Optimal Composition of Chloride Cells for Osmoregulation in a Randomly Fluctuating Environment

Fish live in water with a different osmotic pressure from that in the body. 33 Their gills have chloride cells that transport ions to maintain an appropriate level of 34 osmotic pressure in the body. The direction of ion transport is different between 35 seawater and freshwater. There are two types of chloride cells that specialize in 36 unidirectional transport and generalist cells that can switch their function quickly in 37 response to environmental salinity. In species that experience salinity changes 38 throughout life (euryhaline species), individuals may replace some chloride cells with 39 cells of different types upon a sudden change in environmental salinity. In this paper, 40 we develop a dynamic optimization model for the chloride cell composition of an 41 individual living in an environment with randomly fluctuating salinity. The optimal 42 solution is to minimize the sum of the workload of chloride cells in coping with the 43 difference in osmotic pressure, the maintenance cost, and the temporal cost due to 44 environmental change. The optimal fraction of generalist chloride cells increases with 45 the frequency of salinity changes and the time needed for new cells to be fully 46 functional but decreases with excess maintenance cost.


Division of Mathematical Sciences, 21
Tokyo Woman's Christian University Teleost fish live in an environment whose osmotic pressure differs from that 53 of the blood. To cope with the difference in osmotic pressure between the inside and 54 outside of the body, they are equipped with chloride cells (or ionocytes) at their gills, 55 which transport ions such as Na + and Cl - [1,2,3]. Fishes living in seawater have gill 56 chloride cells specialized for living in seawater, in which the osmotic pressure is higher 57 than that within the body. For example, the osmotic pressure in seawater is 58 approximately 1,000 mOsm/kgH2O and that in the body fluid of bony fishes is 59 approximately 300 mOsm/kgH2O [4]. Their chloride cells retain water and secrete 60 liquid with a high ionic concentration. In contrast, fishes living in freshwater have gill 61 chloride cells that function in the opposite manner because freshwater has lower 62 osmotic pressure than the body. 63 Some fish species change their habitat in the course of their life, accompanied 64 by changes in environmental salinity, exemplified by chum salmon and glass eel that 65 migrate between freshwater and marine water. They are called euryhaline species [3]. 66 Because the osmotic pressure of the blood is lower than that in seawater and higher than 67 that in freshwater, chloride cells need to function in the opposite direction in the two 68 environments [3,5,6]. The fish may replace their chloride cells at their gills. However, it 69 takes some time for new chloride cells to become fully functional. Consequently, the 70 change in the type of chloride cells causes the fish to cope with the osmotic pressure 71 difference with fewer chloride cells, leading to a temporally increased workload for the 72 osmoregulation system. 73 The delay in full functioning of chloride cells when the cells are replaced with 74 new types is a very serious problem for species living in fluctuating salinity, such as 75 fishes in tide pools or in estuaries and killifish, sea bass, and tilapia in brackish water. 76 Let be the osmotic pressure in the body and let and be the osmotic 127 pressures in seawater and in freshwater, respectively. The osmotic pressure in the body 128 is between and ( > > ). The amount of salinity stress is proportional to 129 the difference in osmotic pressure between the inside and outside of the body divided by 130 the number of chloride cells that function to cope with the difference. It is 131 − + in seawater; − + in freshwater.
(1) 132 In seawater, the osmotic pressure difference ( − ) is addressed by seawater-133 adapted cells and generalist cells , the latter working in both environments. In 134 freshwater, the osmotic pressure difference ( − ) is addressed by freshwater-135

adapted cells and generalist cells . 136
To reduce the stress caused by the osmotic pressure difference between the 137 inside and outside of the body, having only generalist chloride cells would seem to be 138 beneficial. However, generalist chloride cells are more costly to maintain than specialist 139

cells. Let
, , and be the daily maintenance costs per cell for seawater-adapted 140 cells, freshwater-adapted cells, and generalist chloride cells, respectively. We assume 141 that > and > . The total maintenance cost per day is + + . 142

Temporally enhanced osmoregulation load due to chloride cell replacement 144
Immediately after a change in the environmental salinity, some chloride cells 145 may be replaced. Newly formed chloride cells become fully functional only after some 146 maturation time. For example, if the environment changes from s to f, the cell 147 composition changes from ( , , ) to ( , , ). Some of the cells in ( , , ) 148 cannot be functional for some period because they must be converted from the previous 149 composition ( , , ). Let be the length of time needed for maturation. The cells 150 of the old composition cope with the new salinity level (freshwater environment) less 7 well than ( , , ) in the period of length . This temporarily enhanced 152 osmoregulation load is an additional cost accompanying the change in chloride cell 153

composition. 154
To be concrete, the daily stress before the environmental change is 155 , and the daily stress after the completion of changes in the chloride 156 cell composition is ( − ) ( + ) ⁄ . During a period of length required for the 157 maturation of new chloride cells, the fish experiences daily stress with ( , , ) in a 158 freshwater environment. Hence, the daily stress is ( − ) ( + ) ⁄ , which is greater 159 than the daily stress after maturation ( − ) ( + ) ⁄ . The additional stress 160 accompanied by the change in cell types is: 161 (2) 162 We assume that the mean duration of each environment is much longer than . 163 164

Dynamic optimization 165
We consider the expected total future load if the current composition of 166 chloride cell types is ( , , ) and the current environment is , where indicates 167 seawater ( = ) or freshwater ( = ). The individual chooses the chloride cell 168 composition immediately after a change in the environmental salinity. Consider a period 169 in which the fish is in seawater. Immediately before the period, the environment was 170 freshwater. The current period will end either when the next environmental change to 171 freshwater occurs or when the fish leaves the habitat, which occur at rates and , 172 respectively. Hence, the length of the current seawater environment follows an 173 exponential distribution with mean 1 ( + ) ⁄ . The expected future total load is 174 where ( , , ) is the chloride cell composition in the current environment (seawater), 178 ( ′, ′, ′) is that in the previous environment (freshwater), and ( ", ", ") is that in 179 the next environment (freshwater). The minimization symbol indicates that the 180 individual chooses the optimal chloride cell composition that achieves the smallest total 181 future load. The choice must be made every time the environmental salinity changes. 182 The first minimization symbol in equation (3) indicates the choice of ( , , ), which is 183 made when the environment changes from freshwater to seawater, and the cell 184 composition ( , , ) chosen is then adopted during the current period of a seawater 185 environment, which lasts 1 ( + ) ⁄ days on average. 186 The first term after the square brace in equation (3) is the excess workload, 187 which arises because the cell composition adopted before the environmental change 188 The third term, written in the next line, is the total workload to be paid after 194 the next environmental transition (from seawater to freshwater), multiplied by a factor 195 , which is the probability that the environmental change stems from the end 196 of the current period rather than the fish leaving the habitat. The terms in the curly brace 197 after the second minimization symbol are mostly the decision to be made in the future. 198 However, there is one term that is affected by ( , , ): the chloride cell composition in 199 the current seawater environment. This term arises because the fish must use the current 200 cell composition until newly formed cells become fully functional. 201 If we move some terms out of the minimization symbols, equation (3) can be 202 rewritten as 203 If we pay attention to the terms depending on the current choice of cell composition 207 ( , , ), we see that the optimal value can be calculated from the following: 208 where minimization is calculated under + + = , ≥ 0, ≥ 0, and ≥ 0. under + + = . 215 Equations (4a) and (4b) are derived from assuming the excess workload 216 caused by the delay of new cell maturation is described by equation (2), which allows 217 us to move the term depending on the cell composition adopted in the current period 218 before the minimization symbol. We cannot perform this operation if the excess 219 workload has a more general functional form. Then, the optimal choice of chloride cell 220 composition cannot be obtained by the minimization of a single quantity, such as in 221 Equations (4a) and (4b). In such a case, we may have to adopt an analysis of iterative 222 calculations, which we explain in Appendix A of the Supporting Information. 223 By numerical analyses, we confirmed that these two methods (iterative and 224 noniterative calculation) give the same results. The method using equations (4a) and 225 (4b) is a much simpler and faster way to compute the optimal chloride cell composition 226 in each environment than the iterative calculation explained in Appendix A. In addition, 227 they sometimes provide simple and explicit mathematical expressions (see below). 228 229 Optimal chloride cell composition 230 Here, we discuss the dependence of the optimal chloride cell composition on 231 different parameters. 232 We first consider the situation in which the two environments are symmetric: 233 the environment switches between seawater and freshwater at an equal rate ( = ), the 234 maintenance costs are the same between the two specialist cell types ( = ), and the 235 magnitude of stress is the same although the direction is different ( − = − ). 236 Individuals contain one specialist cell type adapted to the current environment and the 237 other adapted to the next environment, although the current environment may be either 238 seawater or freshwater. Consequently, in the optimal solution, the number of generalist 239 cells is the same between the two salinity levels ( = ), but the numbers of seawater-240 adapted specialist cells and freshwater-adapted specialist cells are switched between the 241 two situations ( = , and = ). 242 The two specialist chloride cell types differ in number. There are more specialist cells 295 that are adapted to the current salinity than the other type adapted to the future salinity 296 ( > ). Note that the equation of is the same as that in Phase II. When 297 environmental change occurs, some fraction of specialist cells switch to specialist cells 298 of the other type (see Fig. 1c). 299 300

Phase IV (No generalist cells) 301
When − is very large, no generalist cells are formed. Two types of specialist 302 cells exist, but the one adapted to the current environment is more abundant than the 303 other type of specialist (illustrated in Fig. 1d). In Appendix B of the Supplementary 304 Information, we derived the optimal chloride cell composition in general cases. 305 For simplicity, we here show the explicit solution when = , indicating 306 that the maintenance cost of the two specialized cell types is the same. Then, we have 307 the following results. Because we assume that is much shorter than the length of each 308 environment or the length of stay in the habitat, we can assume (2 + ) < 1. If so, 309 we have the optimal chloride cell composition as follows:  If is small and close to , the workload for osmoregulation is smaller in 377 freshwater than in seawater. The opposite is the case when is large and close to . 378 In both extreme cases, the optimal cell composition is dominated by specialized cells water, such as killifish, sea bass, and tilapia, have gill chloride cells of several different 391 types [8,16]. Chloride cells of one of these types include molecular machinery 392 functioning in osmoregulation in both directions, and they activate the function that is 393 adaptive and suppress the function that is not adaptive in the current environment. Upon 394 environmental change, they quickly change their function (less than 24 hours) [7,9]. We 395 called them generalist chloride cells in this paper. In addition, newly developed chloride 396 cells are formed from undifferentiated chloride cells [13]. We expect the latter process 397 to require a longer time to complete. 398 In this paper, we discussed the optimal composition of chloride cells in a 399 randomly fluctuating environment. If the cost in the current environment were the only 400 concern, the optimal strategy would be to have specialist cells that are adaptive in the 401 current environment, similar to species living only in seawater or only in freshwater 402 (stenohaline species). However, this strategy is quite risky if there is a chance for the 403 environmental salinity to change. In this paper, we discussed a model incorporating unpredictable changes in 458 salinity, which is suitable for fishes living in brackish water. In contrast, in some 459 species, the transition from one habitat to another with different salinities is a part of the 460 normal life cycle [20]. For example, catadromous fishes, such as eels, have breeding 461 sites in the sea, swim upstream to grow and then go down to the ocean when they 462 mature, while anadromous fishes, such as salmon, have spawning sites in the stream and 463 growing habitats in the ocean. The timing for the change in salinity should be 464 predictable for each individual, and the fish may be able to change chloride cells before 465 the transition to a habitat with different salinity. 466 Suppose, for example, that the salinity change from freshwater to seawater 467 can be predicted beforehand. Fishes may have all freshwater-adapted chloride cells 468 much before the transition, and as the transition time (e.g., the time of going down to 469 the ocean) is approaching, they may replace those specialist cells with generalist 470 chloride cells. After they went down the ocean, generalist chloride cells may be 471 replaced again by specialist chloride cells adapted in seawater. Calculating the optimal 472 schedule of having different chloride cells given such a predictable change in 473 environmental salinity and comparing the model with the case studied in this paper 474 would be an important theme of future theoretical study. 475 The model studied in this paper includes many simplifying assumptions. First, 476 we assumed a single type of specialized chloride cell adapted to the freshwater 477 environment, but multiple types of chloride cells are adapted to the freshwater 478 environment [21,22]. Second, although we assumed that the number of chloride cells 479 was a fixed constant, the number of mature chloride cells might fluctuate over time. The 480 spatial distribution of chloride cells changes upon an environmental change [13,16]. 481 Third, the generalist cell response to environmental changes depends both on hormones 482 and on the direct change in osmotic pressure experienced by the focal cells, suggesting 483 that hormones might coordinate different cells and tissues [10,11,12]. Although only the 484 gills of fish were discussed in this paper, tissues in the intestine and kidney are known 485 to respond to the change [14], which might affect each other by hormones.   Figure 1