Appendix I: Technical Overview of Model

An individual-based model of Undaria pinnatifida growth and reproduction was developed in

the C++ programming language based on a generic, computational framework for representing large

populations of autonomous agents in a discrete, two-dimensional environment (Murphy et al. 2013).

It is optimised for parallel processing (by domain decomposition) using the Message Passing

Interface in order to be able to efficiently process large populations of independent agents (Gropp et

al. 1996). The main life history stages (gametophytes and sporophytes; Fig. S1) of U. pinnatifida are

represented as distinct agents in the model with unique properties based on experimental data from

the literature. Here we present an overview of the latest version of the UndariaGEN model (v0.6.4),

which incorporates several significant new features/revisions to improve its accuracy and robustness

as compared to the previous version (Murphy et al. 2016a). A more detailed description of the base

model (v1.0) and the process of selection/parameterisation of the input values can be found in

(Murphy et al. 2016a). Empirical data (from a survey in Brest Harbour) was used to validate the

model (Murphy et al. 2016b).

There are two principal types of agents in the model, corresponding to the gametophyte and

sporophyte stages in the life cycle of U. pinnatifida respectively. These have distinct growth

parameters and differ in their response to environmental cues. The growth rate and maturation of

the gametophyte and sporophyte agents are functions of local environmental parameters such as

irradiance, day length and temperature. In order to quantify this relationship, a review of the

literature was carried out to gather quantitative data and build mathematical descriptions of these

interactions at the individual level. The overall population dynamics is therefore an emergent

property of the interactions between these components and the environmental parameters.

1.1. Program structure

The model is constructed using the object-oriented programing paradigm of C++. The initial

phase of the program involves the creation and initialisation of an array of U. pinnatifida agents

which are stored in an array data structure. The input parameters for the simulation are entered via

a text input file (Table S1). These specify physical parameters such as the size and scale of the

environment as well as parameters for the U. pinnatifida agents such as the initial number of

seedlings and their responses to environmental cues.

The main program loop consists of a series of steps representing the main biotic and abiotic

functions of the system (Fig. S2). Each loop represents a discrete hour of real-time during which the

agents grow and interact with their environment. The first step of the loop involves updating the

current environmental conditions (temperature, irradiance and day length). These values are

updated on a daily basis (i.e. every 24 loops of the simulation). Step 2 involves the implementation

of Fick’s first law of diffusion for dispersal of spores. Spore germination to form new gametophyte

agents can occur in the presence of an appropriate substrate for attachment. Steps 4 and 5 involves

the growth and reproduction of sporophyte/gametophyte agents according to their particular life

cycle rules.

The simulation will exit the program loop when certain exit conditions occur. These include

when all agents have died or when a set (user-defined) period of time has elapsed. There are also

processes for displaying a real-time graphical display of the locations of the agents in the

environment, and for recording various diagnostic parameters to text files for later analysis and

graphing of results. The following sections include an overview of the main components of the

model: the environment, and the main agent types (gametophytes/sporophytes).

1.2. The environment

The environment is represented as a discrete, two-dimensional grid with each grid element

(or lattice point) corresponding to an area of 0.25 m2. For this study the modelled environment

represented a port/marina with floating pontoons as potential attachment points for the

macroalgae. A 2-dimensional representation of the environment was chosen since in this situation

the species is established close to the surface, although depth is taken into account when calculating

light availability for photosynthesis. The dispersal of particles (e.g. spores) between lattice positions

in the environment is calculated using a discretised implementation of Fick’s First Law of diffusion

(Ginovart et al. 2002). This is sufficient for modelling dispersal at a local scale since U. pinnatifida

spores lose their fixation ability shortly (a few hours) after release, and thus long distance transport

mediated by hydrodynamic processes can be ignored when the focus is on local population dynamics

(Suto 1950; Thiébaut et al. 1998).

There is a parameter called the light attenuation coefficient for photosynthetically available

radiation (KdPAR) which determines the amount of light attenuation, due to dissolved matter, with

increasing depth in the water column (Saulquin et al. 2013). This is used to calculate the residual

energy available for photosynthesis by U. pinnatifida agents according to the following equation:

E(z )=E (0)e− z K dPAR (1)

where z is the depth (m) of the seaweed below the surface, E(0) is the level of irradiance at the

water surface, and E(z) is the energy available for photosynthesis at depth z. For the case studies in

this paper, the depth (z) was set to 1.0 m and the attenuation coefficient (K dPAR) to 0.4 for conditions

representative of the coastline of Brittany, France (Saulquin et al. 2013).

1.3. Sporophyte agents

1.3.1. Growth

Sporophyte agents are modelled from their initial microscopic, cellular scale up to a

maximum length of 1-3 metres. This represents a particular modelling challenge due to the range of

scales involved (covering several orders of magnitude). Therefore, the growth rate is calculated as a

function of the length of the sporophyte, in order to take into account inherent scaling effects.

Studies from the literature with growth rate estimates of sporophytes in different size classes (from

microscopic sporophytes in culture to mature sporophytes 79 cm in length) were used to calculate

the basic relationship between length and the relative growth rate (Choi et al. 2007; Pang, Lüning

2004; Shao-jun, Chao-yuan 1996). A power law functional relationship was then fitted to this data to

calculate the baseline relative growth rate (RGR_Sbase) per day for sporophyte agents (where

conditions are assumed to be constant with irradiance = 40 mol m-2 s-1, temperature = 15oC, day

length = 12 hours):

RGRSbase=3.615 l−0.407 (2)

Photosynthetic efficiency has also been shown to decrease with increasing sporophyte size,

due to increased thallus complexity/density and increased dry weight:fresh weight ratio (Campbell

et al. 1999; Gómez, Wiencke 1996). Therefore, when determining the photosynthetic rate of

sporophyte agents, plant size must also be taken into account (Table S1). This relationship was

estimated using data from the literature on the photosynthetic efficiency of sporophytes of different

size classes in comparison with microscopic U. pinnatifida gametophytes (Campbell et al. 1999; Choi

et al. 2005). These relationships were used to calculate the parameters (Pmax, , Ic) as a function of

sporophyte length (Fig. S3) in order to generate a unique photosynthesis-irradiance curve for each

agent based on the hyperbolic equation of Jassby, Platt (1976):

RGSI=Pmax ∙(1−exp[−α ∙I−Ic

Pmax ]) (3)

where RG_SI is the relative effect of irradiance on the growth rate of the sporophyte agent, I is

irradiance (mol m-2 s-1), Pmax is the maximum rate of photosynthesis, is the slope of the curve, and

Ic is the compensation point.

In order to determine growth as a function of water temperature, parameters for a thermal

performance curve were estimated by fitting to experimental data on U. pinnatifida sporophytes

(Fig. S4a) (Morita et al. 2003b). This is used to estimate the relative effect of temperature on the

growth rate of the sporophyte:

RGST=S [ 1(1+K1 e−K2 (T b−C T min ) ] ]׿ (4)

where RG_ST is the relative effect of temperature on the sporophyte growth rate, K1, K2 and K3 are

constants, CTmin and CTmax are the lower and upper critical temperature limits respectively, and S is a

scaling factor (Stevenson et al. 1985).

Meanwhile, experimental data from Pang, Lüning (2004) were used to define a hyperbolic

relationship between the growth rate and day light hours (Fig. S4b). In the model, the growth rate of

the sporophyte is also assumed to be proportional to its rate of photosynthesis. Therefore, a

photosynthesis-irradiance curve was defined based on empirical data from Campbell et al. (1999):

RGSDL=Gmax ∙(1−exp [−α ∙I−I c

Gmax ]) (5)

where RG_SDL is the relative effect of day length on the sporophyte growth rate, I is the irradiance

(mol m-2 s-1), Gmax is the maximum rate of growth, is the slope of the curve and Ic is the

compensation point (Jassby, Platt 1976).

In the latest version of the model, an additional parameter has been incorporated to

represent scale effects on photosynthetic performance between individual plants and communities.

It has been demonstrated that the light saturation point (Ik) for macroalgal communities can be

several times higher than for individual thallus pieces tested in the laboratory (Binzer, Middelboe

2005). Therefore, in order to take this into account in the model, we have included an additional

parameter to vary the light saturation point according to the community structure. For the test cases

described in this paper, this value was calculated by fitting to field data collected by Voisin (2007)

from Brest harbour, France (see Table S1, “Community light saturation factor”). The parameter , in

equation (5) above, is divided by the light saturation factor in order to represent the effect of

reduced photosynthetic efficiency on the sporophyte growth rate.

In conclusion, the relative daily growth rate of the sporophyte agent (RGR_S) is calculated as

a function of the combined effects of the three environmental parameters (irradiance, temperature

and day length), described in equations (3)-(5), on the base growth rate, equation (2):

RGRS=RGRSbase × RGST × RG SI × RGSDL (6)

1.3.2. Maturation and spore release

In order to determine the mean size at which sporophytes reach maturity, field data from

Brest harbour, France was used (mean sporophyte length at maturity = 32.7 cm) (Voisin 2007). This

is used in the model to determine the mean size at which maturity is reached (i.e. the formation of a

spore-producing structure called the sporophyll). The probability of spore shedding by mature

sporophytes is described as a function of sea water temperature (Saito 1975). This relationship was

calculated for the model by fitting a logistic function to field data from Suto (1952). Once released

spores are subject to diffusion in the environment according to Fick’s First Law of diffusion (Fick

1855). When they come into contact with a suitable substrate they can attach and germinate to form

new gametophyte agents according to a specified (substrate-specific) probability of

fixation/germination. Otherwise, the spores in the water column are subject to a pre-defined half-

life for degradation over time (Table S1).

1.3.3. Death of sporophyte

The death of a sporophyte agent is determined either when it reaches the natural end of its

lifespan, assumed to be after it has matured and released all its spores, or through premature

death/removal. Field studies have shown that up to 70% of sporophyte recruits die/disappear within

one month of their appearance (Voisin 2007). Therefore, premature death (by various means such as

competition or physical dislodgement) represents a significant proportion of the deaths in a

population. To account for this, field data collected by Voisin (2007) from a population of U.

pinnatifida in Brest, France, was used to create an age to mortality curve (Weibull distribution)

which determines the probability of premature death as a function of the age of the sporophyte.

In the earlier version of the model, there was a single age to mortality curve used to describe

the probability of early death among recruits. However, in the latest version, it has been updated to

take into account seasonal variation in the probability of premature mortality. When the deaths of

young sporophytes recorded in Brest harbour are plotted according to the month they occur in, it is

possible to observe a clear seasonal trend in the age to mortality (Fig. S7). For example, in

July/August there is a peak in the mortality of young recruits (<1 months old). This is probably due to

increased competition/shading from mature macroalgal assemblages inhibiting the establishment of

new recruits.

Therefore, a series of 12 individual Weibull functions (for each month of the year) were

fitted to the mortality data in order to capture the seasonal correlation in the probability of early

death. Cosine curves were used to describe the change in the shape (k) and scale () parameters of

the 12 Weibull curves as a function of the day of the year (Fig. S8):

y=A cos [ω ( x−α ) ]+C (7)

where x is the day of the year, A is the amplitude, is the horizontal phase shift, C is the vertical

offset and is the angular frequency (2/365). This functional relationship could then be used in the

model to generate a probability of early mortality based on the time of the year and the age of the

sporophyte, rather than assume a constant probability.

1.4. Gametophyte agents

1.4.1. Growth

Gametophytes represent the haploid phase of the life cycle of U. pinnatifida with their own

unique growth requirements (Fig. 1 in the main text). They are represented in the model as distinct

agents with independent biological properties. The responses of the gametophytes to temperature,

light and day length were modelled using the same methodology as for the sporophytes, but with

unique parameters estimated from the literature (Fig. S5 & Table S1 ) (Choi et al. 2005; Morita et al.

2003a). However, since gametophytes generally do not grow beyond a few millimetres in size,

scaling effects on photosynthetic efficiency and growth were not applied.

The base growth rate (relative daily increase in size) was calculated using the thermal

performance equation (see eq. 4) of Stevenson et al. (1985) fitted to empirical data from the

literature on the growth of gametophytes (Fig. S5b) (Morita et al. 2003a). Similarly, the relative

effect of day length and data was calculated by fitting to empirical measurements from Choi et al.

(2005). The hyperbolic equation of Jassby, Platt (1976) (see eq. 3) was fitted to this data (Fig. S5a).

The parameters (Pmax, , Ic) for the hyperbolic equation are themselves functions of day length (see

Table S1).

1.4.2.Gametogenesis

The process of gametogenesis (maturation and production of gametes) in gametophytes is

also sensitive to environmental parameters such as temperature and day length. The probability of a

new sporophyte forming is assumed to be a function of the relative proportion of fertile

gametophytes present. Data from the literature on the effects of temperature and different day

length regimes on gametophyte fertility were used to calculate the relative probability of

gametogenesis and subsequent sporophyte formation (Choi et al. 2005; Morita et al. 2003a).

The relationship between temperature and gametophyte fertility is defined by the following

logistic relationship which was fitted to empirical data from Morita et al. (2003a) (Fig. S5a):

T g=1−[ 1(1+e−k ( t−t0 ) ) ] (8)

where Tg is the relative temperature effect on gametogenesis, k is the steepness of the curve, t is the

current water temperature, and to is the temperature at the midpoint of the sigmoid. Similarly, the

effect of day length on fertility was determined by fitting a Weibull curve to data from Choi et al.

(2005) (Fig. S6b).

References:

Binzer T, Middelboe AL (2005) From thallus to communities: scale effects and photosynthetic performance in macroalgae communities. Marine Ecology Progress Series 287:65-75

Campbell SJ, Bité JS, Burridge TR (1999) Seasonal Patterns in the Photosynthetic Capacity, Tissue Pigment and Nutrient Content of Different Developmental Stages of Undaria pinnatifida (Phaeophyta: Laminariales) in Port Phillip Bay, South-Eastern Australia. Botanica Marina. pp. 231

Choi H, Kim Y, Lee S, et al. (2007) Growth and reproductive patterns of Undaria pinnatifida sporophytes in a cultivation farm in Busan, Korea. J Appl Phycol 19:131-138

Choi H, Kim Y, Lee S, et al. (2005) Effects of daylength, irradiance and settlement density on the growth and reproduction of Undaria pinnatifida gametophytes. J Appl Phycol 17:423-430

Fick A (1855) V. On liquid diffusion. Philosophical Magazine Series 4 10:30-39Ginovart M, López D, Valls J (2002) INDISIM, An Individual-based Discrete Simulation Model to Study

Bacterial Cultures. J Theor Biol 214:305-319Gómez I, Wiencke C (1996) Photosynthesis, Dark Respiration and Pigment Contents of

Gametophytes and Sporophytes of the Antarctic Brown Alga Desmarestia menziesii. Botanica Marina 39:149

Gropp W, Lusk E, Doss N, et al. (1996) A high-performance, portable implementation of the MPI message passing interface standard. Parallel Computing 22:789-828

Jassby AD, Platt T (1976) Mathematical formulation of the relationship between photosynthesis and light for phytoplankton. Limnology and Oceanography 21:540-547

Morita T, Kurashima A, Maegawa M (2003a) Temperature requirements for the growth and maturation of the gametophytes of Undaria pinnatifida and U. undarioides (Laminariales, Phaeophyceae). Phycological Research 51:154-160

Morita T, Kurashima A, Maegawa M (2003b) Temperature requirements for the growth of young sporophytes of Undaria pinnatifida and Undaria undarioides (Laminariales, Phaeophyceae). Phycological Research 51:266-270

Murphy JT, Johnson MP, Viard F (2016a) A modelling approach to explore the critical environmental parameters influencing the growth and establishment of the invasive seaweed Undaria pinnatifida in Europe. J Theor Biol 396:105-115

Murphy JT, Johnson MP, Walshe R (2013) MODELING THE IMPACT OF SPATIAL STRUCTURE ON GROWTH DYNAMICS OF INVASIVE PLANT SPECIES. Int J Mod Phys C 24:20

Murphy JT, Voisin M, Johnson M, et al. (2016b) Abundance and recruitment data for Undaria pinnatifida in Brest harbour, France: Model versus field results. Data in brief 7:540-5

Pang S, Lüning K (2004) Photoperiodic long-day control of sporophyll and hair formation in the brown alga Undaria pinnatifida. J Appl Phycol 16:83-92

Saito Y (1975) Undaria. Advance of Phycology in Japan, Junk Publishers, The Hague:304-320Saulquin B, Hamdi A, Gohin F, et al. (2013) Estimation of the diffuse attenuation coefficient KdPAR

using MERIS and application to seabed habitat mapping. Remote Sensing of Environment 128:224-233

Shao-jun P, Chao-yuan W (1996) Study on gametophyte vegetative growth of Undaria pinnatifida and its applications. Chin J Oceanol Limn 14:205-210

Stevenson RD, Peterson CR, Tsuji JS (1985) The thermal dependence of locomotion, tongue flicking, digestion, and oxygen consumption in the wandering garter snake. Physiological Zoology:46-57

Suto S (1950) Studies on shedding, swimming and fixing of the spores of seaweeds. Bulletin of the Japanese Society of Scientific Fisheries 16:1-9

Suto S (1952) On shedding of zoospores in some algae of Laminariaceae-2. Bull. Jap. Soc. scient. Fish 18:1-5

Thiébaut E, Lagadeuc Y, Olivier F, et al. (1998) Do hydrodynamic factors affect the recruitment of marine invertebrates in a macrotidal area? The case study of Pectinaria koreni (Polychaeta) in the Bay of Seine (English Channel). Hydrobiologia 375-376:165-176

Voisin M (2007) Les processus d'invasions biologiques en milieu côtier marin: le cas de l'algue brune Undaria pinnatifida, cultivée et introduite à l'échelle mondiale (PhD diss.). Paris 6, France

Table S1: Input parameters for UndariaGEN simulations of Undaria pinnatifida in simulated port/marina environment. l = sporophyte length (m), d = daylight hours, loop = simulation loop. Ecophysiological parameters derived from literature review (Choi et al. 2005; Morita et al. 2003a, b; Pang, Lüning 2004; Shao-jun, Chao-yuan 1996)

Parameter Type Parameter (units) Input Value

General Length of Simulation Loop (hours)

Environment Size (No. of Cells)

Cell Area (m2)

Substrate depth in water (m)

Attenuation coefficient (KdPAR)

Community light saturation factor

1

514 x 482

0.25

1.0

0.4

3.33

Sporophyte

agents

Initial length, l0 (m) 20.0

Base growth rate 3.615 l-0.407

Day length response (hyperbolic curve):

Pmax

a

Ic

1.56

0.13

0.0

Thermal performance curve :

K1

K2

K3

CTmin

CTmax

Scale

21.09

0.213

0.006

1.62

28.28

3031

Photosynthesis-irradiance curve:

Pmax

a

Ic

0.4ln(l) - 0.596

0.5l-0.33

2.5ln(l) – 19.9

Mean length at maturity (cm) 32.66

Age to Mortality

Cosine curves

(see Fig. 3):

Weibull shape parameter (k):A

C

Weibull scale parameter ():

A

C

0.205

248.5

2.607

0.2

363.2

1.165

Gametophyte

agents

Thermal performance curve :

K1

K2

K3

CTmin

CTmax

Scale

35.67

0.158

0.015

4.45

28.24

10.63

Photosynthesis-irradiance curve:

Pmax

a

Ic

0.29e0.11d

0.029d – 0.2

0.0

Prob. of fertilisation (loop-1) 0.0002

Gametogenesis Temperature response curve (log):

x0

k17.6

0.82

Day length response (Weibull):

4.5

10.96

Spores Half-life (hours)

Release rate (agent-1 loop-1)

Spore stock (agent-1)

Diffusion coefficient

Prob. of germination (loop-1)

24

2.0 x 107

1010

0.15

10-9

Fig. S1: Main stages in the heteromorphic biphasic life cycle of Undaria pinnatifida: Diploid (2N) sporophyte stage (can grow up to 3 metres in length) which matures to form a sporophyll structure that releases spores. The microscopic haploid (N) gametophyte stages (male and female) form after fixation and germination of spores on a suitable substrate. They reproduce sexually to form a new diploid sporophyte generation. Photos of Gametophytes/Spores: Daphné Grulois-Station Biologique Roscoff

Fig. S2: Program flow structure for agent-based model of Undaria pinnatifida population dynamics coded in C++ programming language. One program loop = 1 hour real-time.

Fig. S3: Photosynthesis-irradiance curve parameters (Pmax, , Ic) for hyperbolic equation of Jassby & Platt (1976) calculated as a function of sporophyte length (m).

(a)

(b)

Fig. S4: Relative effect of (a) water temperature (oC) and (b) day length (hours of day light) on the growth rates of sporophyte agents.

(a)

(b)

Fig. S5: Effects of (a) irradiance (mol m-2 s-1) and day length (h), and (b) water temperature (oC) on the relative growth rates of gametophyte agents.

(a)

(b)

Fig. S6: Effects of (a) water temperature (oC), and (b) day length (h) on the relative fertility of gametophyte agents.

(a)

0.6

0.7

0.8

0.9

1

Mor

talit

y (A

ge <

1 m

onth

)

R2=0.85

(b)

Jan Feb

Mar AprMay Jun Jul

AugSe

p Oct NovDec

0

0.1

0.2

0.3

Mor

talit

y (A

ge 1

-2 m

onth

s)

R2=0.73

Fig. S7: Probabilities of premature mortality among juvenile sporophytes of U. pinnatifida calculated from field data in Brest harbour, France (Murphy et al. in press). (a) Probability of mortality among recruits <1 months old. (b) Probability of mortality for sporophytes 1-2 months old. Data points represent 3-month moving average values. Cosine functions were fitted to the data.

0 50 100 150 200 250 300 3502.3

2.4

2.5

2.6

2.7

2.8

lambda Cosine

Wei

bull

Scal

e (l)

(a)y = 0.2cos[(x-248.5)] + 2.6R2 = 0.89

0 50 100 150 200 250 300 3500.9

1

1.1

1.2

1.3

1.4

k Cosine

Day of Year

Wei

bull

shap

e (k

)

(b)y = 0.2cos[(x-363.2)] + 1.2R2 = 0.97

Fig. S8: The shape (k) and scale () parameters for monthly Weibull distributions (fitted to the age to mortality data for juvenile sporophytes from Fig. 2 in the main text) as a function of the time (day) of year. Cosine functions fitted to the data ( = 2/365).