# Modeling Cyanobacteria Vertical Migration

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Background

#### 1.2. Models of Vertical Migration

Equations | Parameters and Definitions | ||
---|---|---|---|

Kromkamp and Walsby [8] | |||

(1) | $\frac{d{\rho}_{c}}{dt}={c}_{1}\left(\frac{I}{{K}_{I}+I}\right)-{c}_{2}{I}_{a}-{c}_{3}$ | ${\rho}_{c}$, density of cyanobacteria colony $t$, time $I$, irradiance at depth of colony ${I}_{a}$, previous irradiance ${K}_{I}=25\text{}\mathsf{\mu}\mathrm{mol}\text{}{\mathrm{m}}^{-2}\text{}{\mathrm{s}}^{-1}$, half-saturation irradiance ${c}_{1}=0.132\text{}\mathrm{kg}\text{}{\mathrm{m}}^{-3}\text{}{\mathrm{min}}^{-1}$ ${c}_{2}=1.67\times {10}^{-5}\text{}\mathrm{kg}\text{}{\mathrm{m}}^{-3}\text{}{\mathrm{min}}^{-1}{\left(\mathsf{\mu}\mathrm{mol}\text{}{\mathrm{m}}^{-2}\text{}{\mathrm{s}}^{-1}\right)}^{-1}$ ${c}_{3}=0.023\text{}\mathrm{kg}\text{}{\mathrm{m}}^{-3}\text{}{\mathrm{min}}^{-1}$ $v,$ settling velocity | $g$, acceleration due to gravity $r,$ cyanobacteria colony radius ${\rho}^{\prime}$, density of water $n,$ viscosity of water $A,$ ratio of cell volume to colony volume $\varphi ,$ form resistance ${z}_{2},$ depth at current timestep ${z}_{1},$ depth at previous timestep $P,$ time interval |

(2) | $v=\frac{2g{r}^{2}\left({\rho}_{c}-{\rho}^{\prime}\right)A}{9\varphi n}$ | ||

(3) | ${z}_{2}=vP+{z}_{1}$ | ||

SCUM96 [21] | |||

(4) | $If\text{}{P}_{qi}-R\le {C}_{gmax},K={P}_{qi}-R\text{}and\text{}B=0$ $If\text{}{P}_{qi}-R{C}_{gmax},K={C}_{gmax}$ $and\text{}B={P}_{qi}-R-K$ $If\text{}{P}_{qi}-R0,K=0\text{}and\text{}B={P}_{qi}-R$ | ${P}_{qi},$ cyanobacteria photosynthetic rate $R,$ respiration rate ${C}_{gmax}$, maximum rate of carbon used for growth $K,$ growth $B,$ ballast ${\rho}_{cel},$ density of a cell ${\rho}_{muc},$ mucilage density $F=0.19$, ratio of cell volume to colony volume | $N=12,032$, number of cells in a colony ${C}_{cel},$ cell carbon content ${V}_{cel},$ cell volume ${\rho}_{c},{\rho}^{\prime},r,$ as defined above |

(5) | ${\rho}_{c}=\left(F\ast N\ast {\rho}_{cel}\right)+\left[\left(1.0-F\right){\rho}_{muc}\right]$ | ||

(6) | ${\rho}_{muc}={\rho}^{\prime}+0.7\text{}\mathrm{kg}/{\mathrm{m}}^{3}$ | ||

(7) | $\mathsf{\Delta}{\rho}_{cel}=\frac{{B}_{g}{C}_{cel}}{{V}_{cel}}$ | ||

(8) | $\mathsf{\Delta}{\rho}_{c}=\frac{N\ast {V}_{cel}\ast \mathsf{\Delta}{\rho}_{cel}}{\frac{4}{3}\pi {r}^{3}}$ | ||

Visser et al. [13] | |||

(9) | $I\ge {I}_{c}$$,\text{}\frac{d{\rho}_{c}}{dt}=\left(\frac{{N}_{0}}{60}\right)I{e}^{-I/{I}_{0}}+c$ | $I,$ irradiance at depth of colony ${I}_{c}=10.9\text{}\mathsf{\mu}\mathrm{mol}\text{}{\mathrm{m}}^{-2}\text{}{\mathrm{s}}^{-1},$ compensation irradiance ${N}_{0}=0.0945\text{}\mathrm{kg}\text{}{\mathrm{m}}^{-3}\text{}{\mathsf{\mu}\mathrm{mol}}^{-1}\text{}{\mathrm{m}}^{2}$ | ${\rho}_{i},$ cell density at end of preceding light period ${f}_{1}=-9.49\times {10}^{-4}\text{}{\mathrm{min}}^{-1}$ |

(10) | $I<{I}_{c}$$,\text{}\frac{d{\rho}_{c}}{dt}={f}_{1}{\rho}_{i}+{f}_{2}$ | ${I}_{0}=277.5\text{}\mathsf{\mu}\mathrm{mol}\text{}{\mathrm{m}}^{-2}\text{}{\mathrm{s}}^{-1}$, light intensity at maximum density $c=-0.0165\text{}\mathrm{kg}\text{}{\mathrm{m}}^{-3}\text{}{\mathrm{min}}^{-1},$ rate of density change when $I=0$ | ${f}_{2}=0.984\text{}\mathrm{kg}\text{}{\mathrm{m}}^{-3}\text{}{\mathrm{min}}^{-1}$ ${\rho}_{c},t,$ as defined above |

Wallace and Hamilton [14] | |||

(11) | $I>0$$,\text{}\frac{d{\rho}_{c}}{dt}=\left({c}_{1}\frac{I}{{K}_{I}+I}-{c}_{3}\right)\left(1-{e}^{-t/{\tau}_{r}}\right)$ | ${K}_{I}=530\text{}\mathsf{\mu}\mathrm{mol}\text{}{\mathrm{m}}^{-2}\text{}{\mathrm{s}}^{-1},$ half-saturation irradiance ${c}_{1}=0.0427\text{}\mathrm{kg}\text{}{\mathrm{m}}^{-3}\text{}{\mathrm{min}}^{-1}$ | ${c}_{3}=4.6\times {10}^{-6}\text{}\mathrm{kg}\text{}{\mathrm{m}}^{-3}\text{}{\mathrm{min}}^{-1}$ ${\tau}_{r}=20\text{}\mathrm{min}$, response time |

(12) | $I=0$$,\text{}\frac{d{\rho}_{c}}{dt}=-{c}_{2}{I}_{a}-{c}_{3}$ | ${c}_{2}=1.67\times {10}^{-5}\text{}\mathrm{kg}\text{}{\mathrm{m}}^{-3}\text{}{\mathrm{min}}^{-1}{\left(\mathsf{\mu}\mathrm{mol}\text{}{\mathrm{m}}^{-2}\text{}{\mathrm{s}}^{-1}\right)}^{-1}$ | ${\rho}_{c},t,I,{I}_{a}$, as defined above |

Belov and Giles [4] | |||

(13) | $V\left(z,t\right)={V}_{0}{e}^{-k\left(h-z\right)}cos\left(\omega t\right)$ | $V\left(t,z\right),\mathrm{velocity}\text{}\mathrm{of}\text{}\mathrm{colony}$ $z,$ depth of colony ${V}_{0}=0.408\text{}\mathrm{m}\text{}{\mathrm{day}}^{-1}$, maximum colony velocity $k=0.1\text{}{\mathrm{m}}^{-1}$, light attenuation coefficien | $h,$ depth of waterbody $\omega =2\pi \text{}{\mathrm{day}}^{-1}$, frequency of daily light cycle $t$, as defined above |

Serizawa et al. [39] | |||

(14) | $V\left(t,z\right)={V}_{m}\left\{F\left(t,z\right)-{F}_{0}\right\}$ | $F\left(t,z\right),$ ballast factor ${V}_{m}=250\text{}\mathrm{m}\text{}{\mathrm{day}}^{-1}$, velocity scale factor ${F}_{0}=0.1$, neutral buoyancy ballast factor $\mu ,$ growth rate | $k=3\text{}{\mathrm{day}}^{-1}$, reciprocol of decay time $\tau ,$ time before present $V\left(t,z\right),t,z$, as defined above |

(15) | $F\left(t,z\right)={{\displaystyle \int}}_{0}^{\infty}\mu \left(t-\tau ,z\right){e}^{-k\tau}d\tau $ | ||

CAEDYM [43] | |||

(16) | $I>0$$,\text{}\frac{d{\rho}_{c}}{dt}={c}_{1}\left(1-{e}^{-I/{I}_{K}}\right)-{c}_{3}$ | ${c}_{1}=0.124\text{}\mathrm{kg}\text{}{\mathrm{m}}^{-3}\text{}{\mathrm{min}}^{-1}$^{a}${c}_{3}=0.023\text{}\mathrm{kg}\text{}{\mathrm{m}}^{-3}\text{}{\mathrm{min}}^{-1}$ ^{a} | ${\rho}_{c},I,t,$ as defined above |

(17) | $I=0$$,\text{}\frac{d{\rho}_{c}}{dt}=-{c}_{3}$ | ${I}_{K}=130\text{}\mathsf{\mu}\mathrm{mol}\text{}{\mathrm{m}}^{-2}\text{}{\mathrm{s}}^{-1}$^{a}, half saturation constant for light-dependent density change |

^{a}Values used in Chung et al. [44].

## 2. Modeling Framework and Available Field Data

#### 2.1. Continuum and Particle Transport Models

#### 2.1.1. Predefined Velocity

#### 2.1.2. Dynamic Velocity

#### 2.2. Field Data

^{7}cells/L) was due to Microcystis aeruginosa. Solar irradiance measured at the surface was recorded every hour. Calculated MRD and depth of maximum chlorophyll a concentration were also reported in the published study (Figure 4).

#### 2.3. Model Setup

^{−1}for all models of Xiangxi Bay. This was done in part to test the models with a lower light attenuation coefficient value, as the values in the Shennong Stream study were relatively high. Each model was run with two different values of ${D}_{z}$,$\text{}{10}^{-5}\text{}{\mathrm{m}}^{2}{\mathrm{s}}^{-1}$ and ${10}^{-4}\text{}{\mathrm{m}}^{2}{\mathrm{s}}^{-1}$.

## 3. Modeling Results

#### 3.1. Shennong Stream Enclosure

^{3}in the continuum framework to 2.80 and 3.18 mg/m

^{3}in the particle-tracking framework.

#### 3.2. Shennong Stream Open Water

^{3}in the continuum framework to 8.50–8.67 mg/m

^{3}in the particle-tracking framework. Differences between frameworks were smaller for both metrics in predefined velocity models.

#### 3.3. Xiangxi Bay

^{−5}m

^{2}s

^{−1}(Table 6). In plots of MRD, these models reproduced the pattern seen in the field data (Figure S17 in Supplementary Materials), while the dynamic velocity models predicted a more shallow MRD than that seen in the data (Figure S18 in Supplementary Materials). The predefined velocity models also show more accurate predictions of depth of maximum chlorophyll a concentration when it is at its deepest between July 2 and 3 (Figure S19 in Supplementary Materials). Most of the dynamic velocity models do not correctly predict this, with the exception being the growth kinetics model (Figure 7).

^{−4}m

^{2}s

^{−1}, there was not a clear distinction between the predefined and dynamic velocity models. The highest errors resulted from the growth kinetics and light function models without time decay (Table 6). Visually, the predefined velocity models seem to capture the shape of the MRD sinusoidal curve, but the dynamic velocity models predict the average depth more accurately (Figures S17 and S18 in Supplementary Materials). The dynamic models show more daily variation in depth of maximum chlorophyll a concentration for both scenarios but only the growth kinetics and light function models with time decay approximate the correct depth on the second day (Figure 7 and Figure S19 in Supplementary Materials).

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Chlorophyll a concentration profiles measured in Shennong Stream enclosure and open water sites; data extracted from Cui et al. [12].

**Figure 3.**Mean residence depth (MRD) calculated from chlorophyll a profiles measured in Shennong Stream open water and enclosure sites; data extracted from Cui et al. [12].

**Figure 4.**Mean residence depth and depth of maximum chlorophyll a calculated from chlorophyll a measurements taken in Xiangxi Bay; data extracted from Wang et al. [49].

**Figure 5.**Time series of observed and predicted mean residence depth of chlorophyll a concentration in the Shennong Stream enclosure site using predefined velocity models (continuum) [4].

**Figure 6.**Time series of observed and predicted mean residence depth of chlorophyll a concentration in the Shennong Stream open water site using dynamic velocity models (particle-tracking) [13].

**Figure 7.**Time series of observed and predicted depth of maximum chlorophyll a concentration in Xiangxi Bay using dynamic velocity models (continuum) [13].

**Figure 8.**Time series of observed and predicted mean residence depth of chlorophyll a concentration in Xiangxi Bay using predefined velocity models (particle-tracking) [4].

**Figure 9.**Time series of observed and predicted mean residence depth of chlorophyll a concentration in Xiangxi Bay using dynamic velocity models (particle-tracking) [13].

Study | Identifier | Minimum Density, $\mathbf{kg}\text{}{\mathbf{m}}^{\mathbf{-}\mathbf{3}}$ | Maximum Density, $\mathbf{kg}\text{}{\mathbf{m}}^{\mathbf{-}\mathbf{3}}$ | Colony Radius, $\mathsf{\mu}\mathbf{m}$ | Saturating Light Intensity, $\mathbf{W}\text{}{\mathbf{m}}^{\mathbf{-}\mathbf{2}}$ | Maximum Growth Rate, ${\mathbf{day}}^{\mathbf{-}\mathbf{1}}$ |
---|---|---|---|---|---|---|

Reynolds [50] | - | - | - | 25–1000 | - | - |

Reynolds et al. [51] | cyanobacteria | - | - | - | - | 0.6–0.8 |

M. aeruginosa | 985 | 1005 | 120–3200 | - | - | |

A. flos-aqua | 920 | 1030 | 28–100 | - | - | |

P. agardhii | 985 | 1085 | 13.7–18.3 | - | - | |

Nakamura et al. [52] | Microcystis sp. | - | - | 10–300 | - | - |

Visser et al. [13] | Microcystis sp. | - | - | - | 139 | - |

Long et al. [53] | M. aeruginosa | - | - | - | - | 1.2 |

Wu and Song [54] | M. aeruginosa | - | - | - | 119–244 | - |

Wu et al. [55] | M. aeruginosa | - | - | - | 65–119 | - |

Zhang et al. [56] | M. aeruginosa | - | - | - | 75–392 | - |

Zhu et al. [17,57] | Microcystis sp. | 967 | 997 | 10–350 | - | - |

Rowe et al. [58] | Microcystis sp. | - | - | 12.5–370, median: 58.5 | - | - |

Variable | Description | Value Range |
---|---|---|

A, m | Migration amplitude | 0.2–1.23 |

$\varphi $, rad | Phase offset | $\pi $ |

$C$ | Light attenuation calibration coefficient | 0.05–0.13 |

${\mu}_{g,max},\text{}{\mathrm{day}}^{-1}$ | Maximum growth rate | 0.7–1.0 |

${\mu}_{m},\text{}{\mathrm{day}}^{-1}$ | Mortality rate | 0.06–0.25 |

${\mu}_{e},\text{}{\mathrm{day}}^{-1}$ | Excretion rate | 0.04 |

${\mu}_{r},\text{}{\mathrm{day}}^{-1}$ | Respiration rate | 0.04 |

${I}_{s},\text{}\mathrm{W}\text{}{\mathrm{m}}^{-2}$ | Saturating light intensity | 100–150 |

${c}_{1},\text{}{\mathrm{day}}^{-1}$ | Coefficient of density increase for light function model | 0.00545–0.02 |

${c}_{2},\text{}{\mathrm{day}}^{-1}$ | Coefficient of density decrease for light function model | 0.00145–0.00518 |

${r}_{c},\text{}\mathsf{\mu}\mathrm{m}$ | Colony radius | 15–64 |

${\rho}_{min},\text{}\mathrm{kg}\text{}{\mathrm{m}}^{-3}$ | Minimum colony density | 920–980 |

${\rho}_{max},\text{}\mathrm{kg}\text{}{\mathrm{m}}^{-3}$ | Maximum colony density | 140–185 |

${\rho}_{0,S},\text{}\mathrm{kg}\text{}{\mathrm{m}}^{-3}$ | Initial colony density at surface | 930–1080 (continuum) 920–980 (particles) |

${\rho}_{0,B},\text{}\mathrm{kg}\text{}{\mathrm{m}}^{-3}$ | Initial colony density at bed | 930–980 (continuum) 995–1010 (particles) |

${\rho}_{i,S},\text{}\mathrm{kg}\text{}{\mathrm{m}}^{-3}$ | Minimum initial colony density at surface for Visser et al. [13] model | 980–1080 |

${\rho}_{i,B},\text{}\mathrm{kg}\text{}{\mathrm{m}}^{-3}$ | Minimum initial colony density at bed for Visser et al. [13] model | 975–980 |

${\rho}_{*},\text{}\mathrm{kg}\text{}{\mathrm{m}}^{-3}$ | Correction for density decrease equation for Visser et al. [13] model | 67 |

$k,\text{}{\mathrm{day}}^{-1}$ | Time decay constant for averaging past densities | 5 |

Mean Residence Depth AME, m | Chlorophyll a Concentration (Profile Average) AME, mg m ^{3} | |||
---|---|---|---|---|

Model | Continuum | Particle Tracking | Continuum | Particle Tracking |

Time-varying velocity | 1.074 | 1.127 | 2.871 | 3.176 |

Belov and Giles [4] | 0.799 | 0.795 | 2.549 | 2.796 |

Growth kinetics | 1.348 | 0.613 | 3.446 | 3.925 |

Growth kinetics with time decay | 1.318 | - | 3.271 | - |

Visser et al. [13] | 1.338 | 0.772 | 3.378 | 3.205 |

Light function | 1.359 | 0.599 | 3.522 | 3.273 |

Light function with time decay | 1.252 | - | 3.201 | - |

Mean Residence Depth AME, m | Chlorophyll a Concentration (Profile Average) AME, mg m ^{3} | |||
---|---|---|---|---|

Model | Continuum | Particle Tracking | Continuum | Particle Tracking |

Time-varying velocity | 1.129 | 1.097 | 8.589 | 8.654 |

Belov and Giles [4] | 1.113 | 1.100 | 8.645 | 8.716 |

Growth kinetics | 1.096 | 0.986 | 7.735 | 8.498 |

Growth kinetics with time decay | 1.064 | - | 7.193 | - |

Visser et al. [13] | 1.093 | 0.993 | 7.589 | 8.670 |

Light function | 1.087 | 0.964 | 7.425 | 8.654 |

Light function with time decay | 1.087 | - | 7.253 | - |

Mean Residence Depth AME, m | Depth of Maximum Chlorophyll a Concentration AME, m | |||||||
---|---|---|---|---|---|---|---|---|

D_{z} = 10^{−5} m^{2} s^{−1} | D_{z} = 10^{−4} m^{2} s^{−1} | D_{z} = 10^{−5} m^{2} s^{−1} | D_{z} = 10^{−4} m^{2} s^{−1} | |||||

Model | Continuum | Particle Tracking | Continuum | Particle Tracking | Continuum | Particle Tracking | Continuum | Particle Tracking |

Time-varying velocity | 0.351 | 0.509 | 0.422 | 0.415 | 0.557 | 0.609 | 0.696 | 0.846 |

Belov and Giles [4] | 0.380 | 0.585 | 0.409 | 0.413 | 0.561 | 0.554 | 0.680 | 0.844 |

Growth kinetics | 1.201 | 0.595 | 0.672 | 0.461 | 0.641 | 1.598 | 0.744 | 1.373 |

Growth kinetics with time decay | 1.299 | - | 0.395 | - | 0.730 | - | 0.667 | - |

Visser et al. [13] | 1.287 | 1.001 | 0.374 | 0.392 | 0.877 | 2.800 | 0.754 | 1.598 |

Light function | 1.410 | 0.887 | 0.629 | 0.371 | 0.693 | 1.837 | 0.802 | 1.454 |

Light function with time decay | 1.650 | - | 0.426 | - | 0.725 | - | 0.687 | - |

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**MDPI and ACS Style**

Overman, C.; Wells, S.
Modeling Cyanobacteria Vertical Migration. *Water* **2022**, *14*, 953.
https://doi.org/10.3390/w14060953

**AMA Style**

Overman C, Wells S.
Modeling Cyanobacteria Vertical Migration. *Water*. 2022; 14(6):953.
https://doi.org/10.3390/w14060953

**Chicago/Turabian Style**

Overman, Corina, and Scott Wells.
2022. "Modeling Cyanobacteria Vertical Migration" *Water* 14, no. 6: 953.
https://doi.org/10.3390/w14060953