Schematic for artificial selection on collectives.

Each selection cycle begins with a total of g Newborn collectives, each with N0 total cells of slow-growing S population (light gray dots) and fast-growing F population (dark gray dots). During maturation (over time τ), S and F cells divide at rates rS and rS + ω (ω > 0), respectively, and S mutates to F at rate µ. In the selection stage, the Adult collective with F frequency f closest to the target composition is chosen to reproduce g Newborns for the next cycle. Newborns are sampled from the chosen Adult (yellow star) with N0 cells per Newborn. The selection cycle is then repeated until the F frequency reaches a steady state, which may or may not be the target composition. To denote a variable x of i-th collective in cycle k at time t (0 ≤ t ≤ τ), we use notation where x ∈ {S, F, s, f}. Note that time t = 0 is for Newborns and t = τ is for Adults.

Nomenclature

Initial and target compositions determine the success of artificial selection on collectives.

(a-c) F frequency of the selected Adult collective (f *) over cycles at different target values (dashed lines). between f L and f H (orange dotted and solid line segments) is inaccessible where selection will fail. a A high target F frequency (e.g. ; magenta dotted line) can be achieved from any initial frequency (black dots). b An intermediate target frequency (e.g. ; green dotted line) is never achievable, as all initial conditions converge to near f H. c A low target frequency (e.g.) is achievable, but only from initial frequencies below f L. For initial frequencies at f L, stochastic outcomes (grey curves) are observed: while some replicates reached the target frequency, others reached f H. For parameters, we used S growth rate rS = 0.5, F growth advantage ω = 0.03, mutation rate µ = 0.0001, maturation time τ≈ 4.8, and N0 = 1000. The number of collectives g = 10. Each black line is averaged from independent 300 realizations. d Inter-collective selection opposes intra-collective selection. We plot probability density distributions of F frequency f during two consecutive cycles when selection is successful. Data correspond to cycles 31 and 32 from the second lowest initial point in c. Δf is a selection progress (see Box.). Black triangle: median. e Two accessible regions (gold). Either high ; region 2) or low starting from low initial f (and ; region 1) can be achieved. We theoretically predict (by numerically integrating Eq. (1)) f H (orange solid line) and f L (orange dotted line), which agree with simulation results (gold regions). f Example trajectories from initial compositions (black dots) to the target compositions (dashed lines). The gold areas indicate the region of initial frequencies where the target frequency can be achieved. g The tension between intra-collective selection and inter-collective selection creates a “waterfall” phenomenon. See the main text for details.

Intra-collective selection and inter-collective selection jointly set the boundaries for selection success.

a The change in F frequency over one cycle. When is sufficiently low or high, inter-collective selection can lower the F frequency to below .The points where Δf = 0 (in the orange line) are denoted as f L and f H, corresponding to the boundaries in Fig. 2. b The distributions of frequency differences obtained by 1000 numerical simulations. The cyan, purple, and black box plots respectively indicate the changes in F frequency after intra-collective selection (the mean frequency among the 100 Adults minus the mean frequency among the 100 Newborns during maturation), after inter-collective selection (the frequency of the 1 selected Adult minus the mean frequency among the 100 Adults), and over one selection cycle (the frequency of the selected Adult of one cycle minus that of the previous cycle). The box ranges from 25% to 75% of the distribution, and the median is indicated by a line across the box. The upper and lower whiskers indicate maximum and minimum values of the distribution. ***: P < 0.001 in an unpaired t-test.

Expanding the success region for artificial collective selection.

a Reducing the population size in Newborn N0 expands the region of success. In gold area, the probability that becomes smaller than in a cycle is more than 50%. We used g = 10 and τ ≈ 4.8. Figures 2-3 correspond to N0 = 1000. Black dotted line indicates the critical Newborn size ,below which all target frequencies can be achieved. b Increasing the total number of collectives g also expands the region of success, although only slightly. We used a fixed Newborn size N0 = 1000. The maturation time τ (τ = log(100)/rS≈ 9.2) is set to be long enough so that an Adult can generate at least 100 Newborns. c Increasing the maturation time τ shrinks the region of success. We used a fixed Newborn size N0 = 1000 and number of collectives g = 10.

In higher dimensions, the success of artificial selection requires the entire evolutionary trajectory remaining in the accessible region.

a During collective maturation, a slow-growing population (S) (with growth rate rS ; light gray) can mutate to a fast-growing population (F) (with growth rate rS + ω; medium gray), which can mutate further into a faster-growing population (FF) (with growth rate rS + 2ω; dark gray). Here, the rates of both mutational steps are µ, and ω > 0. b Evolutionary trajectories from various initial compositions (open circles) to various targets. Intra-collective evolution favors FF over F (vertical blue arrow) over S (horizontal blue arrow). The accessible regions are marked gold (see Sec. I in Supplementary Information). We obtain final compositions starting from several initial compositions while aiming for different target compositions in i, ii, and iii. The evolutionary trajectories are shown in dots with color gradients from the initial to final time. (i) A target composition with a high FF frequency is always achievable. (ii) A target composition with intermediate FF frequency is never achievable. (iii) A target composition with low FF frequency is achievable only if starting from an appropriate initial composition such that the entire trajectory never meanders away from the accessible region. The figures are drawn using mpltern package [36]. c The accessible region in the three-population problem is interpreted as an extension of the two-population problem. First the accessible region between FF and S+F is given, and then the S+F region is stretched into S and F.

Comparison between the calculated Gaussian distribution (“Gauss”, with the mean and variances computed from Eqs. (10),(11),(20),(25)) and simulations using tau-leaping (“tau”). The simulations run 3000 times. The initial number of cells are (S0, F0) = (990, 10), (500, 500), and (10, 990) for each column. The parameters r = 0.5, ω = 0.03, µ = 0.0001, and τ = 4.8 are used.

Congruence between consecutive sampling (MHG for multivariate hypergeometric distribution) and independent binomial (BN) sampling. The initial numbers of cells are S = 8000 and F = 2000 for the left panel, and S = 20 and F = 5 for the right panel. 10000 samples are drawn for each distribution. Here, a parent collective is divided into 10 collectives.

a Trajectories of F frequency for 10 collectives (g = 10) over time. The collective whose frequency is closest to the target value is selected in every cycle (black lines). The gray lines denote the other collectives. For parameters, we used S growth rate rS = 0.5, F growth advantage ω = 0.03, mutation rate µ = 0.0001, maturation time τ ≈ 4.8, and N0 = 1000. b Comparison between frequency trajectories with selection (the chosen one Adult producing all offspring; black) and without selection (each Adult producing one offspring; blue) clearly shows the effect of artificial selection. The black line indicates F frequency of the selected collective at each cycle in a. The blue line indicates the average trajectory without selection (the average of g = 10 individual lineages without inter-collective selection at the end of each cycle).

Color map of the absolute error averaged selected collectives at the end of simulations (k = 1000) and the target frequency . The solid and dashed lines are drawn by the arguments in the main text. For parameters, we used rS = 0.5, ω = 0.03, µ = 0.0001, N0 = 1000, g = 10 and τ ≈ 4.8. The result is the average of 300 independent simulations. Compared to Fig. 2e, this figure has a higher resolution.

The probability density functions of the selected Adult’s F frequency subtracted by . For simulations (blue), at each ,we performed 1000 stochastic simulations. The orange distribution represents Eq. (39) computed by numerical integration. The median values of the distributions are shown in Fig. 4a in the main text.

a Mean (Eq. (33)) and variance (Eq. (34)) of f values of Adult collectives with respect to the Newborn frequency f0. b Scaling relation of F frequency variance (Eq. (41)) with Newborn collective size N0. The initial F frequency is 0.5. The parameters are rS = 0.5, ω = 0.03, µ = 0.0001, and τ ≈ 4.8. c Relation of F frequency variance (Eq. (41)) with maturation time τ. Other parameters are the same as b.

Median (orange) and mean (violet) have similar distributions. We performed 1000 simulations to get probability density. a g = 10, b g = 100, and c g = 1000. Initial F frequency is .The parameters are rS = 0.5, ω = 0.03, µ = 0.0001 and τ = ln[1000]/rS.

Simulation with zero mutation rate. Color map of the absolute error between frequency ⟨f*⟩ of the averaged selected collectives at the end of simulations (k = 1000) and the target frequency .For parameters, we used rS = 0.5, ω = 0.03, µ = 0, N0 = 1000, g = 10 and τ ≈ 4.8.

Change of success region in varying selective advantage ω. rS = 0.5, ω = 0.03, µ = 0.0001, N0 = 1000, g = 10 and τ ≈ 4.8.

Artificial selection also works for deleterious mutation. a Conditional probability density functions of for various values. The left-hand side distribution is obtained from simulations and the right-hand side distribution is numerically obtained by evaluating Eq. (55). Small triangles inside indicate the median values of the distributions. b the median value of distributions at a given .The points where the shifted median becomes zero, Median are denoted as f L and f U, respectively. c The relative error between the target frequency and the ensemble averaged selected frequency is measured after 1000 cycles starting from the initial frequency .Either the lower target frequencies or the higher target frequencies starting from the high initial frequencies can be achieved. The black dashed lines indicate the predicted boundary values f U and f L in a.

Selecting Top-5% outperforms selecting Top 1. We bred 100 collectives and chose either top-1 collective (solid line) or top-5 collectives (dashed line) with f closest to the target value (black dotted line).

a The flow of composition change in F and FF frequencies at each composition (f, h). Top corner indicates that FF cells fix in the collective. Right bottom corner means collectives with only F cells while collectives contain S cells only at left bottom corner. Arrow length means the speed of change. b The accessible regions are marked by the gold area. If the signs of changes in both F frequency and FF frequency after inter-collective selection are opposite to those during maturation, then the given composition is accessible. Otherwise, the composition is not accessible and will change after cycles. Dashed lines are the boundary of accessible region by projecting the collective into a two-population problem (FF vs. S+F). The figures are drawn using mpltern package [7].