassette length scales inversely with the activation threshold $L^{*}\sim\left(\frac{d_{c}}{N_{p}}\right)^{-1}$.

**Acknowledgments: ** VB is supported in part by a Simons Foundation grant in Mathematical Modeling for Living Systems (#400425) for Adaptive Molecular Sensing in the Olfactory and Immune Systems, and by the NSF Center for the Physics of Biological Function (PHY-1734030). AN is supported by the DFG grant (SFB1310) for Predictability in Evolution and the MPRG funding through the Max Planck Society. SG is supported by the NSERC Discovery Grant. VB, SB, and AN thank the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-160761, for hospitality in the initial stages of this work.

Optimal cassette size for detection threshold $d_{c}=1$

In the case of a switch-like receptor where only one complex is needed to detect the phage, i.e. $\alpha(d)=\theta(d-d_{c})$ and $d_{c}=1$ we get

[TABLE]

So the condition for the optimal cassette is

[TABLE]

Introducing $f(x)=e^{-x/\kappa}\left(1-e^{-1/x}\right)$ and $a^{2}=\left(\lambda^{*}\right)^{2}/\kappa$, equation (S2) entails,

[TABLE]

The relationship in equation S3 is only satisfied when $a=1$, implying that the ansatz with $a=1$ and $\lambda^{*}=\sqrt{\kappa}$ is always a solution. Thus, the optimal cassette size follows,

[TABLE]

As discussed in the main text and shown in Fig. 3, for larger detection thresholds $d_{c}\geq 2$, the sublinear relation between the optimal cassette size and the phage diversity is only valid in the regime of low phage diversity $K\ll N_{p}$. For large phage diversity ($K\gg N_{p}$) and as long as $d_{c}\geq 2$, the optimal amount of memory depends only on the CRISPR mechanism via the response threshold $d_{c}$, but does independent on the viral heterogeneity (Methods, Fig. 3).