^a Key Laboratory of Environmental Aquatic Chemistry, State Key Laboratory of Regional Environment and Sustainability, Research Center for Eco-Environmental Sciences, Chinese Academy of Sciences, Beijing 100085, China.
^b Bureau of Hydrology Information Center of Taihu Basin Authority, Shanghai 200434, China.
^c College of Environmental Science and Engineering, Ocean University of China, Qingdao 266100, China.
^d University of Chinese Academy of Sciences, Beijing 100049, China.

^* Corresponding to: Ming Su (mingsu@rcees.ac.cn)

12 figures and 3 tables provided below for the evidence of main text.

Supplementary Fig. 1

Supplementary Fig. 2

Supplementary Fig. 3

Supplementary Fig. 4

Supplementary Fig. 5

Supplementary Fig. 6

Supplementary Fig. 7

Supplementary Fig. 8

Supplementary Fig. 9

Supplementary Fig. 10

Supplementary Fig. 11

Supplementary Fig. 12

Supplementary Table 1

Genus	Strain	Ref.
Planktothricoides	Planktothricoides raciborskii	[1]
Planktothricoides	Planktothricoides raciborskii	[2]
Planktothricoides	Planktothricoides raciborskii	[3]
Planktothricoides	Planktothricoides raciborskii	[4]
Planktothricoides	Planktothricoides raciborskii 1372	[5]
Planktothricoides	Planktothricoides sp.	[6]
Planktothricoides	Planktothricoides sp. SR001	[7]
Pseudanabaena	Pseudanabaena cinerea FACHB1277	[8]
Pseudanabaena	Pseudanabaena cinerea FACHB1278	[9]
Pseudanabaena	Pseudanabaena cinerea FACHB1279	[10]
Pseudanabaena	Pseudanabaena cinerea FACHB1280	[11]
Pseudanabaena	Pseudanabaena galeata	[12]
Pseudanabaena	Pseudanabaena sp.	[13]
Pseudanabaena	Pseudanabaena sp.	[14]
Pseudanabaena	Pseudanabaena sp. dqh15	[15]
Pseudanabaena	Pseudanabaena sp. TH-1	[16]
Pseudanabaena	Pseudanabaena sp. Xili	[17]

Supplementary Table 1: Studies in other literature involving the cultivation of Pseudanabaena and Planktothricoides.

Supplementary Table 2

Comparison	\(t_\text{MIB}\)	\(e_\text{MIB}\)	\(i_\text{MIB}\)	\(f\)
Autumn vs Spring	1.0000	0.0879	1.0000	1.0000
Autumn vs Summer	0.0003^***	<0.0001^***	0.0104^*	1.0000
Spring vs Summer	0.0081^**	0.0398^*	0.0599	1.0000
Autumn vs Winter	0.0022^**	0.7217	<0.0001^***	0.0000^***
Spring vs Winter	<0.0001^***	0.0024^**	<0.0001^***	0.0000^***
Summer vs Winter	<0.0001^***	<0.0001^***	<0.0001^***	0.0000^***

Supplementary Table 2: Statistical comparisons between seasons for various metrics. \(t_\text{MIB}\): total MIB; \(e_\text{MIB}\): extracellular MIB; \(i_\text{MIB}\): intracellular MIB; \(f\): ratio of \(e_\text{MIB}\) and \(t_\text{MIB}\) (\(f = e_\text{MIB}/t_\text{MIB}\)).

Supplementary Table 3

Season	Mean	SD	First quartile	Median	Third quartile
Spring	43.0	77.5	3.16	8.23	33.5
Summer	107	230	6.78	25.9	80.8
Autumn	23.7	47.7	2.80	6.54	17.57
Winter	9.93	16.6	1.10	2.42	5.34

Supplementary Table 3: Statistical Summary of total MIB in Different Seasons (Total n = 372). \(t_\text{MIB}\): total MIB.

Code

require(tidyverse)
require(lubridate)
require(patchwork)
require(drwateR)
require(grid)
require(ggsci)
require(sf)

  logistic_model <- function(
    culturedf,
    main = NULL,
    xlab = "Culture time (d)",
    ylab = "Cell density (cell L<sup>-1</sup>)",
    Kcoef = 0.99,
    N00 = 5e5,
    check = FALSE,
    plot = FALSE,
    returnfitdf = TRUE,
    fitdf_x = seq(
      min(culturedf$julian),
      1.2 * max(culturedf$julian),
      length.out = 100
    )
  ) {
    K <- max(culturedf$n) / Kcoef
    culturedf <- culturedf |>
      dplyr::mutate(z = log(K / n - 1))
    out <- list()
    out[["culturedf"]] <- culturedf
    out[["K"]] <- K
    m <- lm(z ~ julian, data = culturedf)
    r <- -unname(coef(m)[2])
    tstar <- unname(coef(m)[1] / r) #tstar = log(a)/r
    # N0 <- K - K / (1 + exp(-r * tstar)) 
    N0 <- K / (1 + exp(r * tstar)) 
    t00 <- tstar - log(K / N00 - 1) / r
    r2 <- summary(m)$r.squared
    pval <- summary(m)$coefficients[8]
    out[["m"]] <- m
    out[["r"]] <- r
    out[["tstar"]] <- tstar
    out[["N0"]] <- N0
    out[["r2"]] <- r2
    out[["pval"]] <- pval
    out[["t00"]] <- t00
    out[["N00"]] <- N00
    if (isTRUE(check)) {
      p1 <- culturedf |>
        ggplot(aes(julian, z)) +
        geom_point() +
        geom_smooth(method = "lm") +
        labs(x = xlab, y = "log(K / N - 1)", fill = NULL, title = main) +
        dwfun::theme_sci()
      out[["p1"]] <- p1
    }
    if (isTRUE(returnfitdf)) {
      fitdf <- tibble::tibble(julian = fitdf_x) |>
        dplyr::mutate(n = K / (1 + exp(-r * (julian - tstar))))
      out[["fitdf"]] <- fitdf
    }
    if (isTRUE(plot)) {
      p2 <- fitdf |>
        ggplot(aes(julian, n)) +
        geom_point(data = culturedf, aes(y = n)) +
        geom_line() +
        labs(x = xlab, y = ylab, fill = NULL, title = main) +
        dwfun::theme_sci()
      out[["p2"]] <- p2
    }
    return(out)
  }

  culturemdf <- readRDS("culturedf.rds") |>
    tidyr::nest(idf = -c(strain, ppf)) |>
    tidyr::expand_grid(Kcoef = c(0.90, 0.95, 0.99, 1.05, 1.1)) |>
    dplyr::mutate(
      model = purrr::pmap(
        list(idf, ppf, Kcoef),
        \(x, I, Kcoef) {
          logistic_model(
            culturedf = x |>
              dplyr::filter(valid == "GOOD") |>
              dplyr::filter(!is.na(n)),
            Kcoef = Kcoef,
            main = I,
            check = TRUE,
            plot = TRUE,
            returnfitdf = TRUE,
            fitdf_x = 0:40
          )
        }
      )
    ) |>
    dplyr::mutate(culturedf = purrr::map(model, \(x) x[["culturedf"]])) |>
    dplyr::mutate(m = purrr::map(model, \(x) x[["m"]])) |>
    dplyr::mutate(K = purrr::map_dbl(model, \(x) x[["K"]])) |>
    dplyr::mutate(r = purrr::map_dbl(model, \(x) x[["r"]])) |>
    dplyr::mutate(N0 = purrr::map_dbl(model, \(x) x[["N0"]])) |>
    dplyr::mutate(r2 = purrr::map_dbl(model, \(x) x[["r2"]])) |>
    dplyr::mutate(pval = purrr::map_dbl(model, \(x) x[["pval"]])) |>
    dplyr::mutate(tstar = purrr::map_dbl(model, \(x) x[["tstar"]])) |>
    dplyr::mutate(p1 = purrr::map(model, \(x) x[["p1"]])) |>
    dplyr::mutate(p2 = purrr::map(model, \(x) x[["p2"]])) |>
    dplyr::mutate(fitdf = purrr::map(model, \(x) x[["fitdf"]])) |>
    dplyr::group_by(strain, ppf) |>
    dplyr::slice_max(r2, n = 1) |>
    dplyr::arrange(strain, ppf)

    Kprop <- 0.8
   mdf <- culturemdf |>
    unnest(culturedf) |>
    dplyr::mutate(a = (K - N0) / N0) |>
    dplyr::mutate(
      x1 = exp(r * julian) *
        a *
        log(2) /
        (exp(2 * r * julian) + exp(r * julian) * 2 * a + a^2)
    ) |>
    dplyr::mutate(x2 = (K - n) / K) |>
    dplyr::mutate(
      stage = factor(
        n < K * Kprop,
        levels = c("TRUE", "FALSE"),
        labels = c("<80%K", "≥80%K")
      )
    ) |>
    tidyr::nest(
      idf = -c(
        strain,
        ppf,
        stage,
        Kcoef,
        model,
        m,
        r,
        tstar,
        N0,
        r2,
        pval,
        p1,
        p2,
        fitdf
      )
    ) |>
    tidyr::pivot_wider(names_from = stage, values_from = idf) |>
    dplyr::mutate(
      mb = purrr::map(`<80%K`, \(x) {
        x |>
          dplyr::mutate(y = d2t - 1) |>
          lm(formula = y ~ x1 + 0, data = _)
      })
    ) |>
    dplyr::mutate(
      mc = purrr::map(`≥80%K`, \(x) {
        x |>
          lm(formula = d2t ~ x2, data = _)
      })
    ) |>
    dplyr::mutate(b = -purrr::map_dbl(mb, \(x) coef(x)[1])) |>
    dplyr::mutate(c = purrr::map_dbl(mc, \(x) coef(x)[2])) |>
    dplyr::mutate(d = purrr::map_dbl(mc, \(x) coef(x)[1] - 0.5)) |>
    dplyr::relocate(strain, ppf, Kcoef, r, N0, r2, pval, tstar, b, c, d, p1, p2)

  
  Kprop <- 0.8
  mdf <- culturemdf |>
    tidyr::unnest(culturedf) |>
    dplyr::mutate(a = (K - N0) / N0) |>
    dplyr::mutate(
      x1 = exp(r * julian) *
        a *
        log(2) /
        (exp(2 * r * julian) +
          exp(r * julian) * 2 * a +
          a^2)
    ) |>
    dplyr::mutate(x2 = (K - n) / K) |>
    dplyr::mutate(
      stage = factor(
        n < K * Kprop,
        levels = c("TRUE", "FALSE"),
        labels = c("<80%K", "≥80%K")
      )
    ) |>
    tidyr::nest(
      idf = -c(
        strain,
        ppf,
        stage,
        Kcoef,
        model,
        K,
        m,
        r,
        tstar,
        N0,
        deltaT,
        r2,
        pval,
        p1,
        p2,
        fitdf
      )
    ) |>
    tidyr::pivot_wider(names_from = stage, values_from = idf) |>
    dplyr::mutate(
      mb = purrr::map(`<80%K`, \(x) {
        x |>
          dplyr::mutate(y = d2t - 1) |>
          lm(formula = y ~ x1 + 0, data = _)
      })
    ) |>
    dplyr::mutate(
      mc = purrr::map(`≥80%K`, \(x) {
        x |>
          lm(formula = d2t ~ x2, data = _)
      })
    ) |>
    dplyr::mutate(b = -purrr::map_dbl(mb, \(x) coef(x)[1])) |>
    dplyr::mutate(c = purrr::map_dbl(mc, \(x) coef(x)[2])) |>
    dplyr::mutate(d = purrr::map_dbl(mc, \(x) coef(x)[1] - 0.5)) |>
    dplyr::relocate(
      strain,
      ppf,
      Kcoef,
      K,
      r,
      N0,
      deltaT,
      r2,
      pval,
      tstar,
      b,
      c,
      d,
      p1,
      p2
    ) |>
    dplyr::left_join(
      culturemdf_adj |>
        dplyr::select(strain, ppf, culturedf, fitdf)
    ) |>
    dplyr::mutate(theta = (K - N0) / N0) |>
    dplyr::mutate(
      fpreddf = purrr::pmap(
        list(fitdf, K, r, N0, deltaT, theta, b, c, d),
        \(df, K, r, N0, deltaT, theta, alpha1, alpha2, beta) {
          df |>
            dplyr::filter(n < 0.8 * K) |>
            dplyr::mutate(
              f = 1 -
                alpha1 *
                  log(2) *
                  theta *
                  exp(r * julian) /
                  (exp(2 * r * julian) + 2 * theta * exp(r * julian) + theta^2)
            ) |>
            bind_rows(
              df |>
                dplyr::filter(n >= 0.8 * K) |>
                dplyr::mutate(f = 0.5 + (K - n) / K * alpha2 + beta)
            )
        }
      )
    )

  culturemdf |>
    unnest(culturedf) |>
    ggplot(aes(julian, d2t)) +
    geom_point() +
    facet_wrap(~ paste(strain, ppf), ncol = 3, scale = "free") +
    theme_sci(9, 6)

  mdf |>
    unnest(`<80%K`) |>
    dplyr::filter(julian > 0) |>
    ggplot(aes(x1 * log(2) / r, d2t)) +
    geom_point() +
    scale_x_log10(
      breaks = scales::trans_breaks("log10", function(x) 10^x),
      labels = scales::trans_format("log10", scales::math_format(10^.x))
    ) +
    facet_wrap(~ paste0(strain, ppf), ncol = 3, scale = "free") +
    theme_sci(9, 5)

  mdf |>
    tidyr::unnest(fpreddf) |>
    ggplot(aes(julian, f)) +
    geom_point() +
    geom_line(color = "red") +
    facet_wrap(~ paste(strain, ppf), ncol = 3, scale = "fixed") +
    theme_sci(9, 5)

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