- Email: [email protected]
Steepness of thermal gradient is essential to obtain a unified view of thermotaxis in C. elegans
ARTICLE IN PRESS Journal of Theoretical Biology 260 (2009) 56–65
Contents lists available at ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Steepness of thermal gradient is essential to obtain a unified view of thermotaxis in C. elegans Kenichi Nakazato a,, Atsushi Mochizuki a,b,c a
Theoretical Biology Laboratory, RIKEN Advanced Science Institute, Wako 351-0198, Japan Devision of Theoretical Biology, National Institute for Basic Biology, Okazaki 444-8787, Japan c PRESTO, JST, Kawaguchi 332-0012, Japan b
a r t i c l e in f o
a b s t r a c t
Article history: Received 24 March 2009 Received in revised form 28 May 2009 Accepted 28 May 2009 Available online 6 June 2009
One of the adaptive behaviors of animals in their environment is thermotaxis, by which they migrate toward a preferred temperature. This sensorimotor integration is accomplished by choosing one of two behaviors depending on the surrounding temperature, namely thermophilic or cryophilic movement. Caenorhabditis elegans exhibits thermotaxis and its migration behavior has been analyzed experimentally at both the population and individual levels. However, some experimental data are inconsistent especially for thermophilic movement, which is expected to be observed in lower than favorable temperatures. There are no experimental analyzes that find thermophilic tendencies in the individual behavior of worms, despite multiple reports supporting thermophilic movement of the population. Although theoretical methods have been used to study thermotaxis of C. elegans, no mathematical model provides a consistent explanation for this discrepancy. Here we develop a simple biased random walk model, which describes population behavior, but which is based on the results of individual assays. Our model can integrate all previous experiments without any contradiction. We regenerate all the population patterns reported in past studies and give a consistent explanation for the conflicting results. Our results suggest that thermophilic movement is observed, even in individual movements, when the thermal gradient is sufficiently slight. On the contrary, thermophilic movement disappears when the thermal gradient is too steep. The thermal gradient is thus essential for a comprehensive understanding of the experimental studies of thermotaxis in C. elegans. Our model provides insight into an integrative understanding of the neural activity and thermotactic behavior in C. elegans. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Sensorimotor behavior Thermotaxis C. elegans Biased random walk Mathematical modeling
1. Introduction Nematode Caenorhabditis elegans is one of the most wellknown experimental animals in neurobiology, developmental biology, and various other fields. Worms have long since been chosen as convenient experimental animals for the integrated comprehension of genetics, neurobiology, and behavior (Brenner, 1974). The hermaphrodite worm has only 959 cells, of which 302 are neurons. Most importantly, the network structure of these 302 neurons has been fully elucidated (White et al., 1986). In addition, the complete cell lineage (Sulston and Horvitz, 1977) and whole genomic sequence (The C. elegans Sequencing Consortium, 1998) are known. Taking advantage of this knowledge, various sensorimotor behaviors in C. elegans have been investigated intensively (Chalfie et al., 1985; Pierce-Shimomura et al., 1999; Gabel et al., 2007; Ward et al., 2008).
Corresponding author.
E-mail addresses: [email protected] (K. Nakazato), [email protected] (A. Mochizuki). 0022-5193/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2009.05.027
We consider the behavior in a thermal environment known as thermotaxis (Hedgecock and Russell, 1975; Mori and Ohshima, 1995), focusing particularly on the migration behavior in thermotaxis and the mathematical modeling thereof. Worms on a thermal gradient wander around searching for a favorable temperature, and this shows that they have the ability to memorize their environmental temperature and to use this information in their migration (Kimura et al., 2004; Chung et al., 2006; Clark et al., 2006, 2007b; Biron et al., 2006). In addition to searching for the cultivated temperature, worms, once they arrive there, crawl along the isothermal curve (Mori and Ohshima, 1995; Luo et al., 2006). The strategy of searching for the cultivated temperature is considered to be quite simple: worms have a tendency to go down the thermal gradient when situated at a higher temperature than the cultivated temperature and a tendency to go up when at a lower temperature (Hedgecock and Russell, 1975; Mori and Ohshima, 1995; Mohri et al., 2005). However, this thermophilic tendency is now a topic of debate, because the tendency has not been confirmed by several assays (Ryu and Samuel, 2002; Clark et al., 2007a; Yamada and Ohshima, 2003; Ramot et al., 2008b).
ARTICLE IN PRESS K. Nakazato, A. Mochizuki / Journal of Theoretical Biology 260 (2009) 56–65
The navigation behavior of C. elegans has been clarified by a model of a biased random walk (Gray et al., 2005; PierceShimomura et al., 1999; Wiktorsson et al., 2004; Ramot et al., 2008b). The migration movement of C. elegans consists of repeated straight forward runs and turning behavior (Matsuoka et al., 2008; Gray et al., 2005; Pierce-Shimomura et al., 1999; Ramot et al., 2008b). Both types of movement, i.e., straight and turning, may be accompanied by a bias in some direction against the thermal gradient, causing a mean displacement in migration and consequently, thermotactic behavior. Two types of bias are shown in Fig. 1. Run duration bias results in a mean translational displacement. In actual, run of the nematodes is not perfect straight. It may give some angular displacement. However, the effect seems to be small and we ignore it following the previous mathematical studies using random walk model. Reorientation bias causes mean angular displacement and indirectly, translational displacement as well. One of the methods for studying migration behavior is individual movement assays. The above two types of bias in thermotactic behavior have been examined experimentally using individual assays (Ryu and Samuel, 2002; Clark et al., 2007a). Experimental data for run duration times indicate that there is a bias towards cryophilic tendencies in temperatures higher than the cultivated temperature. On the other hand, the thermophilic drive is still under discussion even by the individual movement assays. There are published tracks of worms that appeared to perform thermophilic movement (Hedgecock and Russell, 1975; Mori and Ohshima, 1995). However, some groups have found no evidence of thermophilic bias based on statistical analysis of tracked worms (Ryu and Samuel, 2002; Clark et al., 2007a). Nor are there reports on reorientation bias affecting thermotactic behavior. Population distribution assays have also been used to study thermotactic behavior, however the results of these studies are seemingly inconsistent with one another. In an analysis at the population level, changes in the mean and deviation of a distribution may be used as statistics characterizing thermotactic behavior instead of the distribution itself. Some reports on the population distribution and its statistics state that both cryophilic and thermophilic movements were observed (Hedgecock and Russell, 1975; Ito et al., 2006; Kuhara et al., 2008), thus conflicting with the results of individual movement assays. Thermophilic movements may be hardly observed at least for several minutes in the beginning of assay (Ito et al., 2006; Ramot et al., 2008b). On the other hand, another report asserts the possibility that the thermophilic-like tendency of the population behavior at lower temperatures might be caused by the decrease in motility, in other
Fig. 1. Two types of bias. The first is the bias in run duration time or distance in each direction against thermal gradient. (A) shows that worms crawling up the thermal gradient tend to maintain straight movement longer than worms crawling down the gradient as an example of run duration bias. The other bias is that in reorientation angle. (B) shows an example of reorientation bias: worms prefer the thermophilic direction when changing direction.
57
words a ‘‘cold trap’’ (Yamada and Ohshima, 2003). Their idea is that the motility of worm decreases drastically under a critical temperature then the worms would not move into colder region from moderate temperature. There are two possible approaches for understanding these seemingly contradictory experimental results at different observation levels. One possibility is that a thermophilic-like property is generated in the behavior of the population by some mechanism, such as asymmetric motility between higher and lower temperatures (Yamada and Ohshima, 2003; Dusenbery and Batt, 1980; Anderson et al., 2007), despite there being no thermophilic property in individual behavior. The other possibility is that a thermophilic property at the individual level does exist, but there has not been positive statistical data in the previous experimental methods. In this paper, we aim to give a consistent explanation for the conflicting results by using a theoretical method. We construct a simple biased random walk model of the population behavior, that reflects the results of individual movement assays. Using this model we succeed in giving a consistent explanation for past experiments. At the same time, our results assert the importance of the steepness of the thermal gradient that may change the migration behavior drastically in experiments on thermotaxis. In the next section, we summarize diagrammatically the thermotactic behavior and active ranges of thermoreceptors. We then introduce our biased random walk model, using a variation of the telegrapher’s equation. Numerical results are given in next section. In this section, we explain that past experiments can be regenerated just by taking account of the effect of thermophilic drive in our model. Furthermore, we show that some of the unique results by Yamada–Ohshima can be explained as a consequence of the limited range of thermophilic drive and the ‘‘cold trap’’. In the last section, we summarize our results and discuss the thermophilic drive particularly with respect to those points which have not been considered to be important. Our results together with the experimental results of Paola et al. (in preparation) suggest that thermophilic drives ought to be observed in individual assays as well with different gradient conditions.
2. Model 2.1. Thermotaxis as a biased random walk The duration of movement in a straight line and the reorientation angle of C. elegans are thought to be stochastic in nature (Gray et al., 2005; Pierce-Shimomura et al., 1999, 2005; Ramot et al., 2008b). Individual worms are assumed to run at a constant speed in their straight forward movements around the cultivated temperature (Ryu and Samuel, 2002). Though actual run of the nematodes is not perfectly straight, we assumed it in our model. In this case, the run duration time directly reflects the translational displacement. Reorientation behavior is classified as either small turns or large turns called omega turns or pirouettes (Pierce-Shimomura et al., 1999; Gray et al., 2005). The biased random walk is a mathematical model suited to describing and analyzing directional movement with stochasticity. It has many different mathematical forms corresponding to the different levels of phenomena, such as the Langevin equation or Fokker–Planck equation (Risken, 1989; Schnitzer, 1992), that make it possible to consider the relation between individual and population behaviors of thermotaxis in C. elegans. To formalize the biased random walk model of thermotaxis, we require the probability distribution of the run duration time and the reorientation angle as a function of direction and temperature. Some experimental reports based on the notion of a biased
ARTICLE IN PRESS 58
K. Nakazato, A. Mochizuki / Journal of Theoretical Biology 260 (2009) 56–65
random walk have elucidated features of the distribution functions especially on the cryophilic drive (Ryu and Samuel, 2002; Clark et al., 2007a). According to these reports, worms placed at a higher temperature tend to continue moving down the gradient towards the cultivated temperature. In contrast, no reorientation bias in any direction against the thermal gradient has been observed to date. Regarding thermophilic drive, there are no results from individual movement assays that support the individual thermophilic tendency. Besides the description based on individual behavior, a population based approach is also useful. Modeling a biased random walk in terms of population behavior requires information of the time evolution of the distribution function of the worms’ positions. Results from distribution assays show that a population initially distributed at either lower or higher temperatures shifts towards the cultivated temperature. Although it has been shown that a deterministic mathematical model can be built as for isothermal tracking (Luo et al., 2006), we use a biased random walk without major alteration to the mathematical framework.
2.2. Diagram of thermal behavior Information of environmental temperature is assumed to have been obtained through thermosensory neurons AFD and AWC, and utilized for thermotactic behavior in C. elegans (Kimura et al., 2004; Kuhara et al., 2008; Clark et al., 2006, 2007b; Biron et al., 2006, 2008; Ramot et al., 2008a). AFD and AWC neurons show thermal responses at temperatures down to at most T c 3 in analysis of cellular responses using Ca2þ imaging (Kimura et al., 2004; Clark et al., 2006; Kuhara et al., 2008; Biron et al., 2008). Electrophysiological measurements show the possibility to show thermal responses at temperatures lower than T c 3 after longtime observation by the effect of thermal adaptation (Ramot et al., 2008a). In this study we assume that the active range would not expand in the time scale of our interest ð1 hÞ. In a cold region, where there is no thermal response from AFD nor AWC, run duration and reorientation angle would be homogenous to any direction and thermal gradient, while the active range might differ from behavioral output. In fact no observation has been reported that worms display thermotactic behavior in regions much colder than the cultivated temperature. The diagram of thermotaxis in Fig. 2 is a summary of the known facts. Combining past experiments and considering possible behavior, we divide the environmental space into three sections according to temperature: high, low, and cold temperature regions (HTR, LTR, and CTR) (Fig. 2). The high temperature region is the area in which the temperature is
higher than the cultivated one and in which it is possible to observe a cryophilic drive with run duration bias. In the low temperature region we assume that the worms tend to move in the thermophilic direction in some unknown way. The cold region is the area where thermoreceptors show no response to thermal stimuli and thus we would observe athermotactic behavior. 2.3. Mathematical formulation We use a variation of the telegrapher’s equation to describe the distribution behavior of our biased random walk model (Schnitzer, 1992). We consider a one dimensional space with linear thermal gradient, where position x corresponds to the temperature. The movement of worms is limited to the direction along the thermal gradient. Isothermal tracking is represented as a static state because worms in isothermal tracking do not move in the direction of the thermal gradient. We can now express the time evolution of the distribution as follows: @F @ðvFÞ ¼ þ TF, @t @x T F ¼ ðf ; g; sÞ , 1 0 v 0 0 C B C v¼B @ 0 v 0 A, 0 0 0 0 ða þ cÞ b B a ðb þ cÞ T¼B @ c c
(1) (2) (3) d
1
C d C A, 2d
(4)
where f ðx; tÞ is the distribution of worms crawling up the thermal gradient, gðx; tÞ is the distribution of worms moving down the gradient, and sðx; tÞ is the distribution of worms tracking isothermal curves. The motility vðxÞ of worms depends only on the thermal environment; in other words, motility is a function of position x. T is a matrix consisting of the state transition rates between the three states, f ðx; tÞ; gðx; tÞ, and sðx; tÞ. These transition rates, aðxÞ, bðxÞ, cðxÞ and dðxÞ, are decided from the experimental data on the duration of each state and the destination after the transition events. We assume here that worms at state f ðx; tÞ or gðx; tÞ are rocked into an isothermal tracking state sðx; tÞ at the same rate cðxÞ. After isothermal tracking halts, worms transit to states f ðx; tÞ and gðx; tÞ at the same rate dðxÞ. A thermophilic or cryophilic tendency in this model is represented as the difference of transition rates, bðxÞ aðxÞ or aðxÞ bðxÞ, respectively. Although we must distinguish between run duration bias and reorientation bias to analyze the biased random walk in a real situation, this distinction makes no sense in our simplified model. In our model, run duration bias and reorientation bias are simplified merely as the difference between the two probabilities to reverse the direction, which represents an individual migration tendency. This simplification is appropriate to prevent superfluous hypotheses on the detailed mechanism of a thermophilic drive. Thus, we need only hypothesize the migration tendency at each temperature. Despite this simplification, our model is sufficient to illustrate the distribution behavior of worms on the thermal gradient.
3. Numerical results
Fig. 2. Diagrammatic view of thermotaxis in C. elegans, where the horizontal axis reflects temperature. The environmental space of the worms is divided into three sections along the temperature axis according to thermotactic behavior and active pattern of the thermoreceptors. We consider the effect of isothermal tracking in the parameter settings around the cultivated temperature T c 2 (Fig. 3).
In this section, we explain the past experimental results from the point of view of a biased random walk, using numerical results from our model. Giving numerical results comparable to each experiment, we illustrate the effect of thermophilic drive in each case.
ARTICLE IN PRESS K. Nakazato, A. Mochizuki / Journal of Theoretical Biology 260 (2009) 56–65
3.1. Parameter settings To obtain numerical results comparable to past experiments, we need to decide parameter settings based on the experimental conditions. Parameters to be decided are the velocity matrix v and transition matrix T. Although worms are assumed to move at constant speed at the cultivated temperature, a decrease in motility at abnormal temperatures has been reported (Dusenbery and Batt, 1980; Anderson et al., 2007). We can decide the parameter values of v at each temperature by calculating the average velocity of worms in each state. Parameter v shown in Fig. 3 is calculated from experimental data (Ryu and Samuel, 2002; Luo et al., 2006; Clark et al., 2007a; Dusenbery and Batt, 1980; Anderson et al., 2007). We have to estimate transition matrix T as a function of temperature from the duration in each state, which is obtained from experimental data. We refer to the experimental data in Ryu and Samuel (2002), Luo et al. (2006) and Clark et al. (2007a) and apply the estimation in Matsuoka et al. (2008) in principle. For the low temperature region, we added a hypothetical difference between aðxÞ and bðxÞ to realize a thermophilic bias. Our hypothetical settings for the thermophilic drive are reflected in the modification around the LTR: the parameter bðxÞ is a little higher as shown in Fig. 3. We use the parameter sets, including this hypothetical thermophilic bias ðT d ¼ 5; 10Þ, to obtain numerical results. Furthermore, we use the parameter sets without this hypothesis ðT d ¼ 0Þ to show the effect of the thermophilic drive. Transition rates cðxÞ and dðxÞ are decided using run duration time of isothermal tracking. Full detail of transformation equations and their derivation are shown in Appendix.
3.2. Hedgecock and Russell The first report on thermotaxis in C. elegans is the paper by Hedgecock and Russell (1975) (HR). They reported the results of distribution assays for different temperatures around 12:9, 17:6, 22:8 and 27:1 C with a constant thermal gradient, 0:5 C=cm, using worms cultivated at 20:0 C. A cryophilic tendency was observed at 22:8 and 27:1 C while a thermophilic tendency was observed at 17:6 C. However, neither a thermophilic nor cryophilic tendency was observed at 12:9 C. In Fig. 4, we present numerical results under conditions that simulate the experiments by HR. Figures in the left column, Figs. 4a, d, g and j show results using the parameter sets without
0.03 0.02 0.01 0 5
10
15
20
25
30
35
temperature a(T) v(T)
b(T ) [ T d =10] b(T ) [ T d = 5] b(T ) [ T d = 0]
Cultivated temperature: Tc = 20 Fig. 3. Parameters as functions of temperature. These are calculated using Eqs. (11)–(15) from experimental data of run duration time. T d is a parameter that adjusts the strength of the thermophilic bias.
59
thermophilic bias. No thermophilic tendency in the distribution evolution is observed. Figures in the center column, Figs. 4b, e, h and k, and in the right column, Figs. 4c, f, i and l, show results including thermophilic drive. We can easily recognize thermophilic movement in Figs. 4e and f. The other figures in these columns show no thermophilic tendency because of the range of space in the simulation. The spatial ranges, corresponding to experiments by HR around temperatures 12:9, 22:8 and 27:1 C, do not include the LTR, where thermophilic bias is active. In the cold region (CTR in Fig. 2), the peaks of the distributions shift only slightly to a colder region as shown in Figs. 4a–c. In other words, these figures show athermotactic behavior which is reported by HR. Thus our model can regenerate the results of HR only when using parameter sets with thermophilic drive. 3.3. Kuhara et al. Other distribution assays that support thermophilic drive have been reported by Kuhara et al. (2008) (KO) and Ito et al. (2006) (IM). Both of these studies tested the distribution behavior around the cultivated temperature, however neither tested the region far from the cultivated temperature, where thermoreceptors are inactive or motility is lost. They reported similar results to HR; that is, both thermophilic movement in the HTR and cryophilic movement in the LTR are observed. The numerical results are shown in Fig. 5. We computed these results using a thermal gradient and boundary condition similar to experiments in KO. The figures in the center and right columns, using parameter sets with thermophilic bias, show that experimental results by KO and IM can be regenerated by our model, as we can easily recognize that both peaks (green lines) and averaged positions (red lines) move toward higher temperature than original positions (20) in Figs. 5h and i. The figures in the left column show results using parameter sets without thermophilic bias. However as Fig. 5g shows, a thermophilic shift of peak position may be observed despite the absence of thermophilic bias, a phenomenon that is discussed later. This is consistent with the results by Matsuoka et al. (2008). 3.4. Yamada and Ohshima Some unique results have been reported by Yamada and Ohshima (2003) (YO). They tested distribution assays with a very steep thermal gradient 1:5 C=cm and reported that no convergence of individual worms towards the cultivated temperature was seen. Instead, they observed a widely scattered distribution centered around the LTR and an accumulation of worms with lost motility in the CTR. They concluded that worms have a cryophilic tendency in the HTR but no thermophilic tendency in either the LTR or CTR. In addition, they suggested that the thermophilic tendency observed in other experiments might stem from the ‘‘cold trap’’ in the CTR. We thus evaluate the idea of the cold trap and illustrate the difference between thermophilic bias and a cold trap. To evaluate the effect of the cold trap, we present numerical results in Fig. 6. We compare these results with the normal parameter settings given in Fig. 3 using constant motility, and not a function of temperature. The steepness of the thermal gradient and the boundary conditions are similar to those of YO for easy comparison between our numerical results and the experimental results in Yamada and Ohshima (2003). Figures in the top row show the results with normal parameter sets, in which worms lose their motility in the CTR (Fig. 3). Figs. 6a–c, show that bimodal distributions emerge as time advances, and thus
ARTICLE IN PRESS 60
K. Nakazato, A. Mochizuki / Journal of Theoretical Biology 260 (2009) 56–65
Td = 5
Td = 0
0
60
30 60
14
12
10
0
time
30
time
time
~12.9°C
0
~17.6°C
temperature
30 60
20
18
16
0
time
time
time
30
20
22
0
30
time
time
time
30
60
24
20
temperature
22
60
24
20
26
28
30
24
0
time
time
time
30
22
temperature
0
60
30
temperature
0
20
18
16
temperature
0
60
60
20
18
16
30
temperature
0
14
12
10
0
temperature
~22.8°C
60
temperature
0
60
30
14
12
10
temperature
~27.1°C
Td = 10
30 60
temperature
26
28
30
30 60
temperature
26
28
30
temperature
Thermal gradient: 0.5°C/cm Cultivated temperature: Tc = 20°C Fig. 4. Thermophilic drive is necessary to regenerate the experimental results by Hedgecock and Russell (1975). Results with the condition of the CTR show that worms behave athermotactically regardless of thermophilic drive (a–c). Red, green, gray and purple dashed lines show the mean of population, the position of peaks, initial position and cultivated temperature, respectively. Thermophilic tendency is observed only when the temperature is in the range of the LTR and thermophilic bias is included in our simulation. In our simulations, we use a reflecting boundary at the right and left edges.
thermophilic shifts are not clear, even in the results with thermophilic bias (Figs. 6b and c). These figures suggest that when the thermal gradient is too steep, worms do not converge at the cultivated temperature even under a thermophilic drive. The figures in the bottom row, Figs. 6d–f, illustrate the results with constant motility. These figures show single modal distributions with fat tails in the lower temperature region as contrasted with the figures in the top row. The difference between the figures in the top and bottom rows shows that the cold trap creates bimodal distributions with clear limits in the CTR. On the other hand,
widely scattered distributions are observed both in the top and bottom regardless of thermophilic drive. The significant feature of the experimental conditions of YO is the steepness of the thermal gradient, which determines the spatial width of the LTR. Under the conditions in YO, the width of the LTR become so narrow that worms do not necessarily stay in the LTR or HTR, nor converge at the cultivated temperature, as shown in Fig. 6. Thus we can conclude that the cause of scattered distributions in YO is not the lack of thermophilic bias, but the steepness of the thermal gradient used in their experiments.
ARTICLE IN PRESS K. Nakazato, A. Mochizuki / Journal of Theoretical Biology 260 (2009) 56–65
Td = 0
Td = 5
0
18
20
22
24
e tim
e tim
16
30 60 16
temperature
20
22
60
24
16
18
20
22
24
60 16
temperature
18
20
22
24
60 16
temperature
0
30
tim
18
20
22
24
22
24
30
e
16
20
0
e
tim
e
tim
60
18
temperature
0
30
24
e
16
22
30
tim
e
60
20
0
30
tim
30
18
temperature
0
e
tim
Tc = 20°C
18
30
temperature
0
Tc = 23°C
Td = 10
0
30
e tim
Tc = 17°C
0
60
61
60 16
temperature
18
20
22
24
60 16
temperature
18
20
22
24
temperature
Thermal gradient: 0.44°C/cm Temperature at initial position: 20°C Fig. 5. Thermophilic tendency in the distribution behavior can be seen only if we adopt the parameter sets with thermophilic bias. Sufficiently strong thermophilic bias is needed to observe thermophilic tendency in the distribution behavior. Red, green, gray and purple dashed lines show the mean of population, the position of peaks, initial position and cultivated temperature, respectively.
Furthermore, our results suggest that the experiments of YO are not appropriate to confirm a thermophilic bias in the LTR, as the thermophilic bias would not make a significant difference in the distribution behaviors under steep thermal gradients. There is another experimental report with mathematical analysis using distribution assay with steep thermal gradient 1:15 C=cm (Anderson et al., 2007). The paper showed the result that thermophilic tendency is hardly seen, and we can understand it by our model similarly as the case of Yamada and Ohshima (2003). The difference between the results in Anderson et al. (2007) and Yamada and Ohshima (2003) is that peak of distribution in cold region is not observed in Anderson et al. (2007). This difference may stem from the food condition: worms are on food in Anderson et al. (2007), unlike most thermal preference assays. If we consider that turning behavior of worms is more frequent on food than off food (Gray et al., 2005), then
worms would diffuse slowly and consequently accumulation in cold region would be lost as seen in Anderson et al. (2007).
3.5. Paola et al. (in preparation) As noted in a previous Section 3.4, steepness of thermal gradient is one of the key factors in the observation of thermal tendency in thermotactic behavior. Fig. 7 shows numerical results that illustrate the effect of steepness of the thermal gradient. We have confirmed that our results here basically agree with the experimental results to be published by Paola et al. (in preparation). Thermophilic tendency is clearly observed in Figs. 7b, c, and f, where the thermal gradient is gradual and thermophilic bias is strong. Both peaks and averaged positions move toward higher
ARTICLE IN PRESS 62
K. Nakazato, A. Mochizuki / Journal of Theoretical Biology 260 (2009) 56–65
Td = 0
Td = 5
0
60
8
12
16
20
24
28
0
30
32
time
30
time
time
v = v(T)
0
60
8
12
temperature
16
20
24
28
32
12
16
20
24
8
12
28
32
16
20
24
28
32
28
32
0
30
time
time
time
30
8
60
temperature
0
60
30
temperature
0
v = const.
Td = 10
60
8
12
temperature
16
20
24
28
30
32
temperature
60
8
12
16
20
24
temperature
Thermal gradient: 1.5°C/cm Cultivated temperature: Tc = 20°C Temperature at initial position: 20°C Fig. 6. Convergent distribution around the cultivated temperature is barely confirmed with the steep thermal gradient used in YO regardless of thermophilic bias. Red, green, gray and purple dashed lines show the mean of population, the position of peaks, initial position and cultivated temperature, respectively. Figures in the top row reflect the results with parameter sets shown in Fig. 3(a)–(c). Double peaked distributions, with one peak in the CTR and the other around the cultivated temperature, appear gradually. Figures in the bottom row reflect the results with constant motility (d–f). All distributions in these figures spread widely to colder regions, but not to warmer regions.
temperature than original position (18). On the contrary, thermophilic tendency is barely observed in Figs. 7d, g, and h, where the thermal gradient is steep and thermophilic bias is weak. These results show that a thermophilic tendency in the distribution behavior requires not only a thermophilic bias but also a slight thermal gradient. The steepness of the thermal gradient determines the spatial range of the LTR, which changes the thermal tendency in the distribution behavior of worms. Thermophilic tendency becomes weaker and is ultimately lost, when we steepen the thermal gradient, even with a large thermophilic bias.
4. Summary and discussion Worms are considered to have the ability to migrate towards the cultivated temperature, a phenomenon known as thermotaxis in C. elegans. However, several experimental results, that do not agree with thermophilic drive, have been reported (Ryu and Samuel, 2002; Clark et al., 2007a; Yamada and Ohshima, 2003; Anderson et al., 2007). Mathematical modeling has also been used to study thermotaxis, however none of the models has succeeded in giving a comprehensive explanation for past experimental results. In this paper, we give a consistent explanation for the past experiments using a biased random walk model of thermotaxis in
C. elegans. Our results show that a thermophilic drive is necessary to explain the past experiments without discrepancies. In addition we suggest that the steepness of the thermal gradient or spatial width of the LTR must be taken into account when studying thermotactic behavior. Migration behavior is considered to be accomplished by a balance of thermophilic and cryophilic drive, which is controlled by several neurons (Mori and Ohshima, 1995). Thermoreceptors AFD and AWC are recognized as the gate of thermal information in worms and show different responses to thermal stimuli from environmental temperature (Mori and Ohshima, 1995; Kimura et al., 2004; Kuhara et al., 2008; Clark et al., 2006, 2007b; Biron et al., 2006, 2008; Ramot et al., 2008a). In the cold temperature region the thermoreceptors show no response to thermal stimuli. This inactivity of the thermoreceptors in cold temperatures may cause the athermotactic behavior in the CTR reported in Hedgecock and Russell (1975). In addition to neural activities, motility of worms may affect thermotactic behavior. In fact the cold trap (Yamada and Ohshima, 2003) in the CTR is caused by the loss of motility in cold environments (Dusenbery and Batt, 1980; Anderson et al., 2007). Considering these facts we conclude that thermophilic drive is active only in the LTR and not in the CTR. In Fig. 5g, we observed a shift in the peak of the distribution towards the thermophilic direction even when using a model without thermophilic bias. This result may be considered as
ARTICLE IN PRESS K. Nakazato, A. Mochizuki / Journal of Theoretical Biology 260 (2009) 56–65
Td = 5
Td = 0
20
18
16
0 30
22
time
30
14
60 14
12
21
18
15
14
60 12
21
18
15
24
16
20
24
28
21
18
15
24
temperature
0 30
time
time
time
12
60 12
0
8
30
temperature
30
22
temperature
30
24
0
20
18
16
0
temperature
60
60
time
30
time
time
0.8 °C/cm
22
0
60
30
temperature
0
1.2 °C/cm
20
18
16
temperature
Thermal gradient
Td = 10
0
time
time
0.4 °C/cm
0
60
63
60 8
temperature
12
16
20
24
30
28
temperature
60 8
12
16
20
24
28
temperature
Cultivated temperature: Tc = 20 °C Temperature at initial position: 18 °C Fig. 7. The steepness of thermal gradient affects thermophilic tendency in population distributions. Red, green, gray and purple dashed lines show the mean of population, the position of peaks, initial position and cultivated temperature, respectively. Thermophilic tendency is not clear in the figures in the center (d–f) and the bottom (g–i) rows. The mean position of each distribution, represented by red lines, shows that the tendency of the mean position and that of peaks do not always coincide (a, d, e, and the figures in the bottom row). In cases with very steep gradients, the distributions are characterized by the bimodality thereof.
showing the possibility that the thermophilic movement of the population may be realized even without thermophilic bias of the individuals. However, numerical simulations under the same or similar conditions can realize only a shift in the peak and not a shift in the averaged position of the distribution. In actual experiments conducted by KO and IM, shifts in both peak and mean are observed. We thus conclude that a peak-shift only does not reflect a real thermophilic drive. In general, movement of the mean and that of the peak may be inconsistent. If the distribution of worms covers a wide range of space, movement of the mean may be calculated from the behavior of worms in all three regions HTR, LTR, and CTR. On the other hand, movement of the peaks of the distribution may be determined only from local behavior in either the HTR, LTR, or CTR. Therefore, both the mean and peaks of the distribution
should be measured to analyze thermotactic behavior using the experimental method. Negative results for thermophilic bias are reported in Yamada and Ohshima (2003). The authors suggest that thermotaxis cannot be explained by the balance between thermophilic and cryophilic drive, but by a combination of cryophilic drive and the cold trap. Our results however, suggest that the steep thermal gradient used by Yamada and Ohshima (2003) is not an appropriate condition to deny the thermophilic drive. As a thermophilic bias would act on worms in the LTR only, the thermal gradient should be sufficiently slight for the thermophilic tendency to be realized in an LTR of adequate width. Worms on a steep thermal gradient do not always remain around the cultivated temperature, but occasionally move towards the CTR, where worms are no longer thermotactic and their motility is lost. The frequent excursions of worms result in
ARTICLE IN PRESS 64
K. Nakazato, A. Mochizuki / Journal of Theoretical Biology 260 (2009) 56–65
athermotactic behavior in the CTR when the thermal gradient is extremely steep. In other words, the balance of thermophilic and cryophilic drive alone is not enough to keep the worms around the cultivated temperature under the condition of Yamada and Ohshima (2003). Thermophilic drive is not always observable; it is observable only when the thermal gradient is sufficiently slight. Our results may give theoretical and comprehensive explanation for the report by Ramot et al. (2008b), though direct comparison may not be appropriate because their experimental time (10 min) is short. They tested some different thermal gradients, 0.3, 0.5, 1.0 and 1:5 C=cm, and show that thermophilic tendency can be observed only one case, 0:3 C=cm, in narrow area around the cultivated temperature. Our theoretical results suggest that the narrowness would stem from the spatial width of LTR, in other words active range of thermosensory neurons AFD and AWC. Some papers on individual assays report negative results for thermophilic drive (Ryu and Samuel, 2002; Clark et al., 2007a). The results for these individual assays give experimental data mainly on cryophilic behavior in the HTR and athermotactic behavior in the CTR. There is however no experimental data for the LTR, where we hypothesize thermophilic bias. Our parameter sets for the HTR and CTR are decided from results by Ryu and Samuel (2002) and do not conflict with their results. However, our model describes distribution behavior and not individual behavior, thus direct comparison is not applicable. We have neglected some aspects of thermotactic behavior for simplicity of our model. Sensorimotor responses of worms are assumed to be variable depending on time (Gray et al., 2005). In fact, turn frequency and navigation behavior depend on metabolic conditions (Char et al., 2004). As a result, behavior may change with time under certain experimental conditions. Actually, some population distribution assays suggest that thermophilic movement is slower process than cryophilic movement (Ito et al., 2006; Ramot et al., 2008b). We may consider at least two reasons for the duality of time scale in thermotactic behavior. First, thermophilic bias may be too weak to be observed in a few minutes in the beginning of assays. This is the case of our model and results show that thermophilic tendency may be too weak to be observed in population behavior at the time of 10 (Fig. 5). Second, worms may not choose thermophilic movement for several minutes after transfer to the assay plate for any reason: e.g. Zhao et al. (2003) discusses suppression of reversal behavior for initial 90 s. In this case, our model may need to adopt temporal change of parameters for the detailed modeling. Another point concerns the possibility of deterministic modeling. Luo et al. (2006) studied isothermal tracking using a deterministic model. Thus we should keep in mind that the effect of randomness may be neglected in certain cases. Although these properties must be considered when tackling any detailed modeling, our simple model is sufficient to give a unified view of thermotaxis in C. elegans. Our model unifies current knowledge on thermotactic behavior in C. elegans and offers a testbed for more precise and comprehensive understanding of thermotaxis in C. elegans. To refine our model and gain a deeper understanding of thermotaxis, intensive studies on the LTR would be useful. Furthermore our model can be used as a basis to integrate the knowledge of migration behavior and neural activity in the study of thermotaxis.
Acknowledgments This work has been supported by CREST (Core Research and Evolutional Science and Technology) of JST (Japan Science and Technology). We thank I. Mori and A. Kuhara for many fruitful discussions, D. Saitou for his helpful comments and H. Hamada for his warm encouragement.
Appendix. Derivation of transition rates Transition rates are transformed into duration times using the following equations: 1 1 , 2aðxÞ þ cðxÞ 2aðxÞ 1 1 BðxÞ ¼ , 2bðxÞ þ cðxÞ 2bðxÞ 1 CðxÞ ¼ , 2dðxÞ AðxÞ ¼
(5) (6) (7)
where AðxÞ, BðxÞ and CðxÞ are the run duration times in each state. Here we assume that after turning events from state f ðx; tÞ or gðx; tÞ, the worms are redistributed evenly into states f ðx; tÞ and gðx; tÞ and slightly into state sðx; tÞ. The approximations here are based on the fact that cðxÞ is sufficiently smaller than aðxÞ or bðxÞ, because the initiation event of isothermal tracking is estimated to occur in only about 6% of all turning events (Matsuoka et al., 2008). We assume that run duration times are functions of temperature as shown below: AðxÞ ¼ R RT sðT T c ; kr Þ,
(8)
BðxÞ ¼ AðxÞ þ C d sðT T s ; kc Þ þ T d s2 ðT T s ; Th T; kt Þ,
(9)
where sðx; kÞ is a sigmoidal function
sðx; kÞ ¼
1 , 1 þ expðkxÞ
(10)
and s2 ðx; y; kÞ ¼ sðx; kÞsðy; kÞ. The duration time of isothermal tracking CðxÞ is 80 s (Matsuoka et al., 2008; Luo et al., 2006). The parameters used in these equations are decided from experimental data (Luo et al., 2006) as given in Table 1, except for parameter T d, which represents the strength of the thermophilic bias. Full detailed equations are given below: vðxÞ ¼ vo s2 ðT T ct ; T ht T; kÞ,
(11)
aðxÞ ¼ 1=2AðxÞ,
(12)
bðxÞ ¼ 1=2BðxÞ,
(13)
cðxÞ ¼
as2 ðT T ic ; T ih T; ki Þ AðxÞ þ BðxÞ
dðxÞ ¼ b.
,
(14) (15)
Table 1 Summary of constant parameters. R RT Ts Th T ic T ih T ct T ht k kr kt kc ki
a b vo Cd T c is cultivated temperature.
30 10 T c 2:0 T c þ 2:0 T c 2:0 T c þ 2:0 16 29 0.5 1.0 2.0 2.0 5.0 0.06 0.0125 0.015 20
ARTICLE IN PRESS K. Nakazato, A. Mochizuki / Journal of Theoretical Biology 260 (2009) 56–65
References Anderson, J.L., Albergotti, L., Proulx, S., Peden, C., Huey, R.B., Philips, P.C., 2007. Thermal preference of Caenorhabditis elegans: a null model and empirical tests. J. Exp. Biol. 210, 3107–3116. Biron, D., Shibuya, M., Gabel, C., Wasserman, S.M., Clark, D.A., Brown, A., Sengupta, P., Samuel, A.D.T., 2006. A diacylglycerol kinase modulates longterm thermotactic behavioral plasticity in C. elegans. Nat. Neurosci. 9, 1499–1505. Biron, D., Wasserman, S., Thomas, J.H., Samuel, A.D.T., Sengupta, P., 2008. An olfactory neuron responds stochastically to temperature and modulates Caenorhabditis elegans thermotactic behavior. Proc. Natl. Acad. Sci. USA 105, 11002–11007. Brenner, S., 1974. The genetics of Caenorhabditis elegans. Genetics 77, 71–94. Chalfie, M., Sulston, J.E., White, J.G., Southgate, E., Thomson, J.N., Brenner, S., 1985. The neural circuit for touch sensitivity in Caenorhabditis elegans. J. Neurosci. 5, 956–964. Char, M.Y., Komatsu, H., Fukuto, H.S., Dionne, H.M., Hart, A.C., 2004. Feeding status and serotonin rapidly and reversibly modulate a Caenorhabditis elegans chemosensory circuit. Proc. Natl. Acad. Sci. USA 101, 15512–15517. Chung, S.H., Clark, D.A., Gabel, C.V., Mazur, E., Samuel, A.D.T., 2006. The role of the AFD neuron in C. elegans thermotaxis analyzed using femtosecond laser ablation. BMC Neurosci. 7, 30. Clark, D.A., Biron, D., Sengupta, P., Samuel, A.D.T., 2006. The AFD sensory neurons encode multiple functions underlying thermotactic behavior in Caenorhabditis elegans. J. Neurosci. 26, 7444–7451. Clark, D.A., Gabel, C.V., Lee, T.M., Samuel, A.D.T., 2007a. Short-term adaptation and temporal processing in the cryophilic response of Caenorhabditis elegans. J. Neurophysiol. 97, 1903–1910. Clark, D.A., Gavel, C.V., Gabel, H., Samuel, A.D.T., 2007b. Temporal activity patterns in thermosensory neurons of freely moving Caenorhabditis elegans. J. Neurosci. 27, 6083–6090. Dusenbery, D.B., Batt, J., 1980. Thermal limits and chemotaxis in mutants of the nematode Caenorhabditis elegans. J. Comp. Physiol. 137, 353–356. Gabel, C.V., Gabel, H., Pavlichin, D., Kao, A., Clark, D.A., Samuel, A.D.T., 2007. Neural circuits mediate electrosensory behavior in Caenorhabditis elegans. J. Neurosci. 27, 7586–7596. Gray, J.M., Hill, J.J., Bargmann, C.I., 2005. A circuit for navigation in Caenorhabditis elegans. Proc. Natl. Acad. Sci. USA 102, 3184–3191. Hedgecock, E.M., Russell, R.L., 1975. Normal and mutant thermotaxis in the nematode Caenorhabditis elegans. Proc. Natl. Acad. Sci. USA 72, 4061–4065. Ito, H., Inada, H., Mori, I., 2006. Quantitative analysis of thermotaxis in the nematode Caenorhabditis elegans. J. Neurosci. Methods 154, 45–52. Kimura, K.D., Atsushi, M., Matsumoto, K., Mori, I., 2004. The C. elegans thermosensory neuron AFD respond to warming. Curr. Biol. 14, 1291–1295.
65
Kuhara, A., Okumura, M., Kimata, T., Tanizawa, Y., Takano, R., Inada, K.D.K.H., Matsumoto, K., Mori, I., 2008. Temperature sensing by an olfactory neuron in a circuit controlling behavior of C. elegans. Science 320, 803–807. Luo, L., Clark, D.A., Biron, D., Mahadevan, L., Samuel, A.D.T., 2006. Sensorimotor control during isothermal tracking in Caenorhabditis elegans. J. Exp. Biol. 209, 4652–4662. Matsuoka, T., Gomi, S., Shingai, R., 2008. Simulation of C. elegans thermotactic behavior in a linear thermal gradient using a simple phenomenological motility model. J. Theor. Biol. 250, 230–243. Mohri, A., Kodama, E., Kimura, K.D., Koike, M., Mizuno, T., Mori, I., 2005. Genetic control of temperature preference in the nematode Caenorhabditis elegans. Genetics 169, 1437–1450. Mori, I., Ohshima, Y., 1995. Neural regulation of thermotaxis in Caenorhabditis elegans. Nature 376, 344–348. Pierce-Shimomura, J.T., Dores, M., Lockery, S.R., 2005. Analysis of the effects of turning bias on chemotaxis in C. elegans. J. Exp. Biol. 208, 4727–4733. Pierce-Shimomura, J.T., Morse, T.M., Lockery, S.R., 1999. The fundamental role of pirouettes in Caenorhabditis elegans chemotaxis. J. Neurosci. 19, 9557–9569. Ramot, D., Maclnnis, B.L., Goodman, M.B., 2008a. Bidirectional temperature-sensing by a single thermosensory neuron in C. elegans. Nat. Neurosci. 11, 908–915. Ramot, D., Maclnnis, B.L., Lee, H., Goodman, M.B., 2008b. Thermotaxis is a robust mechanism for thermoregulation in Caenorhabditis elegans nematodes. J. Neurosci. 28, 12546–12557. Risken, H., 1989. The Fokker–Planck Equation: Methods of Solution and Applications. In: Springer Series in Synergetics, vol. 18. Springer, Berlin. Ryu, W.S., Samuel, A.D.T., 2002. Thermotaxis in Caenorhabditis elegans analyzed by measuring responses of defined thermal stimuli. J. Neurosci. 22, 5727–5733. Schnitzer, M.J., 1992. Theory of continuum random walks and application to chemotaxis. Phys. Rev. E 48, 2553–2568. Sulston, J.E., Horvitz, H.R., 1977. Post-embryonic cell lineages of the nematode Caenorhabditis elegans. Dev. Biol. 56, 110–156. The C. elegans Sequencing Consortium, 1998. Genome sequence of the nematode C. elegans. Science 282, 2012–2018. Ward, A., Liu, J., Feng, Z., Zu, X.Z.S., 2008. Light-sensitive neurons and channels mediate phototaxis in C. elegans. Nat. Neurosci. 11, 916–922. White, J.G., Southgate, E., Thomson, J.N., Brenner, S., 1986. The structure of the nervous system of the nematode Caenorhabditis elegans. Phil. Trans. R. Soc. London B 314, 1–340. Wiktorsson, M., Ryden, T., Nilsson, E., Bengtsson, G., 2004. Modeling the movement of a soil insect. J. Theor. Biol. 231, 497–513. Yamada, Y., Ohshima, Y., 2003. Distribution and movement of Caenorhabditis elegans on a thermal gradient. J. Exp. Biol. 206, 2581–2593. Zhao, B., Khare, P., Feldman, L., Dent, J.A., 2003. Reversal frequency in Caenorhabditis elegans represents an integrated response to the state of the animal and its environment. J. Neurosci. 23, 5319–5328.
Contents lists available at ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Steepness of thermal gradient is essential to obtain a unified view of thermotaxis in C. elegans Kenichi Nakazato a,, Atsushi Mochizuki a,b,c a
Theoretical Biology Laboratory, RIKEN Advanced Science Institute, Wako 351-0198, Japan Devision of Theoretical Biology, National Institute for Basic Biology, Okazaki 444-8787, Japan c PRESTO, JST, Kawaguchi 332-0012, Japan b
a r t i c l e in f o
a b s t r a c t
Article history: Received 24 March 2009 Received in revised form 28 May 2009 Accepted 28 May 2009 Available online 6 June 2009
One of the adaptive behaviors of animals in their environment is thermotaxis, by which they migrate toward a preferred temperature. This sensorimotor integration is accomplished by choosing one of two behaviors depending on the surrounding temperature, namely thermophilic or cryophilic movement. Caenorhabditis elegans exhibits thermotaxis and its migration behavior has been analyzed experimentally at both the population and individual levels. However, some experimental data are inconsistent especially for thermophilic movement, which is expected to be observed in lower than favorable temperatures. There are no experimental analyzes that find thermophilic tendencies in the individual behavior of worms, despite multiple reports supporting thermophilic movement of the population. Although theoretical methods have been used to study thermotaxis of C. elegans, no mathematical model provides a consistent explanation for this discrepancy. Here we develop a simple biased random walk model, which describes population behavior, but which is based on the results of individual assays. Our model can integrate all previous experiments without any contradiction. We regenerate all the population patterns reported in past studies and give a consistent explanation for the conflicting results. Our results suggest that thermophilic movement is observed, even in individual movements, when the thermal gradient is sufficiently slight. On the contrary, thermophilic movement disappears when the thermal gradient is too steep. The thermal gradient is thus essential for a comprehensive understanding of the experimental studies of thermotaxis in C. elegans. Our model provides insight into an integrative understanding of the neural activity and thermotactic behavior in C. elegans. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Sensorimotor behavior Thermotaxis C. elegans Biased random walk Mathematical modeling
1. Introduction Nematode Caenorhabditis elegans is one of the most wellknown experimental animals in neurobiology, developmental biology, and various other fields. Worms have long since been chosen as convenient experimental animals for the integrated comprehension of genetics, neurobiology, and behavior (Brenner, 1974). The hermaphrodite worm has only 959 cells, of which 302 are neurons. Most importantly, the network structure of these 302 neurons has been fully elucidated (White et al., 1986). In addition, the complete cell lineage (Sulston and Horvitz, 1977) and whole genomic sequence (The C. elegans Sequencing Consortium, 1998) are known. Taking advantage of this knowledge, various sensorimotor behaviors in C. elegans have been investigated intensively (Chalfie et al., 1985; Pierce-Shimomura et al., 1999; Gabel et al., 2007; Ward et al., 2008).
Corresponding author.
E-mail addresses: [email protected] (K. Nakazato), [email protected] (A. Mochizuki). 0022-5193/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2009.05.027
We consider the behavior in a thermal environment known as thermotaxis (Hedgecock and Russell, 1975; Mori and Ohshima, 1995), focusing particularly on the migration behavior in thermotaxis and the mathematical modeling thereof. Worms on a thermal gradient wander around searching for a favorable temperature, and this shows that they have the ability to memorize their environmental temperature and to use this information in their migration (Kimura et al., 2004; Chung et al., 2006; Clark et al., 2006, 2007b; Biron et al., 2006). In addition to searching for the cultivated temperature, worms, once they arrive there, crawl along the isothermal curve (Mori and Ohshima, 1995; Luo et al., 2006). The strategy of searching for the cultivated temperature is considered to be quite simple: worms have a tendency to go down the thermal gradient when situated at a higher temperature than the cultivated temperature and a tendency to go up when at a lower temperature (Hedgecock and Russell, 1975; Mori and Ohshima, 1995; Mohri et al., 2005). However, this thermophilic tendency is now a topic of debate, because the tendency has not been confirmed by several assays (Ryu and Samuel, 2002; Clark et al., 2007a; Yamada and Ohshima, 2003; Ramot et al., 2008b).
ARTICLE IN PRESS K. Nakazato, A. Mochizuki / Journal of Theoretical Biology 260 (2009) 56–65
The navigation behavior of C. elegans has been clarified by a model of a biased random walk (Gray et al., 2005; PierceShimomura et al., 1999; Wiktorsson et al., 2004; Ramot et al., 2008b). The migration movement of C. elegans consists of repeated straight forward runs and turning behavior (Matsuoka et al., 2008; Gray et al., 2005; Pierce-Shimomura et al., 1999; Ramot et al., 2008b). Both types of movement, i.e., straight and turning, may be accompanied by a bias in some direction against the thermal gradient, causing a mean displacement in migration and consequently, thermotactic behavior. Two types of bias are shown in Fig. 1. Run duration bias results in a mean translational displacement. In actual, run of the nematodes is not perfect straight. It may give some angular displacement. However, the effect seems to be small and we ignore it following the previous mathematical studies using random walk model. Reorientation bias causes mean angular displacement and indirectly, translational displacement as well. One of the methods for studying migration behavior is individual movement assays. The above two types of bias in thermotactic behavior have been examined experimentally using individual assays (Ryu and Samuel, 2002; Clark et al., 2007a). Experimental data for run duration times indicate that there is a bias towards cryophilic tendencies in temperatures higher than the cultivated temperature. On the other hand, the thermophilic drive is still under discussion even by the individual movement assays. There are published tracks of worms that appeared to perform thermophilic movement (Hedgecock and Russell, 1975; Mori and Ohshima, 1995). However, some groups have found no evidence of thermophilic bias based on statistical analysis of tracked worms (Ryu and Samuel, 2002; Clark et al., 2007a). Nor are there reports on reorientation bias affecting thermotactic behavior. Population distribution assays have also been used to study thermotactic behavior, however the results of these studies are seemingly inconsistent with one another. In an analysis at the population level, changes in the mean and deviation of a distribution may be used as statistics characterizing thermotactic behavior instead of the distribution itself. Some reports on the population distribution and its statistics state that both cryophilic and thermophilic movements were observed (Hedgecock and Russell, 1975; Ito et al., 2006; Kuhara et al., 2008), thus conflicting with the results of individual movement assays. Thermophilic movements may be hardly observed at least for several minutes in the beginning of assay (Ito et al., 2006; Ramot et al., 2008b). On the other hand, another report asserts the possibility that the thermophilic-like tendency of the population behavior at lower temperatures might be caused by the decrease in motility, in other
Fig. 1. Two types of bias. The first is the bias in run duration time or distance in each direction against thermal gradient. (A) shows that worms crawling up the thermal gradient tend to maintain straight movement longer than worms crawling down the gradient as an example of run duration bias. The other bias is that in reorientation angle. (B) shows an example of reorientation bias: worms prefer the thermophilic direction when changing direction.
57
words a ‘‘cold trap’’ (Yamada and Ohshima, 2003). Their idea is that the motility of worm decreases drastically under a critical temperature then the worms would not move into colder region from moderate temperature. There are two possible approaches for understanding these seemingly contradictory experimental results at different observation levels. One possibility is that a thermophilic-like property is generated in the behavior of the population by some mechanism, such as asymmetric motility between higher and lower temperatures (Yamada and Ohshima, 2003; Dusenbery and Batt, 1980; Anderson et al., 2007), despite there being no thermophilic property in individual behavior. The other possibility is that a thermophilic property at the individual level does exist, but there has not been positive statistical data in the previous experimental methods. In this paper, we aim to give a consistent explanation for the conflicting results by using a theoretical method. We construct a simple biased random walk model of the population behavior, that reflects the results of individual movement assays. Using this model we succeed in giving a consistent explanation for past experiments. At the same time, our results assert the importance of the steepness of the thermal gradient that may change the migration behavior drastically in experiments on thermotaxis. In the next section, we summarize diagrammatically the thermotactic behavior and active ranges of thermoreceptors. We then introduce our biased random walk model, using a variation of the telegrapher’s equation. Numerical results are given in next section. In this section, we explain that past experiments can be regenerated just by taking account of the effect of thermophilic drive in our model. Furthermore, we show that some of the unique results by Yamada–Ohshima can be explained as a consequence of the limited range of thermophilic drive and the ‘‘cold trap’’. In the last section, we summarize our results and discuss the thermophilic drive particularly with respect to those points which have not been considered to be important. Our results together with the experimental results of Paola et al. (in preparation) suggest that thermophilic drives ought to be observed in individual assays as well with different gradient conditions.
2. Model 2.1. Thermotaxis as a biased random walk The duration of movement in a straight line and the reorientation angle of C. elegans are thought to be stochastic in nature (Gray et al., 2005; Pierce-Shimomura et al., 1999, 2005; Ramot et al., 2008b). Individual worms are assumed to run at a constant speed in their straight forward movements around the cultivated temperature (Ryu and Samuel, 2002). Though actual run of the nematodes is not perfectly straight, we assumed it in our model. In this case, the run duration time directly reflects the translational displacement. Reorientation behavior is classified as either small turns or large turns called omega turns or pirouettes (Pierce-Shimomura et al., 1999; Gray et al., 2005). The biased random walk is a mathematical model suited to describing and analyzing directional movement with stochasticity. It has many different mathematical forms corresponding to the different levels of phenomena, such as the Langevin equation or Fokker–Planck equation (Risken, 1989; Schnitzer, 1992), that make it possible to consider the relation between individual and population behaviors of thermotaxis in C. elegans. To formalize the biased random walk model of thermotaxis, we require the probability distribution of the run duration time and the reorientation angle as a function of direction and temperature. Some experimental reports based on the notion of a biased
ARTICLE IN PRESS 58
K. Nakazato, A. Mochizuki / Journal of Theoretical Biology 260 (2009) 56–65
random walk have elucidated features of the distribution functions especially on the cryophilic drive (Ryu and Samuel, 2002; Clark et al., 2007a). According to these reports, worms placed at a higher temperature tend to continue moving down the gradient towards the cultivated temperature. In contrast, no reorientation bias in any direction against the thermal gradient has been observed to date. Regarding thermophilic drive, there are no results from individual movement assays that support the individual thermophilic tendency. Besides the description based on individual behavior, a population based approach is also useful. Modeling a biased random walk in terms of population behavior requires information of the time evolution of the distribution function of the worms’ positions. Results from distribution assays show that a population initially distributed at either lower or higher temperatures shifts towards the cultivated temperature. Although it has been shown that a deterministic mathematical model can be built as for isothermal tracking (Luo et al., 2006), we use a biased random walk without major alteration to the mathematical framework.
2.2. Diagram of thermal behavior Information of environmental temperature is assumed to have been obtained through thermosensory neurons AFD and AWC, and utilized for thermotactic behavior in C. elegans (Kimura et al., 2004; Kuhara et al., 2008; Clark et al., 2006, 2007b; Biron et al., 2006, 2008; Ramot et al., 2008a). AFD and AWC neurons show thermal responses at temperatures down to at most T c 3 in analysis of cellular responses using Ca2þ imaging (Kimura et al., 2004; Clark et al., 2006; Kuhara et al., 2008; Biron et al., 2008). Electrophysiological measurements show the possibility to show thermal responses at temperatures lower than T c 3 after longtime observation by the effect of thermal adaptation (Ramot et al., 2008a). In this study we assume that the active range would not expand in the time scale of our interest ð1 hÞ. In a cold region, where there is no thermal response from AFD nor AWC, run duration and reorientation angle would be homogenous to any direction and thermal gradient, while the active range might differ from behavioral output. In fact no observation has been reported that worms display thermotactic behavior in regions much colder than the cultivated temperature. The diagram of thermotaxis in Fig. 2 is a summary of the known facts. Combining past experiments and considering possible behavior, we divide the environmental space into three sections according to temperature: high, low, and cold temperature regions (HTR, LTR, and CTR) (Fig. 2). The high temperature region is the area in which the temperature is
higher than the cultivated one and in which it is possible to observe a cryophilic drive with run duration bias. In the low temperature region we assume that the worms tend to move in the thermophilic direction in some unknown way. The cold region is the area where thermoreceptors show no response to thermal stimuli and thus we would observe athermotactic behavior. 2.3. Mathematical formulation We use a variation of the telegrapher’s equation to describe the distribution behavior of our biased random walk model (Schnitzer, 1992). We consider a one dimensional space with linear thermal gradient, where position x corresponds to the temperature. The movement of worms is limited to the direction along the thermal gradient. Isothermal tracking is represented as a static state because worms in isothermal tracking do not move in the direction of the thermal gradient. We can now express the time evolution of the distribution as follows: @F @ðvFÞ ¼ þ TF, @t @x T F ¼ ðf ; g; sÞ , 1 0 v 0 0 C B C v¼B @ 0 v 0 A, 0 0 0 0 ða þ cÞ b B a ðb þ cÞ T¼B @ c c
(1) (2) (3) d
1
C d C A, 2d
(4)
where f ðx; tÞ is the distribution of worms crawling up the thermal gradient, gðx; tÞ is the distribution of worms moving down the gradient, and sðx; tÞ is the distribution of worms tracking isothermal curves. The motility vðxÞ of worms depends only on the thermal environment; in other words, motility is a function of position x. T is a matrix consisting of the state transition rates between the three states, f ðx; tÞ; gðx; tÞ, and sðx; tÞ. These transition rates, aðxÞ, bðxÞ, cðxÞ and dðxÞ, are decided from the experimental data on the duration of each state and the destination after the transition events. We assume here that worms at state f ðx; tÞ or gðx; tÞ are rocked into an isothermal tracking state sðx; tÞ at the same rate cðxÞ. After isothermal tracking halts, worms transit to states f ðx; tÞ and gðx; tÞ at the same rate dðxÞ. A thermophilic or cryophilic tendency in this model is represented as the difference of transition rates, bðxÞ aðxÞ or aðxÞ bðxÞ, respectively. Although we must distinguish between run duration bias and reorientation bias to analyze the biased random walk in a real situation, this distinction makes no sense in our simplified model. In our model, run duration bias and reorientation bias are simplified merely as the difference between the two probabilities to reverse the direction, which represents an individual migration tendency. This simplification is appropriate to prevent superfluous hypotheses on the detailed mechanism of a thermophilic drive. Thus, we need only hypothesize the migration tendency at each temperature. Despite this simplification, our model is sufficient to illustrate the distribution behavior of worms on the thermal gradient.
3. Numerical results
Fig. 2. Diagrammatic view of thermotaxis in C. elegans, where the horizontal axis reflects temperature. The environmental space of the worms is divided into three sections along the temperature axis according to thermotactic behavior and active pattern of the thermoreceptors. We consider the effect of isothermal tracking in the parameter settings around the cultivated temperature T c 2 (Fig. 3).
In this section, we explain the past experimental results from the point of view of a biased random walk, using numerical results from our model. Giving numerical results comparable to each experiment, we illustrate the effect of thermophilic drive in each case.
ARTICLE IN PRESS K. Nakazato, A. Mochizuki / Journal of Theoretical Biology 260 (2009) 56–65
3.1. Parameter settings To obtain numerical results comparable to past experiments, we need to decide parameter settings based on the experimental conditions. Parameters to be decided are the velocity matrix v and transition matrix T. Although worms are assumed to move at constant speed at the cultivated temperature, a decrease in motility at abnormal temperatures has been reported (Dusenbery and Batt, 1980; Anderson et al., 2007). We can decide the parameter values of v at each temperature by calculating the average velocity of worms in each state. Parameter v shown in Fig. 3 is calculated from experimental data (Ryu and Samuel, 2002; Luo et al., 2006; Clark et al., 2007a; Dusenbery and Batt, 1980; Anderson et al., 2007). We have to estimate transition matrix T as a function of temperature from the duration in each state, which is obtained from experimental data. We refer to the experimental data in Ryu and Samuel (2002), Luo et al. (2006) and Clark et al. (2007a) and apply the estimation in Matsuoka et al. (2008) in principle. For the low temperature region, we added a hypothetical difference between aðxÞ and bðxÞ to realize a thermophilic bias. Our hypothetical settings for the thermophilic drive are reflected in the modification around the LTR: the parameter bðxÞ is a little higher as shown in Fig. 3. We use the parameter sets, including this hypothetical thermophilic bias ðT d ¼ 5; 10Þ, to obtain numerical results. Furthermore, we use the parameter sets without this hypothesis ðT d ¼ 0Þ to show the effect of the thermophilic drive. Transition rates cðxÞ and dðxÞ are decided using run duration time of isothermal tracking. Full detail of transformation equations and their derivation are shown in Appendix.
3.2. Hedgecock and Russell The first report on thermotaxis in C. elegans is the paper by Hedgecock and Russell (1975) (HR). They reported the results of distribution assays for different temperatures around 12:9, 17:6, 22:8 and 27:1 C with a constant thermal gradient, 0:5 C=cm, using worms cultivated at 20:0 C. A cryophilic tendency was observed at 22:8 and 27:1 C while a thermophilic tendency was observed at 17:6 C. However, neither a thermophilic nor cryophilic tendency was observed at 12:9 C. In Fig. 4, we present numerical results under conditions that simulate the experiments by HR. Figures in the left column, Figs. 4a, d, g and j show results using the parameter sets without
0.03 0.02 0.01 0 5
10
15
20
25
30
35
temperature a(T) v(T)
b(T ) [ T d =10] b(T ) [ T d = 5] b(T ) [ T d = 0]
Cultivated temperature: Tc = 20 Fig. 3. Parameters as functions of temperature. These are calculated using Eqs. (11)–(15) from experimental data of run duration time. T d is a parameter that adjusts the strength of the thermophilic bias.
59
thermophilic bias. No thermophilic tendency in the distribution evolution is observed. Figures in the center column, Figs. 4b, e, h and k, and in the right column, Figs. 4c, f, i and l, show results including thermophilic drive. We can easily recognize thermophilic movement in Figs. 4e and f. The other figures in these columns show no thermophilic tendency because of the range of space in the simulation. The spatial ranges, corresponding to experiments by HR around temperatures 12:9, 22:8 and 27:1 C, do not include the LTR, where thermophilic bias is active. In the cold region (CTR in Fig. 2), the peaks of the distributions shift only slightly to a colder region as shown in Figs. 4a–c. In other words, these figures show athermotactic behavior which is reported by HR. Thus our model can regenerate the results of HR only when using parameter sets with thermophilic drive. 3.3. Kuhara et al. Other distribution assays that support thermophilic drive have been reported by Kuhara et al. (2008) (KO) and Ito et al. (2006) (IM). Both of these studies tested the distribution behavior around the cultivated temperature, however neither tested the region far from the cultivated temperature, where thermoreceptors are inactive or motility is lost. They reported similar results to HR; that is, both thermophilic movement in the HTR and cryophilic movement in the LTR are observed. The numerical results are shown in Fig. 5. We computed these results using a thermal gradient and boundary condition similar to experiments in KO. The figures in the center and right columns, using parameter sets with thermophilic bias, show that experimental results by KO and IM can be regenerated by our model, as we can easily recognize that both peaks (green lines) and averaged positions (red lines) move toward higher temperature than original positions (20) in Figs. 5h and i. The figures in the left column show results using parameter sets without thermophilic bias. However as Fig. 5g shows, a thermophilic shift of peak position may be observed despite the absence of thermophilic bias, a phenomenon that is discussed later. This is consistent with the results by Matsuoka et al. (2008). 3.4. Yamada and Ohshima Some unique results have been reported by Yamada and Ohshima (2003) (YO). They tested distribution assays with a very steep thermal gradient 1:5 C=cm and reported that no convergence of individual worms towards the cultivated temperature was seen. Instead, they observed a widely scattered distribution centered around the LTR and an accumulation of worms with lost motility in the CTR. They concluded that worms have a cryophilic tendency in the HTR but no thermophilic tendency in either the LTR or CTR. In addition, they suggested that the thermophilic tendency observed in other experiments might stem from the ‘‘cold trap’’ in the CTR. We thus evaluate the idea of the cold trap and illustrate the difference between thermophilic bias and a cold trap. To evaluate the effect of the cold trap, we present numerical results in Fig. 6. We compare these results with the normal parameter settings given in Fig. 3 using constant motility, and not a function of temperature. The steepness of the thermal gradient and the boundary conditions are similar to those of YO for easy comparison between our numerical results and the experimental results in Yamada and Ohshima (2003). Figures in the top row show the results with normal parameter sets, in which worms lose their motility in the CTR (Fig. 3). Figs. 6a–c, show that bimodal distributions emerge as time advances, and thus
ARTICLE IN PRESS 60
K. Nakazato, A. Mochizuki / Journal of Theoretical Biology 260 (2009) 56–65
Td = 5
Td = 0
0
60
30 60
14
12
10
0
time
30
time
time
~12.9°C
0
~17.6°C
temperature
30 60
20
18
16
0
time
time
time
30
20
22
0
30
time
time
time
30
60
24
20
temperature
22
60
24
20
26
28
30
24
0
time
time
time
30
22
temperature
0
60
30
temperature
0
20
18
16
temperature
0
60
60
20
18
16
30
temperature
0
14
12
10
0
temperature
~22.8°C
60
temperature
0
60
30
14
12
10
temperature
~27.1°C
Td = 10
30 60
temperature
26
28
30
30 60
temperature
26
28
30
temperature
Thermal gradient: 0.5°C/cm Cultivated temperature: Tc = 20°C Fig. 4. Thermophilic drive is necessary to regenerate the experimental results by Hedgecock and Russell (1975). Results with the condition of the CTR show that worms behave athermotactically regardless of thermophilic drive (a–c). Red, green, gray and purple dashed lines show the mean of population, the position of peaks, initial position and cultivated temperature, respectively. Thermophilic tendency is observed only when the temperature is in the range of the LTR and thermophilic bias is included in our simulation. In our simulations, we use a reflecting boundary at the right and left edges.
thermophilic shifts are not clear, even in the results with thermophilic bias (Figs. 6b and c). These figures suggest that when the thermal gradient is too steep, worms do not converge at the cultivated temperature even under a thermophilic drive. The figures in the bottom row, Figs. 6d–f, illustrate the results with constant motility. These figures show single modal distributions with fat tails in the lower temperature region as contrasted with the figures in the top row. The difference between the figures in the top and bottom rows shows that the cold trap creates bimodal distributions with clear limits in the CTR. On the other hand,
widely scattered distributions are observed both in the top and bottom regardless of thermophilic drive. The significant feature of the experimental conditions of YO is the steepness of the thermal gradient, which determines the spatial width of the LTR. Under the conditions in YO, the width of the LTR become so narrow that worms do not necessarily stay in the LTR or HTR, nor converge at the cultivated temperature, as shown in Fig. 6. Thus we can conclude that the cause of scattered distributions in YO is not the lack of thermophilic bias, but the steepness of the thermal gradient used in their experiments.
ARTICLE IN PRESS K. Nakazato, A. Mochizuki / Journal of Theoretical Biology 260 (2009) 56–65
Td = 0
Td = 5
0
18
20
22
24
e tim
e tim
16
30 60 16
temperature
20
22
60
24
16
18
20
22
24
60 16
temperature
18
20
22
24
60 16
temperature
0
30
tim
18
20
22
24
22
24
30
e
16
20
0
e
tim
e
tim
60
18
temperature
0
30
24
e
16
22
30
tim
e
60
20
0
30
tim
30
18
temperature
0
e
tim
Tc = 20°C
18
30
temperature
0
Tc = 23°C
Td = 10
0
30
e tim
Tc = 17°C
0
60
61
60 16
temperature
18
20
22
24
60 16
temperature
18
20
22
24
temperature
Thermal gradient: 0.44°C/cm Temperature at initial position: 20°C Fig. 5. Thermophilic tendency in the distribution behavior can be seen only if we adopt the parameter sets with thermophilic bias. Sufficiently strong thermophilic bias is needed to observe thermophilic tendency in the distribution behavior. Red, green, gray and purple dashed lines show the mean of population, the position of peaks, initial position and cultivated temperature, respectively.
Furthermore, our results suggest that the experiments of YO are not appropriate to confirm a thermophilic bias in the LTR, as the thermophilic bias would not make a significant difference in the distribution behaviors under steep thermal gradients. There is another experimental report with mathematical analysis using distribution assay with steep thermal gradient 1:15 C=cm (Anderson et al., 2007). The paper showed the result that thermophilic tendency is hardly seen, and we can understand it by our model similarly as the case of Yamada and Ohshima (2003). The difference between the results in Anderson et al. (2007) and Yamada and Ohshima (2003) is that peak of distribution in cold region is not observed in Anderson et al. (2007). This difference may stem from the food condition: worms are on food in Anderson et al. (2007), unlike most thermal preference assays. If we consider that turning behavior of worms is more frequent on food than off food (Gray et al., 2005), then
worms would diffuse slowly and consequently accumulation in cold region would be lost as seen in Anderson et al. (2007).
3.5. Paola et al. (in preparation) As noted in a previous Section 3.4, steepness of thermal gradient is one of the key factors in the observation of thermal tendency in thermotactic behavior. Fig. 7 shows numerical results that illustrate the effect of steepness of the thermal gradient. We have confirmed that our results here basically agree with the experimental results to be published by Paola et al. (in preparation). Thermophilic tendency is clearly observed in Figs. 7b, c, and f, where the thermal gradient is gradual and thermophilic bias is strong. Both peaks and averaged positions move toward higher
ARTICLE IN PRESS 62
K. Nakazato, A. Mochizuki / Journal of Theoretical Biology 260 (2009) 56–65
Td = 0
Td = 5
0
60
8
12
16
20
24
28
0
30
32
time
30
time
time
v = v(T)
0
60
8
12
temperature
16
20
24
28
32
12
16
20
24
8
12
28
32
16
20
24
28
32
28
32
0
30
time
time
time
30
8
60
temperature
0
60
30
temperature
0
v = const.
Td = 10
60
8
12
temperature
16
20
24
28
30
32
temperature
60
8
12
16
20
24
temperature
Thermal gradient: 1.5°C/cm Cultivated temperature: Tc = 20°C Temperature at initial position: 20°C Fig. 6. Convergent distribution around the cultivated temperature is barely confirmed with the steep thermal gradient used in YO regardless of thermophilic bias. Red, green, gray and purple dashed lines show the mean of population, the position of peaks, initial position and cultivated temperature, respectively. Figures in the top row reflect the results with parameter sets shown in Fig. 3(a)–(c). Double peaked distributions, with one peak in the CTR and the other around the cultivated temperature, appear gradually. Figures in the bottom row reflect the results with constant motility (d–f). All distributions in these figures spread widely to colder regions, but not to warmer regions.
temperature than original position (18). On the contrary, thermophilic tendency is barely observed in Figs. 7d, g, and h, where the thermal gradient is steep and thermophilic bias is weak. These results show that a thermophilic tendency in the distribution behavior requires not only a thermophilic bias but also a slight thermal gradient. The steepness of the thermal gradient determines the spatial range of the LTR, which changes the thermal tendency in the distribution behavior of worms. Thermophilic tendency becomes weaker and is ultimately lost, when we steepen the thermal gradient, even with a large thermophilic bias.
4. Summary and discussion Worms are considered to have the ability to migrate towards the cultivated temperature, a phenomenon known as thermotaxis in C. elegans. However, several experimental results, that do not agree with thermophilic drive, have been reported (Ryu and Samuel, 2002; Clark et al., 2007a; Yamada and Ohshima, 2003; Anderson et al., 2007). Mathematical modeling has also been used to study thermotaxis, however none of the models has succeeded in giving a comprehensive explanation for past experimental results. In this paper, we give a consistent explanation for the past experiments using a biased random walk model of thermotaxis in
C. elegans. Our results show that a thermophilic drive is necessary to explain the past experiments without discrepancies. In addition we suggest that the steepness of the thermal gradient or spatial width of the LTR must be taken into account when studying thermotactic behavior. Migration behavior is considered to be accomplished by a balance of thermophilic and cryophilic drive, which is controlled by several neurons (Mori and Ohshima, 1995). Thermoreceptors AFD and AWC are recognized as the gate of thermal information in worms and show different responses to thermal stimuli from environmental temperature (Mori and Ohshima, 1995; Kimura et al., 2004; Kuhara et al., 2008; Clark et al., 2006, 2007b; Biron et al., 2006, 2008; Ramot et al., 2008a). In the cold temperature region the thermoreceptors show no response to thermal stimuli. This inactivity of the thermoreceptors in cold temperatures may cause the athermotactic behavior in the CTR reported in Hedgecock and Russell (1975). In addition to neural activities, motility of worms may affect thermotactic behavior. In fact the cold trap (Yamada and Ohshima, 2003) in the CTR is caused by the loss of motility in cold environments (Dusenbery and Batt, 1980; Anderson et al., 2007). Considering these facts we conclude that thermophilic drive is active only in the LTR and not in the CTR. In Fig. 5g, we observed a shift in the peak of the distribution towards the thermophilic direction even when using a model without thermophilic bias. This result may be considered as
ARTICLE IN PRESS K. Nakazato, A. Mochizuki / Journal of Theoretical Biology 260 (2009) 56–65
Td = 5
Td = 0
20
18
16
0 30
22
time
30
14
60 14
12
21
18
15
14
60 12
21
18
15
24
16
20
24
28
21
18
15
24
temperature
0 30
time
time
time
12
60 12
0
8
30
temperature
30
22
temperature
30
24
0
20
18
16
0
temperature
60
60
time
30
time
time
0.8 °C/cm
22
0
60
30
temperature
0
1.2 °C/cm
20
18
16
temperature
Thermal gradient
Td = 10
0
time
time
0.4 °C/cm
0
60
63
60 8
temperature
12
16
20
24
30
28
temperature
60 8
12
16
20
24
28
temperature
Cultivated temperature: Tc = 20 °C Temperature at initial position: 18 °C Fig. 7. The steepness of thermal gradient affects thermophilic tendency in population distributions. Red, green, gray and purple dashed lines show the mean of population, the position of peaks, initial position and cultivated temperature, respectively. Thermophilic tendency is not clear in the figures in the center (d–f) and the bottom (g–i) rows. The mean position of each distribution, represented by red lines, shows that the tendency of the mean position and that of peaks do not always coincide (a, d, e, and the figures in the bottom row). In cases with very steep gradients, the distributions are characterized by the bimodality thereof.
showing the possibility that the thermophilic movement of the population may be realized even without thermophilic bias of the individuals. However, numerical simulations under the same or similar conditions can realize only a shift in the peak and not a shift in the averaged position of the distribution. In actual experiments conducted by KO and IM, shifts in both peak and mean are observed. We thus conclude that a peak-shift only does not reflect a real thermophilic drive. In general, movement of the mean and that of the peak may be inconsistent. If the distribution of worms covers a wide range of space, movement of the mean may be calculated from the behavior of worms in all three regions HTR, LTR, and CTR. On the other hand, movement of the peaks of the distribution may be determined only from local behavior in either the HTR, LTR, or CTR. Therefore, both the mean and peaks of the distribution
should be measured to analyze thermotactic behavior using the experimental method. Negative results for thermophilic bias are reported in Yamada and Ohshima (2003). The authors suggest that thermotaxis cannot be explained by the balance between thermophilic and cryophilic drive, but by a combination of cryophilic drive and the cold trap. Our results however, suggest that the steep thermal gradient used by Yamada and Ohshima (2003) is not an appropriate condition to deny the thermophilic drive. As a thermophilic bias would act on worms in the LTR only, the thermal gradient should be sufficiently slight for the thermophilic tendency to be realized in an LTR of adequate width. Worms on a steep thermal gradient do not always remain around the cultivated temperature, but occasionally move towards the CTR, where worms are no longer thermotactic and their motility is lost. The frequent excursions of worms result in
ARTICLE IN PRESS 64
K. Nakazato, A. Mochizuki / Journal of Theoretical Biology 260 (2009) 56–65
athermotactic behavior in the CTR when the thermal gradient is extremely steep. In other words, the balance of thermophilic and cryophilic drive alone is not enough to keep the worms around the cultivated temperature under the condition of Yamada and Ohshima (2003). Thermophilic drive is not always observable; it is observable only when the thermal gradient is sufficiently slight. Our results may give theoretical and comprehensive explanation for the report by Ramot et al. (2008b), though direct comparison may not be appropriate because their experimental time (10 min) is short. They tested some different thermal gradients, 0.3, 0.5, 1.0 and 1:5 C=cm, and show that thermophilic tendency can be observed only one case, 0:3 C=cm, in narrow area around the cultivated temperature. Our theoretical results suggest that the narrowness would stem from the spatial width of LTR, in other words active range of thermosensory neurons AFD and AWC. Some papers on individual assays report negative results for thermophilic drive (Ryu and Samuel, 2002; Clark et al., 2007a). The results for these individual assays give experimental data mainly on cryophilic behavior in the HTR and athermotactic behavior in the CTR. There is however no experimental data for the LTR, where we hypothesize thermophilic bias. Our parameter sets for the HTR and CTR are decided from results by Ryu and Samuel (2002) and do not conflict with their results. However, our model describes distribution behavior and not individual behavior, thus direct comparison is not applicable. We have neglected some aspects of thermotactic behavior for simplicity of our model. Sensorimotor responses of worms are assumed to be variable depending on time (Gray et al., 2005). In fact, turn frequency and navigation behavior depend on metabolic conditions (Char et al., 2004). As a result, behavior may change with time under certain experimental conditions. Actually, some population distribution assays suggest that thermophilic movement is slower process than cryophilic movement (Ito et al., 2006; Ramot et al., 2008b). We may consider at least two reasons for the duality of time scale in thermotactic behavior. First, thermophilic bias may be too weak to be observed in a few minutes in the beginning of assays. This is the case of our model and results show that thermophilic tendency may be too weak to be observed in population behavior at the time of 10 (Fig. 5). Second, worms may not choose thermophilic movement for several minutes after transfer to the assay plate for any reason: e.g. Zhao et al. (2003) discusses suppression of reversal behavior for initial 90 s. In this case, our model may need to adopt temporal change of parameters for the detailed modeling. Another point concerns the possibility of deterministic modeling. Luo et al. (2006) studied isothermal tracking using a deterministic model. Thus we should keep in mind that the effect of randomness may be neglected in certain cases. Although these properties must be considered when tackling any detailed modeling, our simple model is sufficient to give a unified view of thermotaxis in C. elegans. Our model unifies current knowledge on thermotactic behavior in C. elegans and offers a testbed for more precise and comprehensive understanding of thermotaxis in C. elegans. To refine our model and gain a deeper understanding of thermotaxis, intensive studies on the LTR would be useful. Furthermore our model can be used as a basis to integrate the knowledge of migration behavior and neural activity in the study of thermotaxis.
Acknowledgments This work has been supported by CREST (Core Research and Evolutional Science and Technology) of JST (Japan Science and Technology). We thank I. Mori and A. Kuhara for many fruitful discussions, D. Saitou for his helpful comments and H. Hamada for his warm encouragement.
Appendix. Derivation of transition rates Transition rates are transformed into duration times using the following equations: 1 1 , 2aðxÞ þ cðxÞ 2aðxÞ 1 1 BðxÞ ¼ , 2bðxÞ þ cðxÞ 2bðxÞ 1 CðxÞ ¼ , 2dðxÞ AðxÞ ¼
(5) (6) (7)
where AðxÞ, BðxÞ and CðxÞ are the run duration times in each state. Here we assume that after turning events from state f ðx; tÞ or gðx; tÞ, the worms are redistributed evenly into states f ðx; tÞ and gðx; tÞ and slightly into state sðx; tÞ. The approximations here are based on the fact that cðxÞ is sufficiently smaller than aðxÞ or bðxÞ, because the initiation event of isothermal tracking is estimated to occur in only about 6% of all turning events (Matsuoka et al., 2008). We assume that run duration times are functions of temperature as shown below: AðxÞ ¼ R RT sðT T c ; kr Þ,
(8)
BðxÞ ¼ AðxÞ þ C d sðT T s ; kc Þ þ T d s2 ðT T s ; Th T; kt Þ,
(9)
where sðx; kÞ is a sigmoidal function
sðx; kÞ ¼
1 , 1 þ expðkxÞ
(10)
and s2 ðx; y; kÞ ¼ sðx; kÞsðy; kÞ. The duration time of isothermal tracking CðxÞ is 80 s (Matsuoka et al., 2008; Luo et al., 2006). The parameters used in these equations are decided from experimental data (Luo et al., 2006) as given in Table 1, except for parameter T d, which represents the strength of the thermophilic bias. Full detailed equations are given below: vðxÞ ¼ vo s2 ðT T ct ; T ht T; kÞ,
(11)
aðxÞ ¼ 1=2AðxÞ,
(12)
bðxÞ ¼ 1=2BðxÞ,
(13)
cðxÞ ¼
as2 ðT T ic ; T ih T; ki Þ AðxÞ þ BðxÞ
dðxÞ ¼ b.
,
(14) (15)
Table 1 Summary of constant parameters. R RT Ts Th T ic T ih T ct T ht k kr kt kc ki
a b vo Cd T c is cultivated temperature.
30 10 T c 2:0 T c þ 2:0 T c 2:0 T c þ 2:0 16 29 0.5 1.0 2.0 2.0 5.0 0.06 0.0125 0.015 20
ARTICLE IN PRESS K. Nakazato, A. Mochizuki / Journal of Theoretical Biology 260 (2009) 56–65
References Anderson, J.L., Albergotti, L., Proulx, S., Peden, C., Huey, R.B., Philips, P.C., 2007. Thermal preference of Caenorhabditis elegans: a null model and empirical tests. J. Exp. Biol. 210, 3107–3116. Biron, D., Shibuya, M., Gabel, C., Wasserman, S.M., Clark, D.A., Brown, A., Sengupta, P., Samuel, A.D.T., 2006. A diacylglycerol kinase modulates longterm thermotactic behavioral plasticity in C. elegans. Nat. Neurosci. 9, 1499–1505. Biron, D., Wasserman, S., Thomas, J.H., Samuel, A.D.T., Sengupta, P., 2008. An olfactory neuron responds stochastically to temperature and modulates Caenorhabditis elegans thermotactic behavior. Proc. Natl. Acad. Sci. USA 105, 11002–11007. Brenner, S., 1974. The genetics of Caenorhabditis elegans. Genetics 77, 71–94. Chalfie, M., Sulston, J.E., White, J.G., Southgate, E., Thomson, J.N., Brenner, S., 1985. The neural circuit for touch sensitivity in Caenorhabditis elegans. J. Neurosci. 5, 956–964. Char, M.Y., Komatsu, H., Fukuto, H.S., Dionne, H.M., Hart, A.C., 2004. Feeding status and serotonin rapidly and reversibly modulate a Caenorhabditis elegans chemosensory circuit. Proc. Natl. Acad. Sci. USA 101, 15512–15517. Chung, S.H., Clark, D.A., Gabel, C.V., Mazur, E., Samuel, A.D.T., 2006. The role of the AFD neuron in C. elegans thermotaxis analyzed using femtosecond laser ablation. BMC Neurosci. 7, 30. Clark, D.A., Biron, D., Sengupta, P., Samuel, A.D.T., 2006. The AFD sensory neurons encode multiple functions underlying thermotactic behavior in Caenorhabditis elegans. J. Neurosci. 26, 7444–7451. Clark, D.A., Gabel, C.V., Lee, T.M., Samuel, A.D.T., 2007a. Short-term adaptation and temporal processing in the cryophilic response of Caenorhabditis elegans. J. Neurophysiol. 97, 1903–1910. Clark, D.A., Gavel, C.V., Gabel, H., Samuel, A.D.T., 2007b. Temporal activity patterns in thermosensory neurons of freely moving Caenorhabditis elegans. J. Neurosci. 27, 6083–6090. Dusenbery, D.B., Batt, J., 1980. Thermal limits and chemotaxis in mutants of the nematode Caenorhabditis elegans. J. Comp. Physiol. 137, 353–356. Gabel, C.V., Gabel, H., Pavlichin, D., Kao, A., Clark, D.A., Samuel, A.D.T., 2007. Neural circuits mediate electrosensory behavior in Caenorhabditis elegans. J. Neurosci. 27, 7586–7596. Gray, J.M., Hill, J.J., Bargmann, C.I., 2005. A circuit for navigation in Caenorhabditis elegans. Proc. Natl. Acad. Sci. USA 102, 3184–3191. Hedgecock, E.M., Russell, R.L., 1975. Normal and mutant thermotaxis in the nematode Caenorhabditis elegans. Proc. Natl. Acad. Sci. USA 72, 4061–4065. Ito, H., Inada, H., Mori, I., 2006. Quantitative analysis of thermotaxis in the nematode Caenorhabditis elegans. J. Neurosci. Methods 154, 45–52. Kimura, K.D., Atsushi, M., Matsumoto, K., Mori, I., 2004. The C. elegans thermosensory neuron AFD respond to warming. Curr. Biol. 14, 1291–1295.
65
Kuhara, A., Okumura, M., Kimata, T., Tanizawa, Y., Takano, R., Inada, K.D.K.H., Matsumoto, K., Mori, I., 2008. Temperature sensing by an olfactory neuron in a circuit controlling behavior of C. elegans. Science 320, 803–807. Luo, L., Clark, D.A., Biron, D., Mahadevan, L., Samuel, A.D.T., 2006. Sensorimotor control during isothermal tracking in Caenorhabditis elegans. J. Exp. Biol. 209, 4652–4662. Matsuoka, T., Gomi, S., Shingai, R., 2008. Simulation of C. elegans thermotactic behavior in a linear thermal gradient using a simple phenomenological motility model. J. Theor. Biol. 250, 230–243. Mohri, A., Kodama, E., Kimura, K.D., Koike, M., Mizuno, T., Mori, I., 2005. Genetic control of temperature preference in the nematode Caenorhabditis elegans. Genetics 169, 1437–1450. Mori, I., Ohshima, Y., 1995. Neural regulation of thermotaxis in Caenorhabditis elegans. Nature 376, 344–348. Pierce-Shimomura, J.T., Dores, M., Lockery, S.R., 2005. Analysis of the effects of turning bias on chemotaxis in C. elegans. J. Exp. Biol. 208, 4727–4733. Pierce-Shimomura, J.T., Morse, T.M., Lockery, S.R., 1999. The fundamental role of pirouettes in Caenorhabditis elegans chemotaxis. J. Neurosci. 19, 9557–9569. Ramot, D., Maclnnis, B.L., Goodman, M.B., 2008a. Bidirectional temperature-sensing by a single thermosensory neuron in C. elegans. Nat. Neurosci. 11, 908–915. Ramot, D., Maclnnis, B.L., Lee, H., Goodman, M.B., 2008b. Thermotaxis is a robust mechanism for thermoregulation in Caenorhabditis elegans nematodes. J. Neurosci. 28, 12546–12557. Risken, H., 1989. The Fokker–Planck Equation: Methods of Solution and Applications. In: Springer Series in Synergetics, vol. 18. Springer, Berlin. Ryu, W.S., Samuel, A.D.T., 2002. Thermotaxis in Caenorhabditis elegans analyzed by measuring responses of defined thermal stimuli. J. Neurosci. 22, 5727–5733. Schnitzer, M.J., 1992. Theory of continuum random walks and application to chemotaxis. Phys. Rev. E 48, 2553–2568. Sulston, J.E., Horvitz, H.R., 1977. Post-embryonic cell lineages of the nematode Caenorhabditis elegans. Dev. Biol. 56, 110–156. The C. elegans Sequencing Consortium, 1998. Genome sequence of the nematode C. elegans. Science 282, 2012–2018. Ward, A., Liu, J., Feng, Z., Zu, X.Z.S., 2008. Light-sensitive neurons and channels mediate phototaxis in C. elegans. Nat. Neurosci. 11, 916–922. White, J.G., Southgate, E., Thomson, J.N., Brenner, S., 1986. The structure of the nervous system of the nematode Caenorhabditis elegans. Phil. Trans. R. Soc. London B 314, 1–340. Wiktorsson, M., Ryden, T., Nilsson, E., Bengtsson, G., 2004. Modeling the movement of a soil insect. J. Theor. Biol. 231, 497–513. Yamada, Y., Ohshima, Y., 2003. Distribution and movement of Caenorhabditis elegans on a thermal gradient. J. Exp. Biol. 206, 2581–2593. Zhao, B., Khare, P., Feldman, L., Dent, J.A., 2003. Reversal frequency in Caenorhabditis elegans represents an integrated response to the state of the animal and its environment. J. Neurosci. 23, 5319–5328.