Oeriew My generl reserch interests re in mthemticl iology, nonliner prtil differentil equtions, dynmicl systems nd stochstic processes. My thesis reserch hs focused on deeloping nd nlyzing PDE models of cell polriztion, which is crucil in mny cellulr processes such s cell motility, emryogenesis, nd neurite growth nd differentition. During cell polriztion, signling molecules nd the cytoskeleton re loclized t one single site or multiple sites. A key question concerns how signling molecules nd the cytoskeleton ecome symmetriclly distriuted. In my work, I he modeled the dynmics of prticulr signling molecule, ho GTPse Cdc 42, nd its interctions with the cytoskeleton. I he deeloped three different models: stochstic ctie trnsport model [1], n ctie trnsport model coupled with microtuule growth [2] nd PDE model coupled with dely differentil equtions [3]. These models cpture key fetures of cell polriztion in udding yest, neuronl growth cones, nd fission yest, respectiely. In my reserch, I he pplied riety of mthemticl tools, including numericl nlysis, spectrl theory, perturtion methods, nd ifurction theory. The min results of my thesis work re summrized in more detil elow. 1. A stochstic ctie trnsport model My first project concerned the effect of the geometry of cytoskeleton structures on cell polriztion. I considered two types of structures: ctin cytoskeleton in udding yest nd microtuules (s) in neuronl growth cone. A mjor difference etween these two structures is tht ctin filments re nucleted t the cell memrne while s re nucleted from centers within the cytoplsm (centrosomes), see Fig. 1(, ). Experimentl results show tht udding yest cells cn polrize in the sence of externl cues, while such spontneous polriztion is not osered in the neuronl growth cone. My project nlyzed the potentil reltionship etween the geometry of the cytoskeleton nd the ility of cells to polrize. In prticulr, I used stochstic ctie trnsport model in which cytoplsmic esicle contining signling molecules cn rndomly switch etween diffusing stte nd stte of directed motion long filments, see Fig. 1(c). In ddition, the signling molecules within the memrne were modeled y rection eqution. Actin cytoskeleton nucleted t memrne c ctin microtuule D Microtuule nucleted from cytoplsm z x D Figure 1. () Actin cytoskeleton nucleted from the memrne. () Microtuules nucleted from orgnizing sites within the cytoplsm. (c) Stochstic ctie trnsport model in 2D networks. Vesicles (motor/crgo complex) cn rndomly ind to nd unind from s. 1
One mjor chllenge of modeling this system ws how to couple the stochstic model of cytoplsmic esicles with the rection eqution of the memrne-ound molecules. I proceeded y deriing n effectie two-dimensionl dection- eqution for the concentrtion of signling molecules in the cytoplsm using qusi-stedy-stte nlysis. Importntly, I found tht the coefficient is nisotropic nd depends on the density of filments. Using liner stility nlysis, I deried conditions for the growth of precursor pttern for cell polriztion. My results showed tht if filments re nucleted t sites on the cell memrne (see Fig. 1), spontneous cell polriztion cn occur, wheres if the filments nuclete from orgnizing sites within the cytoplsm (see Fig. 1), cell cn only polrize in response to n externl chemicl grdient. This is consistent with experimentl dt on cell polriztion in udding yest nd the neuronl growth cone. 2. Actie trnsport model coupled with microtuule length growth In my second project, I considered more iophysiclly detiled model of the neuronl growth cone y coupling the growth of s with my preious ctie trnsport model, see Fig. (2). This is motited y the question: how would nonuniform length distriution of s gie rise to polrized distriution of memrne ound molecules? I ddressed this question y undertking two mjor tsks: (1) to explicitly model the s growth nd find the length distriution; (2) to formulte nd sole the modified ctie trnsport model, which tkes into ccount the length distriution of s. First, to derie the length distriution of microtuules, I considered c1-stthmin- pthwy in which the growth nd ctstrophe rtes of s re regulted y stthmin, while the stthmin is regulted y c1 t the memrne, see Fig. 2(). This is modeled y modified Dogterom-Leiler model of growth. I used regulr perturtion theory nd numericl simultion to find the resulting nonuniform length distriution of s for specific nonuniform c1 distriution t the memrne. Second, I coupled the erge length of s with the preious two dimensionl ctie trnsport model [1] y modifying the dection term. This sutle modifiction resulted in c1 growth stthmin z=φ(x) c1 D m protein ound to n protein diffusing in memrne protein diffusing in cytosol ctie c1 inctie c1 ctie stthmin inctie stthmin Figure 2. () A 2D model of growth cone with nonuniform distriution of lengths s specified y the interfce z = φ(x). () Sketch of the c1-stthmin- pthwy. c1 proteins (circles) re locted t the leding edge of the growth cone in ctie (solid circles) or inctie form (open circles). The ctie region of c1 genertes grdient in stthmin phosphoryltion such tht the concentrtion of ctie stthmin increses with distnce from the ctie c1 domin. Actie stthmin inhiits the growth of s. 2
two-dimensionl dection eqution with n interfce t the ends of s, see Fig. 2(). Using smoothing method, I soled the two dimensionl model numericlly to otin the concentrtion of memrne-ound molecules. For piece-wise c1 distriution, I found tht the sptil rition of the concentrtion of memrne-ound molecules is reltiely lrger thn tht for sinusoidl c1 distriution. 3. A PDE-DDE model for cell polriztion in fission yest Another well-studied iologicl system for cell polriztion is fission yest. During its cell cycle, fission yest undergoes trnsition from growing t one tip (mono-polr growth) to growing t two tips (ipolr growth), see Fig. 3(). This trnsition is known s new end tke off (NETO). Experimentl eidence suggests tht the concentrtion of signling molecule Cdc42 oscilltes t oth tips of fission yest. Moreoer, the mplitudes chnge s the cell elongtes. In the cse of longer cells exhiiting ipolr growth, the men mplitudes of the oscilltions re the sme t oth ends. On the other hnd, for shorter, less mture cells exhiiting monopolr growth, the mplitude is significntly lrger t the growing end. Drien y the experimentl results, I deeloped sptil-temporl model tht cn cpture the oscilltions of memrne-ound Cdc42 during cell elongtion. One difficulty is to model the of cytoplsmic Cdc42 s well s the oscilltions of memrne-ound Cdc42. This chllenge ws soled using comprtmentl model, see Fig. 3(). It consists of eqution which descries ulk of Cdc42 in the cytoplsm nd pir of dely differentil equtions (DDEs) which represent the inding of Cdc42 to the cell memrne nd re-relese into the cytoplsm i uninding. The PDE is coupled to the DDEs t the ends of the cell i oundry conditions. This noel PDE-DDE model is menle to mthemticl nlysis nd hs rich dynmics. For cell with fixed length, I used liner stility nlysis nd winding numer rguments to inestigte the effect of on the onset of limit cycle oscilltions t the end comprtments. The liner stility nlysis is nontriil due to the nonliner terms nd time dely in the oundry conditions. I showed tht the criticl time dely for the onset of oscilltions i Hopf ifurction increses s the coefficient decreses. Finlly, for cell with growing length, I soled the model using method of lines nd showed tht the system undergoes trnsition from symmetric to symmetric oscilltions. This is consistent with experimentl findings of NETO in fission yest. cell diision L mono-polr growth i-polr growth old end new end New end tke off (NETO) X1 diffusie flux Cytoplsmic C utoctlysis X2 Figure 3. () Switch from mono-polr growth to ipolr growth during fission yest cell cycle. () Comprtmentl model sed on Cdc42 oscilltions. 3
Motited y the PDE-DDE model for fission yest, I he lso extended my results to PDE-DDE model tht consists of eqution nd pir of delyed logistic equtions [4]. Using numericl ifurction nlysis, I he osered noel effects of on oscilltions t the two end comprtments. For exmple, the oscilltions cn chnge from in-phse to nti-phse s the coefficient chnges. 4.1. Cell polriztion. 4. Ongoing nd future work 4.1.1. Actie trnsport model with stochstic interfce. One ssumption in the neuronl growth cone model [2] is tht the s length distriution hs sufficient time to rech stedy stte efore chnges in the c1 distriution long the leding edge. I would like to relx this ssumption nd to consider multiscle model which cn include dynmic growth of s together with the trnsport of signling molecules. One possiility is to design n lgorithm for the dynmic instility of s nd to clculte the length distriution which would gie rise to stochstic interfce. 4.1.2. Modeling ctin cytoskeleton ctiity in fission yest. In ddition to the signling molecule Cdc 42, the ctin cytoskeleton is lso importnt for mediting cell growth nd diision. During the cell cycle, the ctin cytoskeleton (cles nd ptches) undergoes two mjor rerrngements. The first one occurs during NETO in which the ctin distriution chnges from polrized distriution t the old end to polrized distriution t oth ends. The second occurs ner the onset of mitosis: the polrized distriution t oth ends is lost nd ctin is relocted in the middle of the cell to form the ctomyosin contrctile ring. I m interested in exploring how these trnsitions occur nd how the contrctile ring forms. 4.2. PDE-DDE models. For the PDE-DDE models in [4], there re numer of mthemticl questions worth further explortion. First, I pln to crry out more detiled study of the doule Hopf ifurction nd to determine the sin of ttrctions of coexistent in-phse nd nti-phse solution. Second, I would like to find if the result for the delyed logistic eqution holds for more generl clss of DDEs. Third, in order to understnd the effects of diffusie coupling on complex multi-species iochemicl oscilltors, I would like to compre the ehior of the resulting PDE-ODE system to the corresponding PDE-DDE system, with the ltter otined y reducing ech multi-component ODE y DDE long the lines of Nok nd Tyson [5]. 4.3. New lines of reserch. I he een working on modeling the moement of prticles in hlf disk domin which consists of purely diffusie region nd region with finite numer of s, see Fig. 4(). The model for single wedge (Ω = Ω Ω 1 ) is illustrted in Fig. 4(). In Ω 1, prticle cn rndomly switch etween diffusing nd treling llisticlly long. In Ω, prticle undergoes lone. When prticle is ner the oundry x =, it cn e sored (ctited), see Fig. 4(c). I lso intend to incorporte positie feedck for the ctition. I im to nswer the following questions: (1) How does one quntify the role of ctie trnsport on the rril time of prticle to the oundry x =? (2) How would the rril time chnge if the domin is expnding? (3) For model with N wedges, suppose there is positie feedck or n externl grdient, would prticles t certin wedge more likely rrie t its oundry nd form cluster? 4
c D + D m soring(ctition) reflecting Ω α purely D - ε interfce Γ Ω D cytoplsm Ω 1 θ = π / Ν -ε Θ inctie ctie - ε Figure 4. () A disk domin of rdius with finite numer of s of length ɛ. () Prticles t single wedge Ω = Ω Ω 1. (c) Prticles in the sudomin Ω. Ner the oundry x =, prticle cn e sored (ctited). purely 6 purely Figure 5. Escpe prolem in disk domin with finite numer of filments nucleted t the oundry. () One filment t ech site. () Two filments t ech site. I would lso like to explore the prolem in other domins. Inspired y the geometry of the ctin cytoskeleton in udding yest (see Fig. 1), I pln to study n escpe prolem on disk domin tht hs filments centered t its oundry, see Fig. 5. The exit (sorption) sites re tken to e the sme s the centers of the filments. The difference etween the domin in Fig. 4 nd Fig. 5 is the distriution of filments. As shown in Fig. 5, prticle hs to exit the purely diffusie region first efore it cn escpe. A relted question is whether it is less eneficil for prticle to escpe from this type of region compred to the region shown in Fig. 4(). Beyond my current work, I m lso interested in the synchroniztion of coupled oscilltors, nonliner wes, chemicl rection networks, nd mthemticl models for iologicl pttern formtion. eferences [1] Pul C Bressloff nd Bin Xu. Stochstic ctie-trnsport model of cell polriztion. SIAM Journl on Applied Mthemtics, 75(2):652 678, 215. [2] Bin Xu nd Pul C Bressloff. Model of growth cone memrne polriztion i microtuule length regultion. Biophysicl journl, 19(1):223 2214, 215. [3] Bin Xu nd Pul C Bressloff. A PDE-DDE model for cell polriztion in fission yest. SIAM Journl on Applied Mthemtics, 76(5):1844 187, 216. [4] Bin Xu nd Pul C Bressloff. A theory of synchrony for ctie comprtments with delys coupled through ulk. Physic D, in press, 216. [5] Bél Noák nd John J Tyson. Design principles of iochemicl oscilltors. Nture reiews Moleculr cell iology, 9(12):981 991, 28. 5