Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) are key ingredients in a number of models in physics and financial engineering. In particular, ...parabolic PDEs and BSDEs are fundamental tools in pricing and hedging models for financial derivatives. The PDEs and BSDEs appearing in such applications are often high-dimensional and nonlinear. Since explicit solutions of such PDEs and BSDEs are typically not available, it is a very active topic of research to solve such PDEs and BSDEs approximately. In the recent article (E et al., Multilevel Picard iterations for solving smooth semilinear parabolic heat equations,
arXiv:1607.03295
) we proposed a family of approximation methods based on Picard approximations and multilevel Monte Carlo methods and showed under suitable regularity assumptions on the exact solution of a semilinear heat equation that the computational complexity is bounded by
O
(
d
ε
-
(
4
+
δ
)
)
for any
δ
∈
(
0
,
∞
)
where
d
is the dimensionality of the problem and
ε
∈
(
0
,
∞
)
is the prescribed accuracy. In this paper, we test the applicability of this algorithm on a variety of 100-dimensional nonlinear PDEs that arise in physics and finance by means of numerical simulations presenting approximation accuracy against runtime. The simulation results for many of these 100-dimensional example PDEs are very satisfactory in terms of both accuracy and speed. Moreover, we also provide a review of other approximation methods for nonlinear PDEs and BSDEs from the scientific literature.
We introduce a new family of numerical algorithms for approximating solutions of general high-dimensional semilinear parabolic partial differential equations at single space-time points. The ...algorithm is obtained through a delicate combination of the Feynman–Kac and the Bismut–Elworthy–Li formulas, and an approximate decomposition of the Picard fixed-point iteration with multilevel accuracy. The algorithm has been tested on a variety of semilinear partial differential equations that arise in physics and finance, with satisfactory results. Analytical tools needed for the analysis of such algorithms, including a semilinear Feynman–Kac formula, a new class of seminorms and their recursive inequalities, are also introduced. They allow us to prove for semilinear heat equations with gradient-independent nonlinearities that the computational complexity of the proposed algorithm is bounded by
O
(
d
ε
-
(
4
+
δ
)
)
for any
δ
∈
(
0
,
∞
)
under suitable assumptions, where
d
∈
N
is the dimensionality of the problem and
ε
∈
(
0
,
∞
)
is the prescribed accuracy. Moreover, the introduced class of numerical algorithms is also powerful for proving high-dimensional approximation capacities for deep neural networks.
We prove several properties of the EMCEL scheme, which is capable of approximating one-dimensional continuous strong Markov processes in distribution on the path space (the scheme is briefly ...recalled). Special cases include irregular stochastic differential equations and processes with sticky features. In particular, we highlight differences from the Euler scheme in the case of stochastic differential equations and discuss a certain “stabilizing” behavior of the EMCEL scheme like “smoothing and tempered growth behavior”.
Genome integrity is critically dependent on timely DNA replication and accurate chromosome segregation. Replication stress delays replication into G2/M, which in turn impairs proper chromosome ...segregation and inflicts DNA damage on the daughter cells. Here we show that TopBP1 forms foci upon mitotic entry. In early mitosis, TopBP1 marks sites of and promotes unscheduled DNA synthesis. Moreover, TopBP1 is required for focus formation of the structure-selective nuclease and scaffold protein SLX4 in mitosis. Persistent TopBP1 foci transition into 53BP1 nuclear bodies (NBs) in G1 and precise temporal depletion of TopBP1 just before mitotic entry induced formation of 53BP1 NBs in the next cell cycle, showing that TopBP1 acts to reduce transmission of DNA damage to G1 daughter cells. Based on these results, we propose that TopBP1 maintains genome integrity in mitosis by controlling chromatin recruitment of SLX4 and by facilitating unscheduled DNA synthesis.
Protein phosphatase 2A (PP2A) in complex with B55 regulatory subunits reverses cyclin-dependent kinase 1 (Cdk1) phosphorylations at mitotic exit. Interestingly, threonine and serine residues ...phosphorylated by Cdk1 display distinct phosphorylation dynamics, but the biological significance remains unexplored. Here we demonstrate that the phosphothreonine preference of PP2A-B55 provides an essential regulatory element of mitotic exit. To allow rapid activation of the anaphase-promoting complex/cyclosome (APC/C) co-activator Cdc20, inhibitory phosphorylation sites are conserved as threonines while serine substitutions delay dephosphorylation and Cdc20 activation. Conversely, to ensure timely activation of the interphase APC/C co-activator Cdh1, inhibitory phosphorylation sites are conserved as serines, and threonine substitutions result in premature Cdh1 activation. Furthermore, rapid translocation of the chromosomal passenger complex to the central spindle is prevented by mutation of a single phosphorylated threonine to serine in inner centromere protein (INCENP), leading to failure of cytokinesis. Altogether, the findings of our work reveal that the inherent residue preference of a protein phosphatase can provide temporal regulation in biological processes.
Transcription of the Ebola virus genome depends on the viral transcription factor VP30 in its unphosphorylated form, but the underlying molecular mechanism of VP30 dephosphorylation is unknown. Here ...we show that the Ebola virus nucleoprotein (NP) recruits the host PP2A-B56 protein phosphatase through a B56-binding LxxIxE motif and that this motif is essential for VP30 dephosphorylation and viral transcription. The LxxIxE motif and the binding site of VP30 in NP are in close proximity, and both binding sites are required for the dephosphorylation of VP30. We generate a specific inhibitor of PP2A-B56 and show that it suppresses Ebola virus transcription and infection. This work dissects the molecular mechanism of VP30 dephosphorylation by PP2A-B56, and it pinpoints this phosphatase as a potential target for therapeutic intervention.
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•The Ebola virus NP recruits the PP2A-B56 phosphatase through an LxxIxE binding motif•The LxxIxE motif in NP is required for VP30 dephosphorylation and viral transcription•A specific PP2A-B56 inhibitor impairs recombinant Ebola virus transcription and infection
Transcription of the Ebola virus genome depends on the viral transcription factor VP30 in its unphosphorylated form. Kruse et al. show that the Ebola nucleoprotein recruits the host PP2A-B56 phosphatase through a B56-binding LxxIxE motif, leading to VP30 dephosphorylation and viral transcription.
Mutations in the tumour suppressor gene BRCA2 are associated with predisposition to breast and ovarian cancers. BRCA2 has a central role in maintaining genome integrity by facilitating the repair of ...toxic DNA double-strand breaks (DSBs) by homologous recombination (HR). BRCA2 acts by controlling RAD51 nucleoprotein filament formation on resected single-stranded DNA, but how BRCA2 activity is regulated during HR is not fully understood. Here, we delineate a pathway where ATM and ATR kinases phosphorylate a highly conserved region in BRCA2 in response to DSBs. These phosphorylations stimulate the binding of the protein phosphatase PP2A-B56 to BRCA2 through a conserved binding motif. We show that the phosphorylation-dependent formation of the BRCA2-PP2A-B56 complex is required for efficient RAD51 filament formation at sites of DNA damage and HR-mediated DNA repair. Moreover, we find that several cancer-associated mutations in BRCA2 deregulate the BRCA2-PP2A-B56 interaction and sensitize cells to PARP inhibition. Collectively, our work uncovers PP2A-B56 as a positive regulator of BRCA2 function in HR with clinical implications for BRCA2 and PP2A-B56 mutated cancers.
Proper segregation of chromosomes depends on a functional spindle assembly checkpoint (SAC) and requires kinetochore localization of the Bub1 and Mad1/Mad2 checkpoint proteins. Several aspects of ...Mad1/Mad2 kinetochore recruitment in human cells are unclear and in particular the underlying direct interactions. Here we show that conserved domain 1 (CD1) in human Bub1 binds directly to Mad1 and a phosphorylation site exists in CD1 that stimulates Mad1 binding and SAC signalling. Importantly, fusion of minimal kinetochore-targeting Bub1 fragments to Mad1 bypasses the need for CD1, revealing that the main function of Bub1 is to position Mad1 close to KNL1 MELT repeats. Furthermore, we identify residues in Mad1 that are critical for Mad1 functionality, but not Bub1 binding, arguing for a direct role of Mad1 in the checkpoint. This work dissects functionally relevant molecular interactions required for spindle assembly checkpoint signalling at kinetochores in human cells.
BubR1 is a central component of the spindle assembly checkpoint that inhibits progression into anaphase in response to improper kinetochore-microtubule interactions. In addition, BubR1 also helps ...stabilize kinetochore-microtubule interactions by counteracting the Aurora B kinase but the mechanism behind this is not clear. Here we show that BubR1 directly binds to the B56 family of protein phosphatase 2A (PP2A) regulatory subunits through a conserved motif that is phosphorylated by cyclin-dependent kinase 1 (Cdk1) and polo-like kinase 1 (Plk1). Two highly conserved hydrophobic residues surrounding the serine 670 Cdk1 phosphorylation site are required for B56 binding. Mutation of these residues prevents the establishment of a proper metaphase plate and delays cells in mitosis. Furthermore, we show that phosphorylation of serines 670 and 676 stimulates the binding of B56 to BubR1 and that BubR1 targets a pool of B56 to kinetochores. Our data suggest that BubR1 counteracts Aurora B kinase activity at improperly attached kinetochores by recruiting B56-PP2A phosphatase complexes.
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